[ { "name": "Create and Deploy a Website", "source": "Free Code Camp", "link": "http://www.freecodecamp.com/create-and-deploy-a-website", "image": "http://startbootstrap.com/assets/img/templates/landing-page.jpg", "directions": [ "In the next 5 minutes, you'll create a website and deploy it to the internet!" ], "links": ["http://startbootstrap.com/template-overviews/freelancer/", "http://bitballoon.com", "http://atom.io"] }, { "name": "Start Your First Pair Programming Session", "source": "Free Code Camp", "link": "http://www.freecodecamp.com/first_pair_programming_session", "image": "https://screenhero.com/img/anim-collaboration.gif", "directions": [ "What's all this Pair Programming stuff about? Let's find out! We'll use a popular pair programming tool called Screen Hero." ], "links": ["https://screenhero.com/download.html"] }, { "name": "Make Your Static Page Dynamic Using POWR.io", "source": "Free Code Camp", "link": "http://www.freecodecamp.com/first_dynamic_website", "image": "https://s3-us-west-1.amazonaws.com/powr/images/powr_showcase_bg_1.jpg", "directions": [ "The website you created earlier is cool, but it's not very interactive. Let's " ] }, { "name": "Create your first CodePen", "source": "Free Code Camp", "link": "http://www.freecodecamp.com/first_codepen", "image": "https://s3-us-west-1.amazonaws.com/powr/images/powr_showcase_bg_1.jpg", "directions": [ "Let's put those HTML and CSS skills to work!", "You'll create your own databaseless webpage. We'll show you how." ] }, { "name": "Code Your First API Integration", "source": "Free Code Camp", "link": "http://www.freecodecamp.com/", "image": "http://status.twilio.com/images/logo.png", "time": 2, "directions": [ "Now it's time to flex your JavaScript skills and string together an array of data structures (puns intended).", "And what better way than a to build a simple text messaging app?" ] }, { "source": "Project Euler", "directions": [ "If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.", "Find the sum of all the multiples of 3 or 5 below 1000." ], "solution": "233168", "name": "PE1" }, { "source": "Project Euler", "directions": [ "Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be:", "1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...", "By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.", "Note: This problem has been changed recently, please check that you are using the right parameters." ], "solution": "4613732", "name": "PE2" }, { "source": "Project Euler", "directions": [ "The prime factors of 13195 are 5, 7, 13 and 29.", "What is the largest prime factor of the number 600851475143 ?", "Note: This problem has been changed recently, please check that you are using the right number." ], "solution": "6857", "name": "PE3" }, { "source": "Project Euler", "directions": [ "A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 × 99.", "Find the largest palindrome made from the product of two 3-digit numbers." ], "solution": "906609", "name": "PE4" }, { "source": "Project Euler", "directions": [ "2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.", "What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?" ], "solution": "232792560", "name": "PE5" }, { "source": "Project Euler", "directions": [ "The sum of the squares of the first ten natural numbers is,", "12 + 22 + ... + 102 = 385", "The square of the sum of the first ten natural numbers is,", "(1 + 2 + ... + 10)2 = 552 = 3025", "Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 − 385 = 2640.", "Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum." ], "solution": "25164150", "name": "PE6" }, { "source": "Project Euler", "directions": [ "By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.", "What is the 10 001st prime number?" ], "solution": "104743", "name": "PE7" }, { "source": "Project Euler", "directions": [ "The four adjacent digits in the 1000-digit number that have the greatest product are 9 × 9 × 8 × 9 = 5832.", "7316717653133062491922511967442657474235534919493496983520312774506326239578318016984801869478851843858615607891129494954595017379583319528532088055111254069874715852386305071569329096329522744304355766896648950445244523161731856403098711121722383113622298934233803081353362766142828064444866452387493035890729629049156044077239071381051585930796086670172427121883998797908792274921901699720888093776657273330010533678812202354218097512545405947522435258490771167055601360483958644670632441572215539753697817977846174064955149290862569321978468622482839722413756570560574902614079729686524145351004748216637048440319989000889524345065854122758866688116427171479924442928230863465674813919123162824586178664583591245665294765456828489128831426076900422421902267105562632111110937054421750694165896040807198403850962455444362981230987879927244284909188845801561660979191338754992005240636899125607176060588611646710940507754100225698315520005593572972571636269561882670428252483600823257530420752963450", "Find the thirteen adjacent digits in the 1000-digit number that have the greatest product. What is the value of this product?" ], "solution": "23514624000", "name": "PE8" }, { "source": "Project Euler", "directions": [ "A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,", " a2 + b2 = c2", "For example, 32 + 42 = 9 + 16 = 25 = 52.", "There exists exactly one Pythagorean triplet for which a + b + c = 1000.Find the product abc." ], "solution": "31875000", "name": "PE9" }, { "source": "Project Euler", "directions": [ "The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.", "Find the sum of all the primes below two million.", "
Note: This problem has been changed recently, please check that you are using the right parameters.
" ], "solution": "142913828922", "name": "PE10" }, { "source": "Project Euler", "directions": [ "In the 20×20 grid below, four numbers along a diagonal line have been marked in red.", "08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 0849 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 0081 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 6552 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 9122 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 8024 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 5032 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 7067 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 2124 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 7221 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 9578 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 9216 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 5786 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 5819 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 4004 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 6688 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 6904 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 3620 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 1620 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 5401 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48", "The product of these numbers is 26 × 63 × 78 × 14 = 1788696.", "What is the greatest product of four adjacent numbers in the same direction (up, down, left, right, or diagonally) in the 20×20 grid?" ], "solution": "70600674", "name": "PE11" }, { "source": "Project Euler", "directions": [ "The sequence of triangle numbers is generated by adding the natural numbers. So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. The first ten terms would be:", "1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...", "Let us list the factors of the first seven triangle numbers:", " 1: 1 3: 1,3 6: 1,2,3,610: 1,2,5,1015: 1,3,5,1521: 1,3,7,2128: 1,2,4,7,14,28", "We can see that 28 is the first triangle number to have over five divisors.", "What is the value of the first triangle number to have over five hundred divisors?" ], "solution": "76576500", "name": "PE12" }, { "source": "Project Euler", "directions": [ "Work out the first ten digits of the sum of the following one-hundred 50-digit numbers.", "3710728753390210279879799822083759024651013574025046376937677490009712648124896970078050417018260538743249861995247410594742333095130581237266173096299194221336357416157252243056330181107240615490825023067588207539346171171980310421047513778063246676892616706966236338201363784183836841787343617267572811287981284997940806548193159262169127588983273844274228917432520321923589422876796487670272189318474514457360013064390911672168568445887116031532767038648610584302543993961982891759366568675793495162176457141856560629502157223196586755079324193331649063524627419049291014324458138226633479447581789257586771833721766196375159057923972824559883840758203565325359399008402633568948830189458628227828801811993848262820142781941399405675871511700943903539866437282711265382998724078447305319010429358686515506006295864861532075273371959191420517255829716938887077154664991155934876035329217149700569385437007057682668462462149565007647178729443837760453282654108756828443191190634694037855217779295145361232725250002960710750825638156567108852583507214587657617241097644733911060721826523687722363604517423706905851860660448207621209813287860733969412811426604180868306193284608111910615569405126896925193432545172838864191804704929321505864256304948362467221648435076201727918039944693004732956340691157324443869081257945140890577062294291971079282095503768752567877309186254074496984450833039368212618336384825330154686196124348767681297534375946515803862875928784902015216855548287172012192577669547818283375799310361474035685644909552709786479758116726320100436897842553539920931837441497806860984484030981290777917990882187953273644756755908480308708698755139271185451707854416185242432069315033259959406895756536782107074926966537676326235447210697939506796526947425977097391666937630426339870854105268470829908521139942736573411618276031500127165378607361501080857009149939512557028198746004375358290353174347173269321235781549826297425527373079495375976510530594696606768315657437716740187527588902802571733229619176668713819931811048770190271252676802760780030136786809925254634010616328665263627021854049770558562994658063623799314074625596224074486908231174977792365466257246923322810917141914302881971032885978066697608929386382850253334033441306557801612781592181500556186883646842009047023053081172816430487623791969842487255036638784583114876969321549028104240201383351244621814417734706378329949063625966649858761822122522551248676453367720186971698544312419572409913959008952310058822955482553002635207815322967962494816419538682187747608532713228572311042480345612486769706450799523637774242535411291684276865538926205024910326572967237019132757256752856532482582654630922070585965222979886027225833191312637514734199488953476574550118495701454879288984856827726077713721403798879715382982037830314735277215803481445134913732266513813482954382919991818027891652243102739225112286953940957953066405232632538044100059654939159879593635297461521855023713076422551211836938035803885849034169811622207297718615823667842468915799353296192262467957194401269043877107275048102390895523597457231897067725479150615055049539229795309011299675198618808822587531452958409925120382900940777077567211306739708304724483816533873502340845647058077308829591747671403631980081871290118754913105471265819762333104481838626951545633492636657289756340050042846280183517070527831839425882145521227251250327551216035469812005817621652128276527516912968977893223819573432933994643750190783694576588335239988675506164965184775180738168837861091527357929701337621778427521926234019423996391680449839931733127313292418570714734956691667468763466091503591467750499518671430235219628894890102423325116913619626622732674608005915474718307983928685352069469445407247684182252467441716151403642798227334805555621481897142617910342598647204516893989422179826088076852877836461827993463137677543078093633330189826420901084880252167467088321512018588354322381287695278671329612474782464538636993009049310363619763878039621840735723997942234062353938083396513274080111166662789198148808779794187687614423003098449085141160661826293682836764744779239180335110989069790714857869440895529906536404474255760836599766457950966602439640990538960712019821997604759949019723029764913982680032973156037120041377903785566085089252167309393198727502754689069037075394130426523150119480937724504879515095410092164586375471059843679178639167021187492431995700641917969777599028300699153687137119366149528113058763802784107544497330784078992311553556256114232242325503368544248891735344889911501440648020369068063960672322193204149535415031288803395360532993403680069777106505666319548123488067321014673905856855793458140362782270328082616570773948327592232845941706525094512325230608229188020587773197198394501808880724296619808111977715854250201654509041324580978688277894872185961772107838435069186155435662884062257473692284509516208496039801340017239306716668235552452528046097225350353422647252425087405407559178978126433033169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], "solution": "5537376230", "name": "PE13" }, { "source": "Project Euler", "directions": [ "The following iterative sequence is defined for the set of positive integers:", "n → n/2 (n is even)n → 3n + 1 (n is odd)", "Using the rule above and starting with 13, we generate the following sequence:", "13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1", "It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.", "Which starting number, under one million, produces the longest chain?", "NOTE: Once the chain starts the terms are allowed to go above one million." ], "solution": "837799", "name": "PE14" }, { "source": "Project Euler", "directions": [ "Starting in the top left corner of a 2×2 grid, and only being able to move to the right and down, there are exactly 6 routes to the bottom right corner.", "How many such routes are there through a 20×20 grid?" ], "solution": "137846528820", "name": "PE15" }, { "source": "Project Euler", "directions": [ "215 = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.", "What is the sum of the digits of the number 21000?" ], "solution": "1366", "name": "PE16" }, { "source": "Project Euler", "directions": [ "If the numbers 1 to 5 are written out in words: one, two, three, four, five, then there are 3 + 3 + 5 + 4 + 4 = 19 letters used in total.", "If all the numbers from 1 to 1000 (one thousand) inclusive were written out in words, how many letters would be used? ", "NOTE: Do not count spaces or hyphens. For example, 342 (three hundred and forty-two) contains 23 letters and 115 (one hundred and fifteen) contains 20 letters. The use of \"and\" when writing out numbers is in compliance with British usage." ], "solution": "21124", "name": "PE17" }, { "source": "Project Euler", "directions": [ "By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.", "37 42 4 68 5 9 3", "That is, 3 + 7 + 4 + 9 = 23.", "Find the maximum total from top to bottom of the triangle below:", "7595 6417 47 8218 35 87 1020 04 82 47 6519 01 23 75 03 3488 02 77 73 07 63 6799 65 04 28 06 16 70 9241 41 26 56 83 40 80 70 3341 48 72 33 47 32 37 16 94 2953 71 44 65 25 43 91 52 97 51 1470 11 33 28 77 73 17 78 39 68 17 5791 71 52 38 17 14 91 43 58 50 27 29 4863 66 04 68 89 53 67 30 73 16 69 87 40 3104 62 98 27 23 09 70 98 73 93 38 53 60 04 23", "NOTE: As there are only 16384 routes, it is possible to solve this problem by trying every route. However, Problem 67, is the same challenge with a triangle containing one-hundred rows; it cannot be solved by brute force, and requires a clever method! ;o)" ], "solution": "1074", "name": "PE18" }, { "source": "Project Euler", "directions": [ "You are given the following information, but you may prefer to do some research for yourself.", "1 Jan 1900 was a Monday.Thirty days has September,April, June and November.All the rest have thirty-one,Saving February alone,Which has twenty-eight, rain or shine.And on leap years, twenty-nine.A leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400.", "How many Sundays fell on the first of the month during the twentieth century (1 Jan 1901 to 31 Dec 2000)?" ], "solution": "171", "name": "PE19" }, { "source": "Project Euler", "directions": [ "n! means n × (n − 1) × ... × 3 × 2 × 1", "For example, 10! = 10 × 9 × ... × 3 × 2 × 1 = 3628800,and the sum of the digits in the number 10! is 3 + 6 + 2 + 8 + 8 + 0 + 0 = 27.", "Find the sum of the digits in the number 100!" ], "solution": "648", "name": "PE20" }, { "source": "Project Euler", "directions": [ "Let d(n) be defined as the sum of proper divisors of n (numbers less than n which divide evenly into n).If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and each of a and b are called amicable numbers.", "For example, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110; therefore d(220) = 284. The proper divisors of 284 are 1, 2, 4, 71 and 142; so d(284) = 220.", "Evaluate the sum of all the amicable numbers under 10000." ], "solution": "31626", "name": "PE21" }, { "source": "Project Euler", "directions": [ "Using names.txt (right click and 'Save Link/Target As...'), a 46K text file containing over five-thousand first names, begin by sorting it into alphabetical order. Then working out the alphabetical value for each name, multiply this value by its alphabetical position in the list to obtain a name score.", "For example, when the list is sorted into alphabetical order, COLIN, which is worth 3 + 15 + 12 + 9 + 14 = 53, is the 938th name in the list. So, COLIN would obtain a score of 938 × 53 = 49714.", "What is the total of all the name scores in the file?" ], "solution": "871198282", "name": "PE22" }, { "source": "Project Euler", "directions": [ "A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number.", "A number n is called deficient if the sum of its proper divisors is less than n and it is called abundant if this sum exceeds n.", "A number whose proper divisors are less than the number is called deficient and a number whose proper divisors exceed the number is called abundant.
", "As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest number that can be written as the sum of two abundant numbers is 24. By mathematical analysis, it can be shown that all integers greater than 28123 can be written as the sum of two abundant numbers. However, this upper limit cannot be reduced any further by analysis even though it is known that the greatest number that cannot be expressed as the sum of two abundant numbers is less than this limit.", "Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers." ], "solution": "4179871", "name": "PE23" }, { "source": "Project Euler", "directions": [ "A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. The lexicographic permutations of 0, 1 and 2 are:", "012 021 102 120 201 210", "What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?" ], "solution": "2783915460", "name": "PE24" }, { "source": "Project Euler", "directions": [ "The Fibonacci sequence is defined by the recurrence relation:", "Fn = Fn−1 + Fn−2, where F1 = 1 and F2 = 1.", "Hence the first 12 terms will be:", "F1 = 1F2 = 1F3 = 2F4 = 3F5 = 5F6 = 8F7 = 13F8 = 21F9 = 34F10 = 55F11 = 89F12 = 144", "The 12th term, F12, is the first term to contain three digits.", "What is the first term in the Fibonacci sequence to contain 1000 digits?" ], "solution": "4782", "name": "PE25" }, { "source": "Project Euler", "directions": [ "A unit fraction contains 1 in the numerator. The decimal representation of the unit fractions with denominators 2 to 10 are given:", "1/2= 0.51/3= 0.(3)1/4= 0.251/5= 0.21/6= 0.1(6)1/7= 0.(142857)1/8= 0.1251/9= 0.(1)1/10= 0.1", "Where 0.1(6) means 0.166666..., and has a 1-digit recurring cycle. It can be seen that 1/7 has a 6-digit recurring cycle.", "Find the value of d < 1000 for which 1/d contains the longest recurring cycle in its decimal fraction part." ], "solution": "983", "name": "PE26" }, { "source": "Project Euler", "directions": [ "Euler discovered the remarkable quadratic formula:", "n² + n + 41", "It turns out that the formula will produce 40 primes for the consecutive values n = 0 to 39. However, when n = 40, 402 + 40 + 41 = 40(40 + 1) + 41 is divisible by 41, and certainly when n = 41, 41² + 41 + 41 is clearly divisible by 41.", "The incredible formula n² − 79n + 1601 was discovered, which produces 80 primes for the consecutive values n = 0 to 79. The product of the coefficients, −79 and 1601, is −126479.", "Considering quadratics of the form:", "n² + an + b, where |a| < 1000 and |b| < 1000where |n| is the modulus/absolute value of ne.g. |11| = 11 and |−4| = 4", "Find the product of the coefficients, a and b, for the quadratic expression that produces the maximum number of primes for consecutive values of n, starting with n = 0." ], "solution": "-59231", "name": "PE27" }, { "source": "Project Euler", "directions": [ "Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:", "21 22 23 24 2520 7 8 9 1019 6 1 2 1118 5 4 3 1217 16 15 14 13", "It can be verified that the sum of the numbers on the diagonals is 101.", "What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral formed in the same way?" ], "solution": "669171001", "name": "PE28" }, { "source": "Project Euler", "directions": [ "Consider all integer combinations of ab for 2 ≤ a ≤ 5 and 2 ≤ b ≤ 5:", "22=4, 23=8, 24=16, 25=3232=9, 33=27, 34=81, 35=24342=16, 43=64, 44=256, 45=102452=25, 53=125, 54=625, 55=3125", "If they are then placed in numerical order, with any repeats removed, we get the following sequence of 15 distinct terms:", "4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125", "How many distinct terms are in the sequence generated by ab for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100?" ], "solution": "9183", "name": "PE29" }, { "source": "Project Euler", "directions": [ "Surprisingly there are only three numbers that can be written as the sum of fourth powers of their digits:", "1634 = 14 + 64 + 34 + 448208 = 84 + 24 + 04 + 849474 = 94 + 44 + 74 + 44", "As 1 = 14 is not a sum it is not included.", "The sum of these numbers is 1634 + 8208 + 9474 = 19316.", "Find the sum of all the numbers that can be written as the sum of fifth powers of their digits." ], "solution": "443839", "name": "PE30" }, { "source": "Project Euler", "directions": [ "In England the currency is made up of pound, £, and pence, p, and there are eight coins in general circulation:", "1p, 2p, 5p, 10p, 20p, 50p, £1 (100p) and £2 (200p).", "It is possible to make £2 in the following way:", "1×£1 + 1×50p + 2×20p + 1×5p + 1×2p + 3×1p", "How many different ways can £2 be made using any number of coins?" ], "solution": "73682", "name": "PE31" }, { "source": "Project Euler", "directions": [ "We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once; for example, the 5-digit number, 15234, is 1 through 5 pandigital.", "The product 7254 is unusual, as the identity, 39 × 186 = 7254, containing multiplicand, multiplier, and product is 1 through 9 pandigital.", "Find the sum of all products whose multiplicand/multiplier/product identity can be written as a 1 through 9 pandigital.", "HINT: Some products can be obtained in more than one way so be sure to only include it once in your sum." ], "solution": "45228", "name": "PE32" }, { "source": "Project Euler", "directions": [ "The fraction 49/98 is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that 49/98 = 4/8, which is correct, is obtained by cancelling the 9s.", "We shall consider fractions like, 30/50 = 3/5, to be trivial examples.", "There are exactly four non-trivial examples of this type of fraction, less than one in value, and containing two digits in the numerator and denominator.", "If the product of these four fractions is given in its lowest common terms, find the value of the denominator." ], "solution": "100", "name": "PE33" }, { "source": "Project Euler", "directions": [ "145 is a curious number, as 1! + 4! + 5! = 1 + 24 + 120 = 145.", "Find the sum of all numbers which are equal to the sum of the factorial of their digits.", "Note: as 1! = 1 and 2! = 2 are not sums they are not included." ], "solution": "40730", "name": "PE34" }, { "source": "Project Euler", "directions": [ "The number, 197, is called a circular prime because all rotations of the digits: 197, 971, and 719, are themselves prime.", "There are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, and 97.", "How many circular primes are there below one million?" ], "solution": "55", "name": "PE35" }, { "source": "Project Euler", "directions": [ "The decimal number, 585 = 10010010012 (binary), is palindromic in both bases.", "Find the sum of all numbers, less than one million, which are palindromic in base 10 and base 2.", "(Please note that the palindromic number, in either base, may not include leading zeros.)" ], "solution": "872187", "name": "PE36" }, { "source": "Project Euler", "directions": [ "The number 3797 has an interesting property. Being prime itself, it is possible to continuously remove digits from left to right, and remain prime at each stage: 3797, 797, 97, and 7. Similarly we can work from right to left: 3797, 379, 37, and 3.", "Find the sum of the only eleven primes that are both truncatable from left to right and right to left.", "NOTE: 2, 3, 5, and 7 are not considered to be truncatable primes." ], "solution": "748317", "name": "PE37" }, { "source": "Project Euler", "directions": [ "Take the number 192 and multiply it by each of 1, 2, and 3:", "192 × 1 = 192192 × 2 = 384192 × 3 = 576", "By concatenating each product we get the 1 to 9 pandigital, 192384576. We will call 192384576 the concatenated product of 192 and (1,2,3)", "The same can be achieved by starting with 9 and multiplying by 1, 2, 3, 4, and 5, giving the pandigital, 918273645, which is the concatenated product of 9 and (1,2,3,4,5).", "What is the largest 1 to 9 pandigital 9-digit number that can be formed as the concatenated product of an integer with (1,2, ... , n) where n > 1?" ], "solution": "932718654", "name": "PE38" }, { "source": "Project Euler", "directions": [ "If p is the perimeter of a right angle triangle with integral length sides, {a,b,c}, there are exactly three solutions for p = 120.", "{20,48,52}, {24,45,51}, {30,40,50}", "For which value of p ≤ 1000, is the number of solutions maximised?" ], "solution": "840", "name": "PE39" }, { "source": "Project Euler", "directions": [ "An irrational decimal fraction is created by concatenating the positive integers:", "0.123456789101112131415161718192021...", "It can be seen that the 12th digit of the fractional part is 1.", "If dn represents the nth digit of the fractional part, find the value of the following expression.", "d1 × d10 × d100 × d1000 × d10000 × d100000 × d1000000" ], "solution": "210", "name": "PE40" }, { "source": "Project Euler", "directions": [ "We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once. For example, 2143 is a 4-digit pandigital and is also prime.", "What is the largest n-digit pandigital prime that exists?" ], "solution": "7652413", "name": "PE41" }, { "source": "Project Euler", "directions": [ "The nth term of the sequence of triangle numbers is given by, tn = ½n(n+1); so the first ten triangle numbers are:", "1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...", "By converting each letter in a word to a number corresponding to its alphabetical position and adding these values we form a word value. For example, the word value for SKY is 19 + 11 + 25 = 55 = t10. If the word value is a triangle number then we shall call the word a triangle word.", "Using words.txt (right click and 'Save Link/Target As...'), a 16K text file containing nearly two-thousand common English words, how many are triangle words?" ], "solution": "162", "name": "PE42" }, { "source": "Project Euler", "directions": [ "The number, 1406357289, is a 0 to 9 pandigital number because it is made up of each of the digits 0 to 9 in some order, but it also has a rather interesting sub-string divisibility property.", "Let d1 be the 1st digit, d2 be the 2nd digit, and so on. In this way, we note the following:", "d2d3d4=406 is divisible by 2d3d4d5=063 is divisible by 3d4d5d6=635 is divisible by 5d5d6d7=357 is divisible by 7d6d7d8=572 is divisible by 11d7d8d9=728 is divisible by 13d8d9d10=289 is divisible by 17", "Find the sum of all 0 to 9 pandigital numbers with this property." ], "solution": "16695334890", "name": "PE43" }, { "source": "Project Euler", "directions": [ "Pentagonal numbers are generated by the formula, Pn=n(3n−1)/2. The first ten pentagonal numbers are:", "1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...", "It can be seen that P4 + P7 = 22 + 70 = 92 = P8. However, their difference, 70 − 22 = 48, is not pentagonal.", "Find the pair of pentagonal numbers, Pj and Pk, for which their sum and difference are pentagonal and D = |Pk − Pj| is minimised; what is the value of D?" ], "solution": "5482660", "name": "PE44" }, { "source": "Project Euler", "directions": [ "Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:", "Triangle Tn=n(n+1)/2 1, 3, 6, 10, 15, ...Pentagonal Pn=n(3n−1)/2 1, 5, 12, 22, 35, ...Hexagonal Hn=n(2n−1) 1, 6, 15, 28, 45, ...", "It can be verified that T285 = P165 = H143 = 40755.", "Find the next triangle number that is also pentagonal and hexagonal." ], "solution": "1533776805", "name": "PE45" }, { "source": "Project Euler", "directions": [ "It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.", "9 = 7 + 2×1215 = 7 + 2×2221 = 3 + 2×3225 = 7 + 2×3227 = 19 + 2×2233 = 31 + 2×12", "It turns out that the conjecture was false.", "What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?" ], "solution": "5777", "name": "PE46" }, { "source": "Project Euler", "directions": [ "The first two consecutive numbers to have two distinct prime factors are:", "14 = 2 × 715 = 3 × 5", "The first three consecutive numbers to have three distinct prime factors are:", "644 = 2² × 7 × 23645 = 3 × 5 × 43646 = 2 × 17 × 19.", "Find the first four consecutive integers to have four distinct prime factors. What is the first of these numbers?" ], "solution": "134043", "name": "PE47" }, { "source": "Project Euler", "directions": [ "The series, 11 + 22 + 33 + ... + 1010 = 10405071317.", "Find the last ten digits of the series, 11 + 22 + 33 + ... + 10001000." ], "solution": "9110846700", "name": "PE48" }, { "source": "Project Euler", "directions": [ "The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the 4-digit numbers are permutations of one another.", "There are no arithmetic sequences made up of three 1-, 2-, or 3-digit primes, exhibiting this property, but there is one other 4-digit increasing sequence.", "What 12-digit number do you form by concatenating the three terms in this sequence?" ], "solution": "296962999629", "name": "PE49" }, { "source": "Project Euler", "directions": [ "The prime 41, can be written as the sum of six consecutive primes:", "41 = 2 + 3 + 5 + 7 + 11 + 13", "This is the longest sum of consecutive primes that adds to a prime below one-hundred.", "The longest sum of consecutive primes below one-thousand that adds to a prime, contains 21 terms, and is equal to 953.", "Which prime, below one-million, can be written as the sum of the most consecutive primes?" ], "solution": "997651", "name": "PE50" }, { "source": "Project Euler", "directions": [ "By replacing the 1st digit of *57, it turns out that six of the possible values: 157, 257, 457, 557, 757, and 857, are all prime.
", "By replacing the 1st digit of the 2-digit number *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.", "By replacing the 3rd and 4th digits of 56**3 with the same digit, this 5-digit number is the first example having seven primes among the ten generated numbers, yielding the family: 56003, 56113, 56333, 56443, 56663, 56773, and 56993. Consequently 56003, being the first member of this family, is the smallest prime with this property.", "Find the smallest prime which, by replacing part of the number (not necessarily adjacent digits) with the same digit, is part of an eight prime value family." ], "solution": "121313", "name": "PE51" }, { "source": "Project Euler", "directions": [ "It can be seen that the number, 125874, and its double, 251748, contain exactly the same digits, but in a different order.", "Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits." ], "solution": "142857", "name": "PE52" }, { "source": "Project Euler", "directions": [ "There are exactly ten ways of selecting three from five, 12345:", "123, 124, 125, 134, 135, 145, 234, 235, 245, and 345", "In combinatorics, we use the notation, 5C3 = 10.", "In general,", "nCr = n!r!(n−r)!,where r ≤ n, n! = n×(n−1)×...×3×2×1, and 0! = 1.", "It is not until n = 23, that a value exceeds one-million: 23C10 = 1144066.", "How many, not necessarily distinct, values of nCr, for 1 ≤ n ≤ 100, are greater than one-million?" ], "solution": "4075", "name": "PE53" }, { "source": "Project Euler", "directions": [ "In the card game poker, a hand consists of five cards and are ranked, from lowest to highest, in the following way:", "High Card: Highest value card.One Pair: Two cards of the same value.Two Pairs: Two different pairs.Three of a Kind: Three cards of the same value.Straight: All cards are consecutive values.Flush: All cards of the same suit.Full House: Three of a kind and a pair.Four of a Kind: Four cards of the same value.Straight Flush: All cards are consecutive values of same suit.Royal Flush: Ten, Jack, Queen, King, Ace, in same suit.", "The cards are valued in the order:2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace.", "If two players have the same ranked hands then the rank made up of the highest value wins; for example, a pair of eights beats a pair of fives (see example 1 below). But if two ranks tie, for example, both players have a pair of queens, then highest cards in each hand are compared (see example 4 below); if the highest cards tie then the next highest cards are compared, and so on.", "Consider the following five hands dealt to two players:", "Hand Player 1 Player 2 Winner1 5H 5C 6S 7S KDPair of Fives 2C 3S 8S 8D TDPair of Eights Player 22 5D 8C 9S JS ACHighest card Ace 2C 5C 7D 8S QHHighest card Queen Player 13 2D 9C AS AH ACThree Aces 3D 6D 7D TD QDFlush with Diamonds Player 24 4D 6S 9H QH QCPair of QueensHighest card Nine 3D 6D 7H QD QSPair of QueensHighest card Seven Player 15 2H 2D 4C 4D 4SFull HouseWith Three Fours 3C 3D 3S 9S 9DFull Housewith Three Threes Player 1", "The file, poker.txt, contains one-thousand random hands dealt to two players. Each line of the file contains ten cards (separated by a single space): the first five are Player 1's cards and the last five are Player 2's cards. You can assume that all hands are valid (no invalid characters or repeated cards), each player's hand is in no specific order, and in each hand there is a clear winner.", "How many hands does Player 1 win?" ], "solution": "376", "name": "PE54" }, { "source": "Project Euler", "directions": [ "If we take 47, reverse and add, 47 + 74 = 121, which is palindromic.", "Not all numbers produce palindromes so quickly. For example,", "349 + 943 = 1292,1292 + 2921 = 42134213 + 3124 = 7337", "That is, 349 took three iterations to arrive at a palindrome.", "Although no one has proved it yet, it is thought that some numbers, like 196, never produce a palindrome. A number that never forms a palindrome through the reverse and add process is called a Lychrel number. Due to the theoretical nature of these numbers, and for the purpose of this problem, we shall assume that a number is Lychrel until proven otherwise. In addition you are given that for every number below ten-thousand, it will either (i) become a palindrome in less than fifty iterations, or, (ii) no one, with all the computing power that exists, has managed so far to map it to a palindrome. In fact, 10677 is the first number to be shown to require over fifty iterations before producing a palindrome: 4668731596684224866951378664 (53 iterations, 28-digits).", "Surprisingly, there are palindromic numbers that are themselves Lychrel numbers; the first example is 4994.", "How many Lychrel numbers are there below ten-thousand?", "NOTE: Wording was modified slightly on 24 April 2007 to emphasise the theoretical nature of Lychrel numbers." ], "solution": "249", "name": "PE55" }, { "source": "Project Euler", "directions": [ "A googol (10100) is a massive number: one followed by one-hundred zeros; 100100 is almost unimaginably large: one followed by two-hundred zeros. Despite their size, the sum of the digits in each number is only 1.", "Considering natural numbers of the form, ab, where a, b < 100, what is the maximum digital sum?" ], "solution": "972", "name": "PE56" }, { "source": "Project Euler", "directions": [ "It is possible to show that the square root of two can be expressed as an infinite continued fraction.", "√ 2 = 1 + 1/(2 + 1/(2 + 1/(2 + ... ))) = 1.414213...", "By expanding this for the first four iterations, we get:", "1 + 1/2 = 3/2 = 1.51 + 1/(2 + 1/2) = 7/5 = 1.41 + 1/(2 + 1/(2 + 1/2)) = 17/12 = 1.41666...1 + 1/(2 + 1/(2 + 1/(2 + 1/2))) = 41/29 = 1.41379...", "The next three expansions are 99/70, 239/169, and 577/408, but the eighth expansion, 1393/985, is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.", "In the first one-thousand expansions, how many fractions contain a numerator with more digits than denominator?" ], "solution": "153", "name": "PE57" }, { "source": "Project Euler", "directions": [ "Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed.", "37 36 35 34 33 32 3138 17 16 15 14 13 3039 18 5 4 3 12 2940 19 6 1 2 11 2841 20 7 8 9 10 2742 21 22 23 24 25 2643 44 45 46 47 48 49", "It is interesting to note that the odd squares lie along the bottom right diagonal, but what is more interesting is that 8 out of the 13 numbers lying along both diagonals are prime; that is, a ratio of 8/13 ≈ 62%.", "If one complete new layer is wrapped around the spiral above, a square spiral with side length 9 will be formed. If this process is continued, what is the side length of the square spiral for which the ratio of primes along both diagonals first falls below 10%?" ], "solution": "26241", "name": "PE58" }, { "source": "Project Euler", "directions": [ "Each character on a computer is assigned a unique code and the preferred standard is ASCII (American Standard Code for Information Interchange). For example, uppercase A = 65, asterisk (*) = 42, and lowercase k = 107.", "A modern encryption method is to take a text file, convert the bytes to ASCII, then XOR each byte with a given value, taken from a secret key. The advantage with the XOR function is that using the same encryption key on the cipher text, restores the plain text; for example, 65 XOR 42 = 107, then 107 XOR 42 = 65.", "For unbreakable encryption, the key is the same length as the plain text message, and the key is made up of random bytes. The user would keep the encrypted message and the encryption key in different locations, and without both \"halves\", it is impossible to decrypt the message.", "Unfortunately, this method is impractical for most users, so the modified method is to use a password as a key. If the password is shorter than the message, which is likely, the key is repeated cyclically throughout the message. The balance for this method is using a sufficiently long password key for security, but short enough to be memorable.", "Your task has been made easy, as the encryption key consists of three lower case characters. Using cipher.txt (right click and 'Save Link/Target As...'), a file containing the encrypted ASCII codes, and the knowledge that the plain text must contain common English words, decrypt the message and find the sum of the ASCII values in the original text." ], "solution": "107359", "name": "PE59" }, { "source": "Project Euler", "directions": [ "The primes 3, 7, 109, and 673, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime. For example, taking 7 and 109, both 7109 and 1097 are prime. The sum of these four primes, 792, represents the lowest sum for a set of four primes with this property.", "Find the lowest sum for a set of five primes for which any two primes concatenate to produce another prime." ], "solution": "26033", "name": "PE60" }, { "source": "Project Euler", "directions": [ "Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:", "Triangle P3,n=n(n+1)/2 1, 3, 6, 10, 15, ...Square P4,n=n2 1, 4, 9, 16, 25, ...Pentagonal P5,n=n(3n−1)/2 1, 5, 12, 22, 35, ...Hexagonal P6,n=n(2n−1) 1, 6, 15, 28, 45, ...Heptagonal P7,n=n(5n−3)/2 1, 7, 18, 34, 55, ...Octagonal P8,n=n(3n−2) 1, 8, 21, 40, 65, ...", "The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three interesting properties.", "The set is cyclic, in that the last two digits of each number is the first two digits of the next number (including the last number with the first).Each polygonal type: triangle (P3,127=8128), square (P4,91=8281), and pentagonal (P5,44=2882), is represented by a different number in the set.This is the only set of 4-digit numbers with this property.", "Find the sum of the only ordered set of six cyclic 4-digit numbers for which each polygonal type: triangle, square, pentagonal, hexagonal, heptagonal, and octagonal, is represented by a different number in the set." ], "solution": "28684", "name": "PE61" }, { "source": "Project Euler", "directions": [ "The cube, 41063625 (3453), can be permuted to produce two other cubes: 56623104 (3843) and 66430125 (4053). In fact, 41063625 is the smallest cube which has exactly three permutations of its digits which are also cube.", "Find the smallest cube for which exactly five permutations of its digits are cube." ], "solution": "127035954683", "name": "PE62" }, { "source": "Project Euler", "directions": [ "The 5-digit number, 16807=75, is also a fifth power. Similarly, the 9-digit number, 134217728=89, is a ninth power.", "How many n-digit positive integers exist which are also an nth power?" ], "solution": "49", "name": "PE63" }, { "source": "Project Euler", "directions": [ "All square roots are periodic when written as continued fractions and can be written in the form:", "√N = a0 +1 a1 +1 a2 +1 a3 + ...", "For example, let us consider √23:", "√23 = 4 + √23 — 4 = 4 + 1 = 4 + 1 1√23—4 1 + √23 – 37", "If we continue we would get the following expansion:", "√23 = 4 +1 1 +1 3 +1 1 +1 8 + ...", "The process can be summarised as follows:", "a0 = 4, 1√23—4 = √23+47 = 1 + √23—37a1 = 1, 7√23—3 = 7(√23+3)14 = 3 + √23—32a2 = 3, 2√23—3 = 2(√23+3)14 = 1 + √23—47a3 = 1, 7√23—4 = 7(√23+4)7 = 8 + √23—4a4 = 8, 1√23—4 = √23+47 = 1 + √23—37a5 = 1, 7√23—3 = 7(√23+3)14 = 3 + √23—32a6 = 3, 2√23—3 = 2(√23+3)14 = 1 + √23—47a7 = 1, 7√23—4 = 7(√23+4)7 = 8 + √23—4", "It can be seen that the sequence is repeating. For conciseness, we use the notation √23 = [4;(1,3,1,8)], to indicate that the block (1,3,1,8) repeats indefinitely.", "The first ten continued fraction representations of (irrational) square roots are:", "√2=[1;(2)], period=1√3=[1;(1,2)], period=2√5=[2;(4)], period=1√6=[2;(2,4)], period=2√7=[2;(1,1,1,4)], period=4√8=[2;(1,4)], period=2√10=[3;(6)], period=1√11=[3;(3,6)], period=2√12= [3;(2,6)], period=2√13=[3;(1,1,1,1,6)], period=5", "Exactly four continued fractions, for N ≤ 13, have an odd period.", "How many continued fractions for N ≤ 10000 have an odd period?" ], "solution": "1322", "name": "PE64" }, { "source": "Project Euler", "directions": [ "The square root of 2 can be written as an infinite continued fraction.", "√2 = 1 +1 2 +1 2 +1 2 +1 2 + ...", "The infinite continued fraction can be written, √2 = [1;(2)], (2) indicates that 2 repeats ad infinitum. In a similar way, √23 = [4;(1,3,1,8)].", "It turns out that the sequence of partial values of continued fractions for square roots provide the best rational approximations. Let us consider the convergents for √2.", "1 +1= 3/2 2 1 +1= 7/5 2 +1 2 1 +1= 17/12 2 +1 2 +1 2 1 +1= 41/29 2 +1 2 +1 2 +1 2 ", "Hence the sequence of the first ten convergents for √2 are:", "1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, ...", "What is most surprising is that the important mathematical constant,e = [2; 1,2,1, 1,4,1, 1,6,1 , ... , 1,2k,1, ...].", "The first ten terms in the sequence of convergents for e are:", "2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, ...", "The sum of digits in the numerator of the 10th convergent is 1+4+5+7=17.", "Find the sum of digits in the numerator of the 100th convergent of the continued fraction for e." ], "solution": "272", "name": "PE65" }, { "source": "Project Euler", "directions": [ "Consider quadratic Diophantine equations of the form:", "x2 – Dy2 = 1", "For example, when D=13, the minimal solution in x is 6492 – 13×1802 = 1.", "It can be assumed that there are no solutions in positive integers when D is square.", "By finding minimal solutions in x for D = {2, 3, 5, 6, 7}, we obtain the following:", "32 – 2×22 = 122 – 3×12 = 192 – 5×42 = 152 – 6×22 = 182 – 7×32 = 1", "Hence, by considering minimal solutions in x for D ≤ 7, the largest x is obtained when D=5.", "Find the value of D ≤ 1000 in minimal solutions of x for which the largest value of x is obtained." ], "solution": "661", "name": "PE66" }, { "source": "Project Euler", "directions": [ "By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.", "37 42 4 68 5 9 3", "That is, 3 + 7 + 4 + 9 = 23.", "Find the maximum total from top to bottom in triangle.txt (right click and 'Save Link/Target As...'), a 15K text file containing a triangle with one-hundred rows.", "NOTE: This is a much more difficult version of Problem 18. It is not possible to try every route to solve this problem, as there are 299 altogether! If you could check one trillion (1012) routes every second it would take over twenty billion years to check them all. There is an efficient algorithm to solve it. ;o)" ], "solution": "7273", "name": "PE67" }, { "source": "Project Euler", "directions": [ "Consider the following \"magic\" 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine.", "Working clockwise, and starting from the group of three with the numerically lowest external node (4,3,2 in this example), each solution can be described uniquely. For example, the above solution can be described by the set: 4,3,2; 6,2,1; 5,1,3.", "It is possible to complete the ring with four different totals: 9, 10, 11, and 12. There are eight solutions in total.", "TotalSolution Set94,2,3; 5,3,1; 6,1,294,3,2; 6,2,1; 5,1,3102,3,5; 4,5,1; 6,1,3102,5,3; 6,3,1; 4,1,5111,4,6; 3,6,2; 5,2,4111,6,4; 5,4,2; 3,2,6121,5,6; 2,6,4; 3,4,5121,6,5; 3,5,4; 2,4,6", "By concatenating each group it is possible to form 9-digit strings; the maximum string for a 3-gon ring is 432621513.", "Using the numbers 1 to 10, and depending on arrangements, it is possible to form 16- and 17-digit strings. What is the maximum 16-digit string for a \"magic\" 5-gon ring?" ], "solution": "6531031914842725", "name": "PE68" }, { "source": "Project Euler", "directions": [ "Euler's Totient function, φ(n) [sometimes called the phi function], is used to determine the number of numbers less than n which are relatively prime to n. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, φ(9)=6.", "nRelatively Primeφ(n)n/φ(n)211231,221.541,32251,2,3,441.2561,52371,2,3,4,5,661.1666...81,3,5,74291,2,4,5,7,861.5101,3,7,942.5", "It can be seen that n=6 produces a maximum n/φ(n) for n ≤ 10.", "Find the value of n ≤ 1,000,000 for which n/φ(n) is a maximum." ], "solution": "510510", "name": "PE69" }, { "source": "Project Euler", "directions": [ "Euler's Totient function, φ(n) [sometimes called the phi function], is used to determine the number of positive numbers less than or equal to n which are relatively prime to n. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, φ(9)=6.The number 1 is considered to be relatively prime to every positive number, so φ(1)=1. ", "Interestingly, φ(87109)=79180, and it can be seen that 87109 is a permutation of 79180.", "Find the value of n, 1 < n < 107, for which φ(n) is a permutation of n and the ratio n/φ(n) produces a minimum." ], "solution": "8319823", "name": "PE70" }, { "source": "Project Euler", "directions": [ "Consider the fraction, n/d, where n and d are positive integers. If n0. ( | is the bitwise-OR operator)", "It can be seen that eventually there will be an index N such that xi = 232 -1 (a bit-pattern of all ones) for all i ≥ N.", "Find the expected value of N. Give your answer rounded to 10 digits after the decimal point." ], "solution": "6.3551758451", "name": "PE323" }, { "source": "Project Euler", "directions": [ "Let f(n) represent the number of ways one can fill a 3×3×n tower with blocks of 2×1×1. You're allowed to rotate the blocks in any way you like; however, rotations, reflections etc of the tower itself are counted as distinct.", "For example (with q = 100000007) :f(2) = 229,f(4) = 117805,f(10) mod q = 96149360,f(103) mod q = 24806056,f(106) mod q = 30808124.", "Find f(1010000) mod 100000007." ], "solution": "96972774", "name": "PE324" }, { "source": "Project Euler", "directions": [ "A game is played with two piles of stones and two players. At her turn, a player removes a number of stones from the larger pile. The number of stones she removes must be a positive multiple of the number of stones in the smaller pile.", "E.g., let the ordered pair(6,14) describe a configuration with 6 stones in the smaller pile and 14 stones in the larger pile, then the first player can remove 6 or 12 stones from the larger pile.", "The player taking all the stones from a pile wins the game.", "A winning configuration is one where the first player can force a win. For example, (1,5), (2,6) and (3,12) are winning configurations because the first player can immediately remove all stones in the second pile.", "A losing configuration is one where the second player can force a win, no matter what the first player does. For example, (2,3) and (3,4) are losing configurations: any legal move leaves a winning configuration for the second player.", "Define S(N) as the sum of (xi+yi) for all losing configurations (xi,yi), 0 < xi < yi ≤ N. We can verify that S(10) = 211 and S(104) = 230312207313.", "Find S(1016) mod 710." ], "solution": "54672965", "name": "PE325" }, { "source": "Project Euler", "directions": [ "Let an be a sequence recursively defined by: . ", "So the first 10 elements of an are: 1,1,0,3,0,3,5,4,1,9.", "Let f(N,M) represent the number of pairs (p,q) such that: ", "It can be seen that f(10,10)=4 with the pairs (3,3), (5,5), (7,9) and (9,10).", "You are also given that f(104,103)=97158.", "Find f(1012,106).", " " ], "solution": "1966666166408794329", "name": "PE326" }, { "source": "Project Euler", "directions": [ "A series of three rooms are connected to each other by automatic doors.", "Each door is operated by a security card. Once you enter a room the door automatically closes and that security card cannot be used again. A machine at the start will dispense an unlimited number of cards, but each room (including the starting room) contains scanners and if they detect that you are holding more than three security cards or if they detect an unattended security card on the floor, then all the doors will become permanently locked. However, each room contains a box where you may safely store any number of security cards for use at a later stage.", "If you simply tried to travel through the rooms one at a time then as you entered room 3 you would have used all three cards and would be trapped in that room forever!", "However, if you make use of the storage boxes, then escape is possible. For example, you could enter room 1 using your first card, place one card in the storage box, and use your third card to exit the room back to the start. Then after collecting three more cards from the dispensing machine you could use one to enter room 1 and collect the card you placed in the box a moment ago. You now have three cards again and will be able to travel through the remaining three doors. This method allows you to travel through all three rooms using six security cards in total.", "It is possible to travel through six rooms using a total of 123 security cards while carrying a maximum of 3 cards.", "Let C be the maximum number of cards which can be carried at any time.", "Let R be the number of rooms to travel through.", "Let M(C,R) be the minimum number of cards required from the dispensing machine to travel through R rooms carrying up to a maximum of C cards at any time.", "For example, M(3,6)=123 and M(4,6)=23.And, ΣM(C,6)=146 for 3 ≤ C ≤ 4.", "You are given that ΣM(C,10)=10382 for 3 ≤ C ≤ 10.", "Find ΣM(C,30) for 3 ≤ C ≤ 40." ], "solution": "34315549139516", "name": "PE327" }, { "source": "Project Euler", "directions": [ "We are trying to find a hidden number selected from the set of integers {1, 2, ..., n} by asking questions. Each number (question) we ask, has a cost equal to the number asked and we get one of three possible answers:", " \"Your guess is lower than the hidden number\", or \"Yes, that's it!\", or \"Your guess is higher than the hidden number\".", "Given the value of n, an optimal strategy minimizes the total cost (i.e. the sum of all the questions asked) for the worst possible case. E.g.", "If n=3, the best we can do is obviously to ask the number \"2\". The answer will immediately lead us to find the hidden number (at a total cost = 2).", "If n=8, we might decide to use a \"binary search\" type of strategy: Our first question would be \"4\" and if the hidden number is higher than 4 we will need one or two additional questions.Let our second question be \"6\". If the hidden number is still higher than 6, we will need a third question in order to discriminate between 7 and 8.Thus, our third question will be \"7\" and the total cost for this worst-case scenario will be 4+6+7=17.", "We can improve considerably the worst-case cost for n=8, by asking \"5\" as our first question.If we are told that the hidden number is higher than 5, our second question will be \"7\", then we'll know for certain what the hidden number is (for a total cost of 5+7=12).If we are told that the hidden number is lower than 5, our second question will be \"3\" and if the hidden number is lower than 3 our third question will be \"1\", giving a total cost of 5+3+1=9.Since 12>9, the worst-case cost for this strategy is 12. That's better than what we achieved previously with the \"binary search\" strategy; it is also better than or equal to any other strategy.So, in fact, we have just described an optimal strategy for n=8.", "Let C(n) be the worst-case cost achieved by an optimal strategy for n, as described above.Thus C(1) = 0, C(2) = 1, C(3) = 2 and C(8) = 12.Similarly, C(100) = 400 and C(n) = 17575.", "Find C(n)." ], "solution": "260511850222", "name": "PE328" }, { "source": "Project Euler", "directions": [ "Susan has a prime frog.Her frog is jumping around over 500 squares numbered 1 to 500.He can only jump one square to the left or to the right, with equal probability, and he cannot jump outside the range [1;500].(if it lands at either end, it automatically jumps to the only available square on the next move.)", "When he is on a square with a prime number on it, he croaks 'P' (PRIME) with probability 2/3 or 'N' (NOT PRIME) with probability 1/3 just before jumping to the next square.When he is on a square with a number on it that is not a prime he croaks 'P' with probability 1/3 or 'N' with probability 2/3 just before jumping to the next square.", "Given that the frog's starting position is random with the same probability for every square, and given that she listens to his first 15 croaks, what is the probability that she hears the sequence PPPPNNPPPNPPNPN?", "Give your answer as a fraction p/q in reduced form." ], "solution": "199740353/29386561536000", "name": "PE329" }, { "source": "Project Euler", "directions": [ "An infinite sequence of real numbers ", "a", "(", "n", ") is defined for all integers ", "n", " as follows:", "For example,", "a(0) = 11! + 12! + 13! + ... = e − 1 ", "a(1) = e − 11! + 12! + 13! + ... = 2e − 3 ", "a(2) = 2e − 31! + e − 12! + 13! + ... = 72 e − 6 ", "with e = 2.7182818... being Euler's constant.", "It can be shown that a(n) is of the form A(n) e + B(n)n! for integers A(n) and B(n). ", "For example a(10) = 328161643 e − 65269448610! .", "Find A(109) + B(109) and give your answer mod 77 777 777." ], "solution": "15955822", "name": "PE330" }, { "source": "Project Euler", "directions": [ "N×N disks are placed on a square game board. Each disk has a black side and white side.", "At each turn, you may choose a disk and flip all the disks in the same row and the same column as this disk: thus 2×N-1 disks are flipped. The game ends when all disks show their white side. The following example shows a game on a 5×5 board.", "It can be proven that 3 is the minimal number of turns to finish this game.", "The bottom left disk on the N×N board has coordinates (0,0);the bottom right disk has coordinates (N-1,0) and the top left disk has coordinates (0,N-1). ", "Let CN be the following configuration of a board with N×N disks:A disk at (x,y) satisfying , shows its black side; otherwise, it shows its white side. C5 is shown above.", "Let T(N) be the minimal number of turns to finish a game starting from configuration CN or 0 if configuration CN is unsolvable.We have shown that T(5)=3. You are also given that T(10)=29 and T(1 000)=395253.", "Find ." ], "solution": "467178235146843549", "name": "PE331" }, { "source": "Project Euler", "directions": [ "A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in three vertices.", "Let C(r) be the sphere with the centre (0,0,0) and radius r.Let Z(r) be the set of points on the surface of C(r) with integer coordinates.Let T(r) be the set of spherical triangles with vertices in Z(r).Degenerate spherical triangles, formed by three points on the same great arc, are not included in T(r).Let A(r) be the area of the smallest spherical triangle in T(r).", "For example A(14) is 3.294040 rounded to six decimal places.", "Find A(r). Give your answer rounded to six decimal places." ], "solution": "2717.751525", "name": "PE332" }, { "source": "Project Euler", "directions": [ "All positive integers can be partitioned in such a way that each and every term of the partition can be expressed as 2ix3j, where i,j ≥ 0.", "Let's consider only those such partitions where none of the terms can divide any of the other terms.For example, the partition of 17 = 2 + 6 + 9 = (21x30 + 21x31 + 20x32) would not be valid since 2 can divide 6. Neither would the partition 17 = 16 + 1 = (24x30 + 20x30) since 1 can divide 16. The only valid partition of 17 would be 8 + 9 = (23x30 + 20x32).", "Many integers have more than one valid partition, the first being 11 having the following two partitions.11 = 2 + 9 = (21x30 + 20x32)11 = 8 + 3 = (23x30 + 20x31)", "Let's define P(n) as the number of valid partitions of n. For example, P(11) = 2.", "Let's consider only the prime integers q which would have a single valid partition such as P(17).", "The sum of the primes q <100 such that P(q)=1 equals 233.", "Find the sum of the primes q <1000000 such that P(q)=1." ], "solution": "3053105", "name": "PE333" }, { "source": "Project Euler", "directions": [ "In Plato's heaven, there exist an infinite number of bowls in a straight line.Each bowl either contains some or none of a finite number of beans.A child plays a game, which allows only one kind of move: removing two beans from any bowl, and putting one in each of the two adjacent bowls. The game ends when each bowl contains either one or no beans.", "For example, consider two adjacent bowls containing 2 and 3 beans respectively, all other bowls being empty. The following eight moves will finish the game:", "You are given the following sequences:", " t0 = 123456. ", " ti = ti-12 , if ti-1 is even ti-12 926252, if ti-1 is odd where ⌊x⌋ is the floor function and is the bitwise XOR operator. ", " bi = ( ti mod 211) + 1. ", "The first two terms of the last sequence are b1 = 289 and b2 = 145.If we start with b1 and b2 beans in two adjacent bowls, 3419100 moves would be required to finish the game.", "Consider now 1500 adjacent bowls containing b1, b2,..., b1500 beans respectively, all other bowls being empty. Find how many moves it takes before the game ends." ], "solution": "150320021261690835", "name": "PE334" }, { "source": "Project Euler", "directions": [ "Whenever Peter feels bored, he places some bowls, containing one bean each, in a circle. After this, he takes all the beans out of a certain bowl and drops them one by one in the bowls going clockwise. He repeats this, starting from the bowl he dropped the last bean in, until the initial situation appears again. For example with 5 bowls he acts as follows:", "So with 5 bowls it takes Peter 15 moves to return to the initial situation.", "Let M(x) represent the number of moves required to return to the initial situation, starting with x bowls. Thus, M(5) = 15. It can also be verified that M(100) = 10920.", "Find M(2k+1). Give your answer modulo 79." ], "solution": "5032316", "name": "PE335" }, { "source": "Project Euler", "directions": [ "A train is used to transport four carriages in the order: ABCD. However, sometimes when the train arrives to collect the carriages they are not in the correct order. To rearrange the carriages they are all shunted on to a large rotating turntable. After the carriages are uncoupled at a specific point the train moves off the turntable pulling the carriages still attached with it. The remaining carriages are rotated 180 degrees. All of the carriages are then rejoined and this process is repeated as often as necessary in order to obtain the least number of uses of the turntable.Some arrangements, such as ADCB, can be solved easily: the carriages are separated between A and D, and after DCB are rotated the correct order has been achieved.", "However, Simple Simon, the train driver, is not known for his efficiency, so he always solves the problem by initially getting carriage A in the correct place, then carriage B, and so on.", "Using four carriages, the worst possible arrangements for Simon, which we shall call maximix arrangements, are DACB and DBAC; each requiring him five rotations (although, using the most efficient approach, they could be solved using just three rotations). The process he uses for DACB is shown below.", "It can be verified that there are 24 maximix arrangements for six carriages, of which the tenth lexicographic maximix arrangement is DFAECB.", "Find the 2011th lexicographic maximix arrangement for eleven carriages." ], "solution": "CAGBIHEFJDK", "name": "PE336" }, { "source": "Project Euler", "directions": [ "Let {a1, a2,..., an} be an integer sequence of length n such that:", "a1 = 6for all 1 ≤ i n : φ(ai) i+1) i i+1 1", "Let S(N) be the number of such sequences with an ≤ N.For example, S(10) = 4: {6}, {6, 8}, {6, 8, 9} and {6, 10}.We can verify that S(100) = 482073668 and S(10 000) mod 108 = 73808307.", "Find S(20 000 000) mod 108.", "1 φ denotes Euler's totient function." ], "solution": "85068035", "name": "PE337" }, { "source": "Project Euler", "directions": [ "A rectangular sheet of grid paper with integer dimensions w × h is given. Its grid spacing is 1.When we cut the sheet along the grid lines into two pieces and rearrange those pieces without overlap, we can make new rectangles with different dimensions.", "For example, from a sheet with dimensions 9 × 4 , we can make rectangles with dimensions 18 × 2, 12 × 3 and 6 × 6 by cutting and rearranging as below:", "Similarly, from a sheet with dimensions 9 × 8 , we can make rectangles with dimensions 18 × 4 and 12 × 6 .", "For a pair w and h, let F(w,h) be the number of distinct rectangles that can be made from a sheet with dimensions w × h .For example, F(2,1) = 0, F(2,2) = 1, F(9,4) = 3 and F(9,8) = 2. Note that rectangles congruent to the initial one are not counted in F(w,h).Note also that rectangles with dimensions w × h and dimensions h × w are not considered distinct.", "For an integer N, let G(N) be the sum of F(w,h) for all pairs w and h which satisfy 0 < h ≤ w ≤ N.We can verify that G(10) = 55, G(103) = 971745 and G(105) = 9992617687.", "Find G(1012). Give your answer modulo 108." ], "solution": "15614292", "name": "PE338" }, { "source": "Project Euler", "directions": [ "\"And he came towards a valley, through which ran a river; and the borders of the valley were wooded, and on each side of the river were level meadows. And on one side of the river he saw a flock of white sheep, and on the other a flock of black sheep. And whenever one of the white sheep bleated, one of the black sheep would cross over and become white; and when one of the black sheep bleated, one of the white sheep would cross over and become black.\"en.wikisource.org", "Initially each flock consists of n sheep. Each sheep (regardless of colour) is equally likely to be the next sheep to bleat. After a sheep has bleated and a sheep from the other flock has crossed over, Peredur may remove a number of white sheep in order to maximize the expected final number of black sheep. Let E(n) be the expected final number of black sheep if Peredur uses an optimal strategy.", "You are given that E(5) = 6.871346 rounded to 6 places behind the decimal point.Find E(10 000) and give your answer rounded to 6 places behind the decimal point." ], "solution": "19823.542204", "name": "PE339" }, { "source": "Project Euler", "directions": [ "For fixed integers a, b, c, define the crazy function F(n) as follows:F(n) = n - c for all n > b F(n) = F(a + F(a + F(a + F(a + n)))) for all n ≤ b.", "Also, define S(a, b, c) = .", "For example, if a = 50, b = 2000 and c = 40, then F(0) = 3240 and F(2000) = 2040.Also, S(50, 2000, 40) = 5204240.", "Find the last 9 digits of S(217, 721, 127)." ], "solution": "291504964", "name": "PE340" }, { "source": "Project Euler", "directions": [ "The Golomb's self-describing sequence {G(n)} is the only nondecreasing sequence of natural numbers such that n appears exactly G(n) times in the sequence. The values of G(n) for the first few n are", "n123456789101112131415…G(n)122334445556666…", "You are given that G(103) = 86, G(106) = 6137.You are also given that ΣG(n3) = 153506976 for 1 ≤ n < 103.", "Find ΣG(n3) for 1 ≤ n < 106." ], "solution": "56098610614277014", "name": "PE341" }, { "source": "Project Euler", "directions": [ "Consider the number 50.502 = 2500 = 22 × 54, so φ(2500) = 2 × 4 × 53 = 8 × 53 = 23 × 53. 1So 2500 is a square and φ(2500) is a cube.", "Find the sum of all numbers n, 1 < n < 1010 such that φ(n2) is a cube.", "1 φ denotes Euler's totient function." ], "solution": "5943040885644", "name": "PE342" }, { "source": "Project Euler", "directions": [ "For any positive integer k, a finite sequence ai of fractions xi/yi is defined by:a1 = 1/k andai = (xi-1+1)/(yi-1-1) reduced to lowest terms for i>1.When ai reaches some integer n, the sequence stops. (That is, when yi=1.)Define f(k) = n. For example, for k = 20:", "1/20 → 2/19 → 3/18 = 1/6 → 2/5 → 3/4 → 4/3 → 5/2 → 6/1 = 6", "So f(20) = 6.", "Also f(1) = 1, f(2) = 2, f(3) = 1 and Σf(k3) = 118937 for 1 ≤ k ≤ 100.", "Find Σf(k3) for 1 ≤ k ≤ 2×106." ], "solution": "269533451410884183", "name": "PE343" }, { "source": "Project Euler", "directions": [ "One variant of N.G. de Bruijn's silver dollar game can be described as follows:", "On a strip of squares a number of coins are placed, at most one coin per square. Only one coin, called the silver dollar, has any value. Two players take turns making moves. At each turn a player must make either a regular or a special move.", "A regular move consists of selecting one coin and moving it one or more squares to the left. The coin cannot move out of the strip or jump on or over another coin.", "Alternatively, the player can choose to make the special move of pocketing the leftmost coin rather than making a regular move. If no regular moves are possible, the player is forced to pocket the leftmost coin.", "The winner is the player who pockets the silver dollar.", "A winning configuration is an arrangement of coins on the strip where the first player can force a win no matter what the second player does.", "Let W(n,c) be the number of winning configurations for a strip of n squares, c worthless coins and one silver dollar.", "You are given that W(10,2) = 324 and W(100,10) = 1514704946113500.", "Find W(1 000 000, 100) modulo the semiprime 1000 036 000 099 (= 1 000 003 · 1 000 033)." ], "solution": "65579304332", "name": "PE344" }, { "source": "Project Euler", "directions": [ "We define the Matrix Sum of a matrix as the maximum sum of matrix elements with each element being the only one in his row and column. For example, the Matrix Sum of the matrix below equals 3315 ( = 863 + 383 + 343 + 959 + 767):", " 7 53 183 439 863497 383 563 79 973287 63 343 169 583627 343 773 959 943767 473 103 699 303", "Find the Matrix Sum of:", " 7 53 183 439 863 497 383 563 79 973 287 63 343 169 583627 343 773 959 943 767 473 103 699 303 957 703 583 639 913447 283 463 29 23 487 463 993 119 883 327 493 423 159 743217 623 3 399 853 407 103 983 89 463 290 516 212 462 350960 376 682 962 300 780 486 502 912 800 250 346 172 812 350870 456 192 162 593 473 915 45 989 873 823 965 425 329 803973 965 905 919 133 673 665 235 509 613 673 815 165 992 326322 148 972 962 286 255 941 541 265 323 925 281 601 95 973445 721 11 525 473 65 511 164 138 672 18 428 154 448 848414 456 310 312 798 104 566 520 302 248 694 976 430 392 198184 829 373 181 631 101 969 613 840 740 778 458 284 760 390821 461 843 513 17 901 711 993 293 157 274 94 192 156 574 34 124 4 878 450 476 712 914 838 669 875 299 823 329 699815 559 813 459 522 788 168 586 966 232 308 833 251 631 107813 883 451 509 615 77 281 613 459 205 380 274 302 35 805" ], "solution": "13938", "name": "PE345" }, { "source": "Project Euler", "directions": [ "The number 7 is special, because 7 is 111 written in base 2, and 11 written in base 6 (i.e. 710 = 116 = 1112). In other words, 7 is a repunit in at least two bases b > 1. ", "We shall call a positive integer with this property a strong repunit. It can be verified that there are 8 strong repunits below 50: {1,7,13,15,21,31,40,43}. Furthermore, the sum of all strong repunits below 1000 equals 15864.", "Find the sum of all strong repunits below 10", "12", "." ], "solution": "336108797689259276", "name": "PE346" }, { "source": "Project Euler", "directions": [ "The largest integer ≤ 100 that is only divisible by both the primes 2 and 3 is 96, as 96=32*3=25*3.For two distinct primes p and q let M(p,q,N) be the largest positive integer ≤N only divisibleby both p and q and M(p,q,N)=0 if such a positive integer does not exist.", "E.g. M(2,3,100)=96. M(3,5,100)=75 and not 90 because 90 is divisible by 2 ,3 and 5.Also M(2,73,100)=0 because there does not exist a positive integer ≤ 100 that is divisible by both 2 and 73.", "Let S(N) be the sum of all distinct M(p,q,N).S(100)=2262.", "Find S(10 000 000)." ], "solution": "11109800204052", "name": "PE347" }, { "source": "Project Euler", "directions": [ "Many numbers can be expressed as the sum of a square and a cube. Some of them in more than one way.", "Consider the palindromic numbers that can be expressed as the sum of a square and a cube, both greater than 1, in exactly 4 different ways.For example, 5229225 is a palindromic number and it can be expressed in exactly 4 different ways:", "22852 + 20322232 + 66318102 + 125311972 + 1563", " ", "Find the sum of the five smallest such palindromic numbers." ], "solution": "1004195061", "name": "PE348" }, { "source": "Project Euler", "directions": [ "An ant moves on a regular grid of squares that are coloured either black or white. The ant is always oriented in one of the cardinal directions (left, right, up or down) and moves from square to adjacent square according to the following rules:- if it is on a black square, it flips the color of the square to white, rotates 90 degrees counterclockwise and moves forward one square.- if it is on a white square, it flips the color of the square to black, rotates 90 degrees clockwise and moves forward one square.", "Starting with a grid that is entirely white, how many squares are black after 1018 moves of the ant?" ], "solution": "115384615384614952", "name": "PE349" }, { "source": "Project Euler", "directions": [ "A list of size n is a sequence of n natural numbers. Examples are (2,4,6), (2,6,4), (10,6,15,6), and (11).", "The greatest common divisor, or gcd, of a list is the largest natural number that divides all entries of the list. Examples: gcd(2,6,4) = 2, gcd(10,6,15,6) = 1 and gcd(11) = 11.", "The least common multiple, or lcm, of a list is the smallest natural number divisible by each entry of the list. Examples: lcm(2,6,4) = 12, lcm(10,6,15,6) = 30 and lcm(11) = 11.", "Let f(G, L, N) be the number of lists of size N with gcd ≥ G and lcm ≤ L. For example:", "f(10, 100, 1) = 91.f(10, 100, 2) = 327.f(10, 100, 3) = 1135.f(10, 100, 1000) mod 1014 = 3286053.", "Find f(106, 1012, 1018) mod 1014." ], "solution": "84664213", "name": "PE350" }, { "source": "Project Euler", "directions": [ "A hexagonal orchard of order n is a triangular lattice made up of points within a regular hexagon with side n. The following is an example of a hexagonal orchard of order 5:", "Highlighted in green are the points which are hidden from the center by a point closer to it. It can be seen that for a hexagonal orchard of order 5, 30 points are hidden from the center.", "Let H(n) be the number of points hidden from the center in a hexagonal orchard of order n.", "H(5) = 30. H(10) = 138. H(1 000) = 1177848.", "Find H(100 000 000)." ], "solution": "11762187201804552", "name": "PE351" }, { "source": "Project Euler", "directions": [ "Each one of the 25 sheep in a flock must be tested for a rare virus, known to affect 2% of the sheep population.An accurate and extremely sensitive PCR test exists for blood samples, producing a clear positive / negative result, but it is very time-consuming and expensive.", "Because of the high cost, the vet-in-charge suggests that instead of performing 25 separate tests, the following procedure can be used instead:The sheep are split into 5 groups of 5 sheep in each group. For each group, the 5 samples are mixed together and a single test is performed. Then,", "If the result is negative, all the sheep in that group are deemed to be virus-free.If the result is positive, 5 additional tests will be performed (a separate test for each animal) to determine the affected individual(s).", "Since the probability of infection for any specific animal is only 0.02, the first test (on the pooled samples) for each group will be:", "Negative (and no more tests needed) with probability 0.985 = 0.9039207968.Positive (5 additional tests needed) with probability 1 - 0.9039207968 = 0.0960792032.", "Thus, the expected number of tests for each group is 1 + 0.0960792032 × 5 = 1.480396016.Consequently, all 5 groups can be screened using an average of only 1.480396016 × 5 = 7.40198008 tests, which represents a huge saving of more than 70% !", "Although the scheme we have just described seems to be very efficient, it can still be improved considerably (always assuming that the test is sufficiently sensitive and that there are no adverse effects caused by mixing different samples). E.g.:", "We may start by running a test on a mixture of all the 25 samples. It can be verified that in about 60.35% of the cases this test will be negative, thus no more tests will be needed. Further testing will only be required for the remaining 39.65% of the cases.If we know that at least one animal in a group of 5 is infected and the first 4 individual tests come out negative, there is no need to run a test on the fifth animal (we know that it must be infected).We can try a different number of groups / different number of animals in each group, adjusting those numbers at each level so that the total expected number of tests will be minimised.", "To simplify the very wide range of possibilities, there is one restriction we place when devising the most cost-efficient testing scheme: whenever we start with a mixed sample, all the sheep contributing to that sample must be fully screened (i.e. a verdict of infected / virus-free must be reached for all of them) before we start examining any other animals.", "For the current example, it turns out that the most cost-efficient testing scheme (we'll call it the ", "optimal strategy", ") requires an average of just ", "4.155452", " tests!", "Using the optimal strategy, let T(s,p) represent the average number of tests needed to screen a flock of s sheep for a virus having probability p to be present in any individual.Thus, rounded to six decimal places, T(25, 0.02) = 4.155452 and T(25, 0.10) = 12.702124.", "Find ΣT(10000, p) for p=0.01, 0.02, 0.03, ... 0.50.Give your answer rounded to six decimal places." ], "solution": "378563.260589", "name": "PE352" }, { "source": "Project Euler", "directions": [ "A moon could be described by the sphere C(r) with centre (0,0,0) and radius r. ", "There are stations on the moon at the points on the surface of C(r) with integer coordinates. The station at (0,0,r) is called North Pole station, the station at (0,0,-r) is called South Pole station.", "All stations are connected with each other via the shortest road on the great arc through the stations. A journey between two stations is risky. If d is the length of the road between two stations, (d/(π r))2 is a measure for the risk of the journey (let us call it the risk of the road). If the journey includes more than two stations, the risk of the journey is the sum of risks of the used roads.", "A direct journey from the North Pole station to the South Pole station has the length πr and risk 1. The journey from the North Pole station to the South Pole station via (0,r,0) has the same length, but a smaller risk: (½πr/(πr))2+(½πr/(πr))2=0.5.", "The minimal risk of a journey from the North Pole station to the South Pole station on C(r) is M(r).", "You are given that M(7)=0.1784943998 rounded to 10 digits behind the decimal point. ", "Find ∑M(2n-1) for 1≤n≤15.", "Give your answer rounded to 10 digits behind the decimal point in the form a.bcdefghijk." ], "solution": "1.2759860331", "name": "PE353" }, { "source": "Project Euler", "directions": [ "Consider a honey bee's honeycomb where each cell is a perfect regular hexagon with side length 1.", "One particular cell is occupied by the queen bee.For a positive real number L, let B(L) count the cells with distance L from the queen bee cell (all distances are measured from centre to centre); you may assume that the honeycomb is large enough to accommodate for any distance we wish to consider. For example, B(√3) = 6, B(√21) = 12 and B(111 111 111) = 54.", "Find the number of L ≤ 5·1011 such that B(L) = 450." ], "solution": "58065134", "name": "PE354" }, { "source": "Project Euler", "directions": [ "Define Co(n) to be the maximal possible sum of a set of mutually co-prime elements from {1, 2, ..., n}. For example Co(10) is 30 and hits that maximum on the subset {1, 5, 7, 8, 9}.", "You are given that Co(30) = 193 and Co(100) = 1356. ", "Find Co(200000)." ], "solution": "1726545007", "name": "PE355" }, { "source": "Project Euler", "directions": [ "Let an be the largest real root of a polynomial g(x) = x3 - 2n·x2 + n.For example, a2 = 3.86619826...", "Find the last eight digits of.", "Note: represents the floor function." ], "solution": "28010159", "name": "PE356" }, { "source": "Project Euler", "directions": [ "Consider the divisors of 30: 1,2,3,5,6,10,15,30.It can be seen that for every divisor d of 30, d+30/d is prime.", "Find the sum of all positive integers n not exceeding 100 000 000such thatfor every divisor d of n, d+n/d is prime." ], "solution": "1739023853137", "name": "PE357" }, { "source": "Project Euler", "directions": [ "A cyclic number with n digits has a very interesting property:When it is multiplied by 1, 2, 3, 4, ... n, all the products have exactly the same digits, in the same order, but rotated in a circular fashion!", "The smallest cyclic number is the 6-digit number 142857 :142857 × 1 = 142857142857 × 2 = 285714142857 × 3 = 428571142857 × 4 = 571428142857 × 5 = 714285142857 × 6 = 857142 ", "The next cyclic number is 0588235294117647 with 16 digits :0588235294117647 × 1 = 05882352941176470588235294117647 × 2 = 11764705882352940588235294117647 × 3 = 1764705882352941...0588235294117647 × 16 = 9411764705882352", "Note that for cyclic numbers, leading zeros are important.", "There is only one cyclic number for which, the eleven leftmost digits are 00000000137 and the five rightmost digits are 56789 (i.e., it has the form 00000000137...56789 with an unknown number of digits in the middle). Find the sum of all its digits." ], "solution": "3284144505", "name": "PE358" }, { "source": "Project Euler", "directions": [ "An infinite number of people (numbered 1, 2, 3, etc.) are lined up to get a room at Hilbert's newest infinite hotel. The hotel contains an infinite number of floors (numbered 1, 2, 3, etc.), and each floor contains an infinite number of rooms (numbered 1, 2, 3, etc.). ", "Initially the hotel is empty. Hilbert declares a rule on how the nth person is assigned a room: person n gets the first vacant room in the lowest numbered floor satisfying either of the following:", "the floor is emptythe floor is not empty, and if the latest person taking a room in that floor is person m, then m + n is a perfect square", "Person 1 gets room 1 in floor 1 since floor 1 is empty.Person 2 does not get room 2 in floor 1 since 1 + 2 = 3 is not a perfect square.Person 2 instead gets room 1 in floor 2 since floor 2 is empty.Person 3 gets room 2 in floor 1 since 1 + 3 = 4 is a perfect square.", "Eventually, every person in the line gets a room in the hotel.", "Define P(f, r) to be n if person n occupies room r in floor f, and 0 if no person occupies the room. Here are a few examples:P(1, 1) = 1P(1, 2) = 3P(2, 1) = 2P(10, 20) = 440P(25, 75) = 4863P(99, 100) = 19454", "Find the sum of all P(f, r) for all positive f and r such that f × r = 71328803586048 and give the last 8 digits as your answer." ], "solution": "40632119", "name": "PE359" }, { "source": "Project Euler", "directions": [ "Given two points (x1,y1,z1) and (x2,y2,z2) in three dimensional space, the Manhattan distance between those points is defined as |x1-x2|+|y1-y2|+|z1-z2|.", "Let C(r) be a sphere with radius r and center in the origin O(0,0,0).Let I(r) be the set of all points with integer coordinates on the surface of C(r).Let S(r) be the sum of the Manhattan distances of all elements of I(r) to the origin O.", "E.g. S(45)=34518.", "Find S(1010)." ], "solution": "878825614395267072", "name": "PE360" }, { "source": "Project Euler", "directions": [ "The Thue-Morse sequence {Tn} is a binary sequence satisfying:", "T0 = 0T2n = TnT2n+1 = 1 - Tn", "The first several terms of {Tn} are given as follows:01101001100101101001011001101001....", "We define {An} as the sorted sequence of integers such that the binary expression of each element appears as a subsequence in {Tn}.For example, the decimal number 18 is expressed as 10010 in binary. 10010 appears in {Tn} (T8 to T12), so 18 is an element of {An}.The decimal number 14 is expressed as 1110 in binary. 1110 never appears in {Tn}, so 14 is not an element of {An}.", "The first several terms of An are given as follows:", "n0123456789101112…An012345691011121318…", "We can also verify that A100 = 3251 and A1000 = 80852364498.", "Find the last 9 digits of ." ], "solution": "178476944", "name": "PE361" }, { "source": "Project Euler", "directions": [ "Consider the number 54.54 can be factored in 7 distinct ways into one or more factors larger than 1:54, 2×27, 3×18, 6×9, 3×3×6, 2×3×9 and 2×3×3×3.If we require that the factors are all squarefree only two ways remain: 3×3×6 and 2×3×3×3.", "Let's call Fsf(n) the number of ways n can be factored into one or more squarefree factors larger than 1, soFsf(54)=2.", "Let S(n) be ∑Fsf(k) for k=2 to n.", "S(100)=193.", "Find S(10 000 000 000). " ], "solution": "457895958010", "name": "PE362" }, { "source": "Project Euler", "directions": [ "A cubic Bézier curve is defined by four points: P0, P1, P2 and P3.", "The curve is constructed as follows:On the segments P0P1, P1P2 and P2P3 the points Q0,Q1 and Q2 are drawn such thatP0Q0 / P0P1 = P1Q1 / P1P2 = P2Q2 / P2P3 = t (t in [0,1]).On the segments Q0Q1 and Q1Q2 the points R0 and R1 are drawn such thatQ0R0 / Q0Q1 = Q1R1 / Q1Q2 = t for the same value of t.On the segment R0R1 the point B is drawn such that R0B / R0R1 = t for the same value of t.The Bézier curve defined by the points P0, P1, P2, P3 is the locus of B as Q0 takes all possible positions on the segment P0P1.(Please note that for all points the value of t is the same.)", "At this (external) web address you will find an applet that allows you to drag the points P0, P1, P2 and P3 to see what the Bézier curve (green curve) defined by those points looks like. You can also drag the point Q0 along the segment P0P1.", "From the construction it is clear that the Bézier curve will be tangent to the segments P0P1 in P0 and P2P3 in P3.", "A cubic Bézier curve with P0=(1,0), P1=(1,v), P2=(v,1) and P3=(0,1) is used to approximate a quarter circle.The value v > 0 is chosen such that the area enclosed by the lines OP0, OP3 and the curve is equal to π/4 (the area of the quarter circle).", "By how many percent does the length of the curve differ from the length of the quarter circle?", "That is, if L is the length of the curve, calculate 100 × L − π/2π/2", "Give your answer rounded to 10 digits behind the decimal point." ], "solution": "0.0000372091", "name": "PE363" }, { "source": "Project Euler", "directions": [ "There are N seats in a row. N people come after each other to fill the seats according to the following rules:", "If there is any seat whose adjacent seat(s) are not occupied take such a seat.If there is no such seat and there is any seat for which only one adjacent seat is occupied take such a seat.Otherwise take one of the remaining available seats. ", "Let T(", "N", ") be the number of possibilities that ", "N", " seats are occupied by ", "N", " people with the given rules.", " The following figure shows T(4)=8.", "We can verify that T(10) = 61632 and T(1 000) mod 100 000 007 = 47255094.", "Find T(1 000 000) mod 100 000 007." ], "solution": "44855254", "name": "PE364" }, { "source": "Project Euler", "directions": [ "The binomial coeffient C(1018,109) is a number with more than 9 billion (9×109) digits.", "Let M(n,k,m) denote the binomial coefficient C(n,k) modulo m.", "Calculate ∑M(1018,109,p*q*r) for 10001/2", "If the first player picks die B and the second player picks die C we getP(second player wins) = 7/12 > 1/2", "If the first player picks die C and the second player picks die A we getP(second player wins) = 25/36 > 1/2", "So whatever die the first player picks, the second player can pick another die and have a larger than 50% chance of winning.A set of dice having this property is called a nontransitive set of dice.", "We wish to investigate how many sets of nontransitive dice exist. We will assume the following conditions:", "There are three six-sided dice with each side having between 1 and N pips, inclusive.Dice with the same set of pips are equal, regardless of which side on the die the pips are located.The same pip value may appear on multiple dice; if both players roll the same value neither player wins.The sets of dice {A,B,C}, {B,C,A} and {C,A,B} are the same set.", "For N = 7 we find there are 9780 such sets.How many are there for N = 30 ?" ], "solution": "973059630185670", "name": "PE376" }, { "source": "Project Euler", "directions": [ "There are 16 positive integers that do not have a zero in their digits and that have a digital sum equal to 5, namely: 5, 14, 23, 32, 41, 113, 122, 131, 212, 221, 311, 1112, 1121, 1211, 2111 and 11111.Their sum is 17891.", "Let f(n) be the sum of all positive integers that do not have a zero in their digits and have a digital sum equal to n.", "Find $\\displaystyle \\sum_{i=1}^{17} f(13^i)$.Give the last 9 digits as your answer." ], "solution": "732385277", "name": "PE377" }, { "source": "Project Euler", "directions": [ "Let T(n) be the nth triangle number, so T(n) =n (n+1)2.", "Let dT(n) be the number of divisors of T(n).E.g.:T(7) = 28 and dT(7) = 6.", "Let Tr(n) be the number of triples (i, j, k) such that 1 ≤ i and dT(i) > dT(j) > dT(k).Tr(20) = 14, Tr(100) = 5772 and Tr(1000) = 11174776.", "Find Tr(60 000 000). Give the last 18 digits of your answer." ], "solution": "147534623725724718", "name": "PE378" }, { "source": "Project Euler", "directions": [ "Let f(n) be the number of couples (x,y) with x and y positive integers, x ≤ y and the least common multiple of x and y equal to n.", "Let g be the summatory function of f, i.e.: g(n) = ∑ f(i) for 1 ≤ i ≤ n.", "You are given that g(106) = 37429395.", "Find g(1012)." ], "solution": "132314136838185", "name": "PE379" }, { "source": "Project Euler", "directions": [ "An m×n maze is an m×n rectangular grid with walls placed between grid cells such that there is exactly one path from the top-left square to any other square. The following are examples of a 9×12 maze and a 15×20 maze:", "Let C(m,n) be the number of distinct m×n mazes. Mazes which can be formed by rotation and reflection from another maze are considered distinct.", "It can be verified that C(1,1) = 1, C(2,2) = 4, C(3,4) = 2415, and C(9,12) = 2.5720e46 (in scientific notation rounded to 5 significant digits).Find C(100,500) and write your answer in scientific notation rounded to 5 significant digits.", "When giving your answer, use a lowercase e to separate mantissa and exponent.E.g. if the answer is 1234567891011 then the answer format would be 1.2346e12.", " " ], "solution": "6.3202e25093", "name": "PE380" }, { "source": "Project Euler", "directions": [ "For a prime p let S(p) = (∑(p-k)!) mod(p) for 1 ≤ k ≤ 5.", "For example, if p=7,(7-1)! + (7-2)! + (7-3)! + (7-4)! + (7-5)! = 6! + 5! + 4! + 3! + 2! = 720+120+24+6+2 = 872. As 872 mod(7) = 4, S(7) = 4.", "It can be verified that ∑S(p) = 480 for 5 ≤ p < 100.", "Find ∑S(p) for 5 ≤ p < 108." ], "solution": "139602943319822", "name": "PE381" }, { "source": "Project Euler", "directions": [ "A polygon is a flat shape consisting of straight line segments that are joined to form a closed chain or circuit. A polygon consists of at least three sides and does not self-intersect.", "A set S of positive numbers is said to generate a polygon P if:", " no two sides of P are the same length, the length of every side of P is in S, and S contains no other value.", "For example:The set {3, 4, 5} generates a polygon with sides 3, 4, and 5 (a triangle).The set {6, 9, 11, 24} generates a polygon with sides 6, 9, 11, and 24 (a quadrilateral).The sets {1, 2, 3} and {2, 3, 4, 9} do not generate any polygon at all.", "Consider the sequence s, defined as follows:", "s1 = 1, s2 = 2, s3 = 3sn = sn-1 + sn-3 for n > 3.", "Let Un be the set {s1, s2, ..., sn}. For example, U10 = {1, 2, 3, 4, 6, 9, 13, 19, 28, 41}.Let f(n) be the number of subsets of Un which generate at least one polygon.For example, f(5) = 7, f(10) = 501 and f(25) = 18635853.", "Find the last 9 digits of f(1018)." ], "solution": "697003956", "name": "PE382" }, { "source": "Project Euler", "directions": [ "Let f5(n) be the largest integer x for which 5x divides n.For example, f5(625000) = 7.", "Let T5(n) be the number of integers i which satisfy f5((2·i-1)!) 5(i!) and 1 ≤ i ≤ n.It can be verified that T5(103) = 68 and T5(109) = 2408210.", "Find T5(1018)." ], "solution": "22173624649806", "name": "PE383" }, { "source": "Project Euler", "directions": [ "Define the sequence a(n) as the number of adjacent pairs of ones in the binary expansion of n (possibly overlapping).E.g.: a(5) = a(1012) = 0, a(6) = a(1102) = 1, a(7) = a(1112) = 2", "Define the sequence b(n) = (-1)a(n).This sequence is called the Rudin-Shapiro sequence.", "Also consider the summatory sequence of b(n): .", "The first couple of values of these sequences are:n   0   1   2   3   4   5   6   7a(n)   0   0   0   1   0   0   1   2b(n)   1   1   1   -1   1   1   -1   1s(n)   1   2   3   2   3   4   3   4", "The sequence s(n) has the remarkable property that all elements are positive and every positive integer k occurs exactly k times.", "Define g(t,c), with 1 ≤ c ≤ t, as the index in s(n) for which t occurs for the c'th time in s(n).E.g.: g(3,3) = 6, g(4,2) = 7 and g(54321,12345) = 1220847710.", "Let F(n) be the fibonacci sequence defined by:F(0)=F(1)=1 andF(n)=F(n-1)+F(n-2) for n>1.", "Define GF(t)=g(F(t),F(t-1)).", "Find ΣGF(t) for 2≤t≤45." ], "solution": "3354706415856332783", "name": "PE384" }, { "source": "Project Euler", "directions": [ "For any triangle T in the plane, it can be shown that there is a unique ellipse with largest area that is completely inside T.", "For a given n, consider triangles T such that:- the vertices of T have integer coordinates with absolute value ≤ n, and - the foci1 of the largest-area ellipse inside T are (√13,0) and (-√13,0).Let A(n) be the sum of the areas of all such triangles.", "For example, if n = 8, there are two such triangles. Their vertices are (-4,-3),(-4,3),(8,0) and (4,3),(4,-3),(-8,0), and the area of each triangle is 36. Thus A(8) = 36 + 36 = 72.", "It can be verified that A(10) = 252, A(100) = 34632 and A(1000) = 3529008.", "Find A(1 000 000 000).", "1The foci (plural of focus) of an ellipse are two points A and B such that for every point P on the boundary of the ellipse, AP + PB is constant." ], "solution": "3776957309612153700", "name": "PE385" }, { "source": "Project Euler", "directions": [ "Let n be an integer and S(n) be the set of factors of n.", "A subset A of S(n) is called an antichain of S(n) if A contains only one element or if none of the elements of A divides any of the other elements of A.", "For example: S(30) = {1, 2, 3, 5, 6, 10, 15, 30}{2, 5, 6} is not an antichain of S(30).{2, 3, 5} is an antichain of S(30).", "Let N(n) be the maximum length of an antichain of S(n).", "Find ΣN(n) for 1 ≤ n ≤ 108" ], "solution": "528755790", "name": "PE386" }, { "source": "Project Euler", "directions": [ "A Harshad or Niven number is a number that is divisible by the sum of its digits.201 is a Harshad number because it is divisible by 3 (the sum of its digits.)When we truncate the last digit from 201, we get 20, which is a Harshad number.When we truncate the last digit from 20, we get 2, which is also a Harshad number.Let's call a Harshad number that, while recursively truncating the last digit, always results in a Harshad number a right truncatable Harshad number.", " ", "Also:201/3=67 which is prime.Let's call a Harshad number that, when divided by the sum of its digits, results in a prime a strong Harshad number.", "Now take the number 2011 which is prime.When we truncate the last digit from it we get 201, a strong Harshad number that is also right truncatable.Let's call such primes strong, right truncatable Harshad primes.", "You are given that the sum of the strong, right truncatable Harshad primes less than 10000 is 90619.", "Find the sum of the strong, right truncatable Harshad primes less than 1014." ], "solution": "696067597313468", "name": "PE387" }, { "source": "Project Euler", "directions": [ "Consider all lattice points (a,b,c) with 0 ≤ a,b,c ≤ N.", "From the origin O(0,0,0) all lines are drawn to the other lattice points.Let D(N) be the number of distinct such lines.", "You are given that D(1 000 000) = 831909254469114121.", "Find D(1010). Give as your answer the first nine digits followed by the last nine digits." ], "solution": "831907372805129931", "name": "PE388" }, { "source": "Project Euler", "directions": [ "An unbiased single 4-sided die is thrown and its value, T, is noted.T unbiased 6-sided dice are thrown and their scores are added together. The sum, C, is noted.C unbiased 8-sided dice are thrown and their scores are added together. The sum, O, is noted.O unbiased 12-sided dice are thrown and their scores are added together. The sum, D, is noted.D unbiased 20-sided dice are thrown and their scores are added together. The sum, I, is noted.Find the variance of I, and give your answer rounded to 4 decimal places." ], "solution": "2406376.3623", "name": "PE389" }, { "source": "Project Euler", "directions": [ "Consider the triangle with sides √5, √65 and √68.It can be shown that this triangle has area 9.", "S(n) is the sum of the areas of all triangles with sides √(1+b2), √(1+c2) and √(b2+c2) (for positive integers b and c ) that have an integral area not exceeding n.", "The example triangle has b=2 and c=8.", "S(106)=18018206.", "Find S(1010)." ], "solution": "2919133642971", "name": "PE390" }, { "source": "Project Euler", "directions": [ "Let sk be the number of 1’s when writing the numbers from 0 to k in binary.For example, writing 0 to 5 in binary, we have 0, 1, 10, 11, 100, 101. There are seven 1’s, so s5 = 7.The sequence S = {sk : k ≥ 0} starts {0, 1, 2, 4, 5, 7, 9, 12, ...}.", "A game is played by two players. Before the game starts, a number n is chosen. A counter c starts at 0. At each turn, the player chooses a number from 1 to n (inclusive) and increases c by that number. The resulting value of c must be a member of S. If there are no more valid moves, the player loses.", "For example:Let n = 5. c starts at 0.Player 1 chooses 4, so c becomes 0 + 4 = 4.Player 2 chooses 5, so c becomes 4 + 5 = 9.Player 1 chooses 3, so c becomes 9 + 3 = 12.etc.Note that c must always belong to S, and each player can increase c by at most n.", "Let M(n) be the highest number the first player can choose at her first turn to force a win, and M(n) = 0 if there is no such move. For example, M(2) = 2, M(7) = 1 and M(20) = 4.", "Given Σ(M(n))3 = 8150 for 1 ≤ n ≤ 20.", "Find Σ(M(n))3 for 1 ≤ n ≤ 1000." ], "solution": "61029882288", "name": "PE391" }, { "source": "Project Euler", "directions": [ "A rectilinear grid is an orthogonal grid where the spacing between the gridlines does not have to be equidistant.An example of such grid is logarithmic graph paper.", "Consider rectilinear grids in the Cartesian coordinate system with the following properties:", "The gridlines are parallel to the axes of the Cartesian coordinate system.There are N+2 vertical and N+2 horizontal gridlines. Hence there are (N+1) x (N+1) rectangular cells.The equations of the two outer vertical gridlines are x = -1 and x = 1.The equations of the two outer horizontal gridlines are y = -1 and y = 1.The grid cells are colored red if they overlap with the unit circle, black otherwise.", "For this problem we would like you to find the postions of the remaining N inner horizontal and N inner vertical gridlines so that the area occupied by the red cells is minimized.", "E.g. here is a picture of the solution for N = 10:", "The area occupied by the red cells for N = 10 rounded to 10 digits behind the decimal point is 3.3469640797.", "Find the positions for N = 400. Give as your answer the area occupied by the red cells rounded to 10 digits behind the decimal point." ], "solution": "3.1486734435", "name": "PE392" }, { "source": "Project Euler", "directions": [ "An n×n grid of squares contains n2 ants, one ant per square.All ants decide to move simultaneously to an adjacent square (usually 4 possibilities, except for ants on the edge of the grid or at the corners).We define f(n) to be the number of ways this can happen without any ants ending on the same square and without any two ants crossing the same edge between two squares.", "You are given that f(4) = 88.Find f(10)." ], "solution": "112398351350823112", "name": "PE393" }, { "source": "Project Euler", "directions": [ "Jeff eats a pie in an unusual way.The pie is circular. He starts with slicing an initial cut in the pie along a radius.While there is at least a given fraction F of pie left, he performs the following procedure:- He makes two slices from the pie centre to any point of what is remaining of the pie border, any point on the remaining pie border equally likely. This will divide the remaining pie into three pieces. - Going counterclockwise from the initial cut, he takes the first two pie pieces and eats them.When less than a fraction F of pie remains, he does not repeat this procedure. Instead, he eats all of the remaining pie.", "For x ≥ 1, let E(x) be the expected number of times Jeff repeats the procedure above with F = 1/x.It can be verified that E(1) = 1, E(2) ≈ 1.2676536759, and E(7.5) ≈ 2.1215732071.", "Find E(40) rounded to 10 decimal places behind the decimal point." ], "solution": "3.2370342194", "name": "PE394" }, { "source": "Project Euler", "directions": [ "The Pythagorean tree is a fractal generated by the following procedure:", "Start with a unit square. Then, calling one of the sides its base (in the animation, the bottom side is the base):", " Attach a right triangle to the side opposite the base, with the hypotenuse coinciding with that side and with the sides in a 3-4-5 ratio. Note that the smaller side of the triangle must be on the 'right' side with respect to the base (see animation). Attach a square to each leg of the right triangle, with one of its sides coinciding with that leg. Repeat this procedure for both squares, considering as their bases the sides touching the triangle.", "The resulting figure, after an infinite number of iterations, is the Pythagorean tree.", "It can be shown that there exists at least one rectangle, whose sides are parallel to the largest square of the Pythagorean tree, which encloses the Pythagorean tree completely.", "Find the smallest area possible for such a bounding rectangle, and give your answer rounded to 10 decimal places." ], "solution": "28.2453753155", "name": "PE395" }, { "source": "Project Euler", "directions": [ "For any positive integer n, the nth weak Goodstein sequence {g1, g2, g3, ...} is defined as:", " g1 = n for k > 1, gk is obtained by writing gk-1 in base k, interpreting it as a base k + 1 number, and subtracting 1.", "The sequence terminates when g", "k", " becomes 0.", "For example, the 6th weak Goodstein sequence is {6, 11, 17, 25, ...}:", " g1 = 6. g2 = 11 since 6 = 1102, 1103 = 12, and 12 - 1 = 11. g3 = 17 since 11 = 1023, 1024 = 18, and 18 - 1 = 17. g4 = 25 since 17 = 1014, 1015 = 26, and 26 - 1 = 25.", "and so on.", "It can be shown that every weak Goodstein sequence terminates.", "Let G(n) be the number of nonzero elements in the nth weak Goodstein sequence.It can be verified that G(2) = 3, G(4) = 21 and G(6) = 381.It can also be verified that ΣG(n) = 2517 for 1 ≤ n < 8.", "Find the last 9 digits of ΣG(n) for 1 ≤ n < 16." ], "solution": "173214653", "name": "PE396" }, { "source": "Project Euler", "directions": [ "On the parabola y = x2/k, three points A(a, a2/k), B(b, b2/k) and C(c, c2/k) are chosen.", "Let F(K, X) be the number of the integer quadruplets (k, a, b, c) such that at least one angle of the triangle ABC is 45-degree, with 1 ≤ k ≤ K and -X ≤ a < b < c ≤ X.", "For example, F(1, 10) = 41 and F(10, 100) = 12492.Find F(106, 109)." ], "solution": "141630459461893728", "name": "PE397" }, { "source": "Project Euler", "directions": [ "Inside a rope of length n, n-1 points are placed with distance 1 from each other and from the endpoints. Among these points, we choose m-1 points at random and cut the rope at these points to create m segments.", "Let E(n, m) be the expected length of the second-shortest segment.For example, E(3, 2) = 2 and E(8, 3) = 16/7.Note that if multiple segments have the same shortest length the length of the second-shortest segment is defined as the same as the shortest length.", "Find E(107, 100).Give your answer rounded to 5 decimal places behind the decimal point." ], "solution": "2010.59096", "name": "PE398" }, { "source": "Project Euler", "directions": [ "The first 15 fibonacci numbers are:1,1,2,3,5,8,13,21,34,55,89,144,233,377,610.It can be seen that 8 and 144 are not squarefree: 8 is divisible by 4 and 144 is divisible by 4 and by 9. So the first 13 squarefree fibonacci numbers are:1,1,2,3,5,13,21,34,55,89,233,377 and 610.", "The 200th squarefree fibonacci number is:971183874599339129547649988289594072811608739584170445.The last sixteen digits of this number are: 1608739584170445 and in scientific notation this number can be written as 9.7e53.", "Find the 100 000 000th squarefree fibonacci number.Give as your answer its last sixteen digits followed by a comma followed by the number in scientific notation (rounded to one digit after the decimal point).For the 200th squarefree number the answer would have been: 1608739584170445,9.7e53", "Note: For this problem, assume that for every prime p, the first fibonacci number divisible by p is not divisible by p2 (this is part of Wall's conjecture). This has been verified for primes ≤ 3·1015, but has not been proven in general.If it happens that the conjecture is false, then the accepted answer to this problem isn't guaranteed to be the 100 000 000th squarefree fibonacci number, rather it represents only a lower bound for that number." ], "solution": "1508395636674243,6.5e27330467", "name": "PE399" }, { "source": "Project Euler", "directions": [ "A Fibonacci tree is a binary tree recursively defined as:", "T(0) is the empty tree.T(1) is the binary tree with only one node.T(k) consists of a root node that has T(k-1) and T(k-2) as children.", "On such a tree two players play a take-away game. On each turn a player selects a node and removes that node along with the subtree rooted at that node.The player who is forced to take the root node of the entire tree loses.", "Here are the winning moves of the first player on the first turn for T(k) from k=1 to k=6.", "Let ", "f", "(", "k", ") be the number of winning moves of the first player (i.e. the moves for which the second player has no winning strategy) on the first turn of the game when this game is played on T(", "k", ").", "For example, f(5) = 1 and f(10) = 17.", "Find f(10000). Give the last 18 digits of your answer." ], "solution": "438505383468410633", "name": "PE400" }, { "source": "Project Euler", "directions": [ "The divisors of 6 are 1,2,3 and 6.The sum of the squares of these numbers is 1+4+9+36=50.", "Let sigma2(n) represent the sum of the squares of the divisors of n.Thus sigma2(6)=50.", "Let SIGMA2 represent the summatory function of sigma2, that is SIGMA2(n)=∑sigma2(i) for i=1 to n.", "The first 6 values of SIGMA2 are: 1,6,16,37,63 and 113.", "Find SIGMA2(1015) modulo 109. " ], "solution": "281632621", "name": "PE401" }, { "source": "Project Euler", "directions": [ "It can be shown that the polynomial n4 + 4n3 + 2n2 + 5n is a multiple of 6 for every integer n. It can also be shown that 6 is the largest integer satisfying this property.", "Define M(a, b, c) as the maximum m such that n4 + an3 + bn2 + cn is a multiple of m for all integers n. For example, M(4, 2, 5) = 6.", "Also, define S(N) as the sum of M(a, b, c) for all 0 < a, b, c ≤ N.", "We can verify that S(10) = 1972 and S(10000) = 2024258331114.", "Let Fk be the Fibonacci sequence:F0 = 0, F1 = 1 andFk = Fk-1 + Fk-2 for k ≥ 2.", "Find the last 9 digits of Σ S(Fk) for 2 ≤ k ≤ 1234567890123." ], "solution": "356019862", "name": "PE402" }, { "source": "Project Euler", "directions": [ "For integers a and b, we define D(a, b) as the domain enclosed by the parabola y = x2 and the line y = a·x + b:D(a, b) = { (x, y) | x2 ≤ y ≤ a·x + b }.", "L(a, b) is defined as the number of lattice points contained in D(a, b).For example, L(1, 2) = 8 and L(2, -1) = 1.", "We also define S(N) as the sum of L(a, b) for all the pairs (a, b) such that the area of D(a, b) is a rational number and |a|,|b| ≤ N.We can verify that S(5) = 344 and S(100) = 26709528.", "Find S(1012). Give your answer mod 108." ], "solution": "18224771", "name": "PE403" }, { "source": "Project Euler", "directions": [ "Ea is an ellipse with an equation of the form x2 + 4y2 = 4a2.Ea' is the rotated image of Ea by θ degrees counterclockwise around the origin O(0, 0) for 0° < θ < 90°.", "b is the distance to the origin of the two intersection points closest to the origin and c is the distance of the two other intersection points.We call an ordered triplet (a, b, c) a canonical ellipsoidal triplet if a, b and c are positive integers.For example, (209, 247, 286) is a canonical ellipsoidal triplet.", "Let C(N) be the number of distinct canonical ellipsoidal triplets (a, b, c) for a ≤ N.It can be verified that C(103) = 7, C(104) = 106 and C(106) = 11845.", "Find C(1017)." ], "solution": "1199215615081353", "name": "PE404" }, { "source": "Project Euler", "directions": [ "We wish to tile a rectangle whose length is twice its width.Let T(0) be the tiling consisting of a single rectangle.For n > 0, let T(n) be obtained from T(n-1) by replacing all tiles in the following manner:", "The following animation demonstrates the tilings T(n) for n from 0 to 5:", "Let f(n) be the number of points where four tiles meet in T(n).For example, f(1) = 0, f(4) = 82 and f(109) mod 177 = 126897180.", "Find f(10k) for k = 1018, give your answer modulo 177." ], "solution": "237696125", "name": "PE405" }, { "source": "Project Euler", "directions": [ "We are trying to find a hidden number selected from the set of integers {1, 2, ..., n} by asking questions. Each number (question) we ask, we get one of three possible answers:", " \"Your guess is lower than the hidden number\" (and you incur a cost of a), or \"Your guess is higher than the hidden number\" (and you incur a cost of b), or \"Yes, that's it!\" (and the game ends).", "Given the value of n, a, and b, an optimal strategy minimizes the total cost for the worst possible case.", "For example, if n = 5, a = 2, and b = 3, then we may begin by asking \"2\" as our first question.", "If we are told that 2 is higher than the hidden number (for a cost of b=3), then we are sure that \"1\" is the hidden number (for a total cost of 3).If we are told that 2 is lower than the hidden number (for a cost of a=2), then our next question will be \"4\".If we are told that 4 is higher than the hidden number (for a cost of b=3), then we are sure that \"3\" is the hidden number (for a total cost of 2+3=5).If we are told that 4 is lower than the hidden number (for a cost of a=2), then we are sure that \"5\" is the hidden number (for a total cost of 2+2=4).Thus, the worst-case cost achieved by this strategy is 5. It can also be shown that this is the lowest worst-case cost that can be achieved. So, in fact, we have just described an optimal strategy for the given values of n, a, and b.", "Let C(n, a, b) be the worst-case cost achieved by an optimal strategy for the given values of n, a, and b.", "Here are a few examples:C(5, 2, 3) = 5C(500, √2, √3) = 13.22073197...C(20000, 5, 7) = 82C(2000000, √5, √7) = 49.63755955...", "Let Fk be the Fibonacci numbers: Fk = Fk-1 + Fk-2 with base cases F1 = F2 = 1.Find ∑1≤k≤30 C(1012, √k, √Fk), and give your answer rounded to 8 decimal places behind the decimal point." ], "solution": "36813.12757207", "name": "PE406" }, { "source": "Project Euler", "directions": [ "If we calculate a2 mod 6 for 0 ≤ a ≤ 5 we get: 0,1,4,3,4,1.", "The largest value of a such that a2 ≡ a mod 6 is 4.Let's call M(n) the largest value of a < n such that a2 ≡ a (mod n).So M(6) = 4.", "Find ∑M(n) for 1 ≤ n ≤ 107." ], "solution": "39782849136421", "name": "PE407" }, { "source": "Project Euler", "directions": [ "Let's call a lattice point (x, y) inadmissible if x, y and x + y are all positive perfect squares.For example, (9, 16) is inadmissible, while (0, 4), (3, 1) and (9, 4) are not.", "Consider a path from point (x1, y1) to point (x2, y2) using only unit steps north or east.Let's call such a path admissible if none of its intermediate points are inadmissible.", "Let P(n) be the number of admissible paths from (0, 0) to (n, n).It can be verified that P(5) = 252, P(16) = 596994440 and P(1000) mod 1 000 000 007 = 341920854.", "Find P(10 000 000) mod 1 000 000 007." ], "solution": "299742733", "name": "PE408" }, { "source": "Project Euler", "directions": [ "Let n be a positive integer. Consider nim positions where:", "There are n non-empty piles.Each pile has size less than 2n.No two piles have the same size.", "Let W(n) be the number of winning nim positions satisfying the aboveconditions (a position is winning if the first player has a winning strategy). For example, W(1) = 1, W(2) = 6, W(3) = 168, W(5) = 19764360 and W(100) mod 1 000 000 007 = 384777056.", "Find W(10 000 000) mod 1 000 000 007." ], "solution": "253223948", "name": "PE409" }, { "source": "Project Euler", "directions": [ "Let C be the circle with radius r, x2 + y2 = r2. We choose two points P(a, b) and Q(-a, c) so that the line passing through P and Q is tangent to C.", "For example, the quadruplet (r, a, b, c) = (2, 6, 2, -7) satisfies this property.", "Let F(R, X) be the number of the integer quadruplets (r, a, b, c) with this property, and with 0 < r ≤ R and 0 < a ≤ X.", "We can verify that F(1, 5) = 10, F(2, 10) = 52 and F(10, 100) = 3384.Find F(108, 109) + F(109, 108)." ], "solution": "799999783589946560", "name": "PE410" }, { "source": "Project Euler", "directions": [ "Let n be a positive integer. Suppose there are stations at the coordinates (x, y) = (2i mod n, 3i mod n) for 0 ≤ i ≤ 2n. We will consider stations with the same coordinates as the same station.", "We wish to form a path from (0, 0) to (n, n) such that the x and y coordinates never decrease.Let S(n) be the maximum number of stations such a path can pass through.", "For example, if n = 22, there are 11 distinct stations, and a valid path can pass through at most 5 stations. Therefore, S(22) = 5.The case is illustrated below, with an example of an optimal path:", "It can also be verified that S(123) = 14 and S(10000) = 48.", "Find ∑ S(k5) for 1 ≤ k ≤ 30." ], "solution": "9936352", "name": "PE411" }, { "source": "Project Euler", "directions": [ "For integers m, n (0 ≤ n m), let L(m, n) be an m×m grid with the top-right n×n grid removed.", "For example, L(5, 3) looks like this:", "We want to number each cell of L(m, n) with consecutive integers 1, 2, 3, ... such that the number in every cell is smaller than the number below it and to the left of it.", "For example, here are two valid numberings of L(5, 3):", "Let LC(m, n) be the number of valid numberings of L(m, n).It can be verified that LC(3, 0) = 42, LC(5, 3) = 250250, LC(6, 3) = 406029023400 and LC(10, 5) mod 76543217 = 61251715.", "Find LC(10000, 5000) mod 76543217." ], "solution": "38788800", "name": "PE412" }, { "source": "Project Euler", "directions": [ "We say that a d-digit positive number (no leading zeros) is a one-child number if exactly one of its sub-strings is divisible by d.", "For example, 5671 is a 4-digit one-child number. Among all its sub-strings 5, 6, 7, 1, 56, 67, 71, 567, 671 and 5671, only 56 is divisible by 4.Similarly, 104 is a 3-digit one-child number because only 0 is divisible by 3.1132451 is a 7-digit one-child number because only 245 is divisible by 7.", "Let F(N) be the number of the one-child numbers less than N.We can verify that F(10) = 9, F(103) = 389 and F(107) = 277674.", "Find F(1019)." ], "solution": "3079418648040719", "name": "PE413" }, { "source": "Project Euler", "directions": [ "6174 is a remarkable number; if we sort its digits in increasing order and subtract that number from the number you get when you sort the digits in decreasing order, we get 7641-1467=6174.Even more remarkable is that if we start from any 4 digit number and repeat this process of sorting and subtracting, we'll eventually end up with 6174 or immediately with 0 if all digits are equal. This also works with numbers that have less than 4 digits if we pad the number with leading zeroes until we have 4 digits.E.g. let's start with the number 0837:8730-0378=83528532-2358=6174", "6174 is called the Kaprekar constant. The process of sorting and subtracting and repeating this until either 0 or the Kaprekar constant is reached is called the Kaprekar routine.", "We can consider the Kaprekar routine for other bases and number of digits. Unfortunately, it is not guaranteed a Kaprekar constant exists in all cases; either the routine can end up in a cycle for some input numbers or the constant the routine arrives at can be different for different input numbers.However, it can be shown that for 5 digits and a base b = 6t+3≠9, a Kaprekar constant exists.E.g. base 15: (10,4,14,9,5)15base 21: (14,6,20,13,7)21", "Define Cb to be the Kaprekar constant in base b for 5 digits.Define the function sb(i) to be", " 0 if i = Cb or if i written in base b consists of 5 identical digits the number of iterations it takes the Kaprekar routine in base b to arrive at Cb, otherwise", "Note that we can define ", "sb(i)", " for all integers ", "i", " < ", "b", "5", ". If ", "i", " written in base ", "b", " takes less than 5 digits, the number is padded with leading zero digits until we have 5 digits before applying the Kaprekar routine.", "Define S(b) as the sum of sb(i) for 0 < i < b5.E.g. S(15) = 5274369 S(111) = 400668930299", "Find the sum of S(6k+3) for 2 ≤ k ≤ 300.Give the last 18 digits as your answer." ], "solution": "552506775824935461", "name": "PE414" }, { "source": "Project Euler", "directions": [ "A set of lattice points S is called a titanic set if there exists a line passing through exactly two points in S.", "An example of a titanic set is S = {(0, 0), (0, 1), (0, 2), (1, 1), (2, 0), (1, 0)}, where the line passing through (0, 1) and (2, 0) does not pass through any other point in S.", "On the other hand, the set {(0, 0), (1, 1), (2, 2), (4, 4)} is not a titanic set since the line passing through any two points in the set also passes through the other two.", "For any positive integer N, let T(N) be the number of titanic sets S whose every point (x, y) satisfies 0 ≤ x, y ≤ N.It can be verified that T(1) = 11, T(2) = 494, T(4) = 33554178, T(111) mod 108 = 13500401 and T(105) mod 108 = 63259062.", "Find T(1011) mod 108." ], "solution": "55859742", "name": "PE415" }, { "source": "Project Euler", "directions": [ "A row of n squares contains a frog in the leftmost square. By successive jumps the frog goes to the rightmost square and then back to the leftmost square. On the outward trip he jumps one, two or three squares to the right, and on the homeward trip he jumps to the left in a similar manner. He cannot jump outside the squares. He repeats the round-trip travel m times.", "Let F(m, n) be the number of the ways the frog can travel so that at most one square remains unvisited.For example, F(1, 3) = 4, F(1, 4) = 15, F(1, 5) = 46, F(2, 3) = 16 and F(2, 100) mod 109 = 429619151.", "Find the last 9 digits of F(10, 1012)." ], "solution": "898082747", "name": "PE416" }, { "source": "Project Euler", "directions": [ "A unit fraction contains 1 in the numerator. The decimal representation of the unit fractions with denominators 2 to 10 are given:", "1/2= 0.51/3= 0.(3)1/4= 0.251/5= 0.21/6= 0.1(6)1/7= 0.(142857)1/8= 0.1251/9= 0.(1)1/10= 0.1", "Where 0.1(6) means 0.166666..., and has a 1-digit recurring cycle. It can be seen that 1/7 has a 6-digit recurring cycle.", "Unit fractions whose denominator has no other prime factors than 2 and/or 5 are not considered to have a recurring cycle.We define the length of the recurring cycle of those unit fractions as 0. ", "Let L(n) denote the length of the recurring cycle of 1/n.You are given that ∑L(n) for 3 ≤ n ≤ 1 000 000 equals 55535191115.", "Find ∑L(n) for 3 ≤ n ≤ 100 000 000" ], "solution": "446572970925740", "name": "PE417" }, { "source": "Project Euler", "directions": [ "Let n be a positive integer. An integer triple (a, b, c) is called a factorisation triple of n if:", " 1 ≤ a ≤ b ≤ c a·b·c = n.", "Define f(n) to be a + b + c for the factorisation triple (a, b, c) of n which minimises c / a. One can show that this triple is unique.", "For example, f(165) = 19, f(100100) = 142 and f(20!) = 4034872.", "Find f(43!)." ], "solution": "1177163565297340320", "name": "PE418" }, { "source": "Project Euler", "directions": [ "The look and say sequence goes 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...The sequence starts with 1 and all other members are obtained by describing the previous member in terms of consecutive digits.It helps to do this out loud:1 is 'one one' → 1111 is 'two ones' → 2121 is 'one two and one one' → 1211 1211 is 'one one, one two and two ones' → 111221111221 is 'three ones, two twos and one one' → 312211...", "Define A(n), B(n) and C(n) as the number of ones, twos and threes in the n'th element of the sequence respectively.One can verify that A(40) = 31254, B(40) = 20259 and C(40) = 11625.", "Find A(n), B(n) and C(n) for n = 1012. Give your answer modulo 230 and separate your values for A, B and C by a comma. E.g. for n = 40 the answer would be 31254,20259,11625" ], "solution": "998567458,1046245404,43363922", "name": "PE419" }, { "source": "Project Euler", "directions": [ "A positive integer matrix is a matrix whose elements are all positive integers.Some positive integer matrices can be expressed as a square of a positive integer matrix in two different ways. Here is an example:", "We define F(N) as the number of the 2x2 positive integer matrices which have a trace less than N and which can be expressed as a square of a positive integer matrix in two different ways.We can verify that F(50) = 7 and F(1000) = 1019.", "Find F(107)." ], "solution": "145159332", "name": "PE420" }, { "source": "Project Euler", "directions": [ "Numbers of the form n15+1 are composite for every integer n > 1.For positive integers n and m let s(n,m) be defined as the sum of the distinct prime factors of n15+1 not exceeding m.", "E.g. 2", "15", "+1 = 3×3×11×331.", "So ", "s", "(2,10) = 3 and ", "s", "(2,1000) = 3+11+331 = 345.", "Also 10", "15", "+1 = 7×11×13×211×241×2161×9091.", "So ", "s", "(10,100) = 31 and ", "s", "(10,1000) = 483.", "Find ∑ s(n,108) for 1 ≤ n ≤ 1011." ], "solution": "2304215802083466198", "name": "PE421" }, { "source": "Project Euler", "directions": [ "Let H be the hyperbola defined by the equation 12x2 + 7xy - 12y2 = 625.", "Next, define X as the point (7, 1). It can be seen that X is in H.", "Now we define a sequence of points in H, {Pi : i ≥ 1}, as:", " P1 = (13, 61/4). P2 = (-43/6, -4). For i > 2, Pi is the unique point in H that is different from Pi-1 and such that line PiPi-1 is parallel to line Pi-2X. It can be shown that Pi is well-defined, and that its coordinates are always rational.", "You are given that P3 = (-19/2, -229/24), P4 = (1267/144, -37/12) and P7 = (17194218091/143327232, 274748766781/1719926784).", "Find Pn for n = 1114 in the following format:If Pn = (a/b, c/d) where the fractions are in lowest terms and the denominators are positive, then the answer is (a + b + c + d) mod 1 000 000 007.", "For n = 7, the answer would have been: 806236837." ], "solution": "92060460", "name": "PE422" }, { "source": "Project Euler", "directions": [ "Let n be a positive integer.A 6-sided die is thrown n times. Let c be the number of pairs of consecutive throws that give the same value.", "For example, if n = 7 and the values of the die throws are (1,1,5,6,6,6,3), then the following pairs of consecutive throws give the same value:(1,1,5,6,6,6,3)(1,1,5,6,6,6,3)(1,1,5,6,6,6,3)Therefore, c = 3 for (1,1,5,6,6,6,3).", "Define C(n) as the number of outcomes of throwing a 6-sided die n times such that c does not exceed π(n).1For example, C(3) = 216, C(4) = 1290, C(11) = 361912500 and C(24) = 4727547363281250000.", "Define S(L) as ∑ C(n) for 1 ≤ n ≤ L.For example, S(50) mod 1 000 000 007 = 832833871.", "Find S(50 000 000) mod 1 000 000 007.", "1 π denotes the prime-counting function, i.e. π(n) is the number of primes ≤ n." ], "solution": "653972374", "name": "PE423" }, { "source": "Project Euler", "directions": [ "The above is an example of a cryptic kakuro (also known as cross sums, or even sums cross) puzzle, with its final solution on the right. (The common rules of kakuro puzzles can be found easily on numerous internet sites. Other related information can also be currently found at krazydad.com whose author has provided the puzzle data for this challenge.)", "The downloadable text file (kakuro200.txt) contains the description of 200 such puzzles, a mix of 5x5 and 6x6 types. The first puzzle in the file is the above example which is coded as follows:", "6,X,X,(vCC),(vI),X,X,X,(hH),B,O,(vCA),(vJE),X,(hFE,vD),O,O,O,O,(hA),O,I,(hJC,vB),O,O,(hJC),H,O,O,O,X,X,X,(hJE),O,O,X", "The first character is a numerical digit indicating the size of the information grid. It would be either a 6 (for a 5x5 kakuro puzzle) or a 7 (for a 6x6 puzzle) followed by a comma (,). The extra top line and left column are needed to insert information.", "The content of each cell is then described and followed by a comma, going left to right and starting with the top line.X = Gray cell, not required to be filled by a digit.O (upper case letter)= White empty cell to be filled by a digit.A = Or any one of the upper case letters from A to J to be replaced by its equivalent digit in the solved puzzle.( ) = Location of the encrypted sums. Horizontal sums are preceded by a lower case \"h\" and vertical sums are preceded by a lower case \"v\". Those are followed by one or two upper case letters depending if the sum is a single digit or double digit one. For double digit sums, the first letter would be for the \"tens\" and the second one for the \"units\". When the cell must contain information for both a horizontal and a vertical sum, the first one is always for the horizontal sum and the two are separated by a comma within the same set of brackets, ex.: (hFE,vD). Each set of brackets is also immediately followed by a comma.", "The description of the last cell is followed by a Carriage Return/Line Feed (CRLF) instead of a comma.", "The required answer to each puzzle is based on the value of each letter necessary to arrive at the solution and according to the alphabetical order. As indicated under the example puzzle, its answer would be 8426039571. At least 9 out of the 10 encrypting letters are always part of the problem description. When only 9 are given, the missing one must be assigned the remaining digit.", "You are given that the sum of the answers for the first 10 puzzles in the file is 64414157580.", "Find the sum of the answers for the 200 puzzles." ], "solution": "1059760019628", "name": "PE424" }, { "source": "Project Euler", "directions": [ "Two positive numbers A and B are said to be connected (denoted by \"A ↔ B\") if one of these conditions holds:(1) A and B have the same length and differ in exactly one digit; for example, 123 ↔ 173.(2) Adding one digit to the left of A (or B) makes B (or A); for example, 23 ↔ 223 and 123 ↔ 23.", "We call a prime P a 2's relative if there exists a chain of connected primes between 2 and P and no prime in the chain exceeds P.", "For example, 127 is a 2's relative. One of the possible chains is shown below:2 ↔ 3 ↔ 13 ↔ 113 ↔ 103 ↔ 107 ↔ 127However, 11 and 103 are not 2's relatives.", "Let F(N) be the sum of the primes ≤ N which are not 2's relatives.We can verify that F(103) = 431 and F(104) = 78728.", "Find F(107)." ], "solution": "46479497324", "name": "PE425" }, { "source": "Project Euler", "directions": [ "Consider an infinite row of boxes. Some of the boxes contain a ball. For example, an initial configuration of 2 consecutive occupied boxes followed by 2 empty boxes, 2 occupied boxes, 1 empty box, and 2 occupied boxes can be denoted by the sequence (2, 2, 2, 1, 2), in which the number of consecutive occupied and empty boxes appear alternately.", "A turn consists of moving each ball exactly once according to the following rule: Transfer the leftmost ball which has not been moved to the nearest empty box to its right.", "After one turn the sequence (2, 2, 2, 1, 2) becomes (2, 2, 1, 2, 3) as can be seen below; note that we begin the new sequence starting at the first occupied box.", "A system like this is called a Box-Ball System or BBS for short.", "It can be shown that after a sufficient number of turns, the system evolves to a state where the consecutive numbers of occupied boxes is invariant. In the example below, the consecutive numbers of occupied boxes evolves to [1, 2, 3]; we shall call this the final state.", "We define the sequence {ti}:", "s0 = 290797sk+1 = sk2 mod 50515093tk = (sk mod 64) + 1", "Starting from the initial configuration (t0, t1, …, t10), the final state becomes [1, 3, 10, 24, 51, 75].Starting from the initial configuration (t0, t1, …, t10 000 000), find the final state.Give as your answer the sum of the squares of the elements of the final state. For example, if the final state is [1, 2, 3] then 14 ( = 12 + 22 + 32) is your answer." ], "solution": "31591886008", "name": "PE426" }, { "source": "Project Euler", "directions": [ "A sequence of integers S = {si} is called an n-sequence if it has n elements and each element si satisfies 1 ≤ si ≤ n. Thus there are nn distinct n-sequences in total.For example, the sequence S = {1, 5, 5, 10, 7, 7, 7, 2, 3, 7} is a 10-sequence.", "For any sequence S, let L(S) be the length of the longest contiguous subsequence of S with the same value.For example, for the given sequence S above, L(S) = 3, because of the three consecutive 7's.", "Let f(n) = ∑ L(S) for all n-sequences S.", "For example, f(3) = 45, f(7) = 1403689 and f(11) = 481496895121.", "Find f(7 500 000) mod 1 000 000 009." ], "solution": "97138867", "name": "PE427" }, { "source": "Project Euler", "directions": [ "Let a, b and c be positive numbers.Let W, X, Y, Z be four collinear points where |WX| = a, |XY| = b, |YZ| = c and |WZ| = a + b + c.Let Cin be the circle having the diameter XY.Let Cout be the circle having the diameter WZ.", "The triplet (a, b, c) is called a necklace triplet if you can place k ≥ 3 distinct circles C1, C2, ..., Ck such that:", "Ci has no common interior points with any Cj for 1 ≤ i, j ≤ k and i ≠ j,Ci is tangent to both Cin and Cout for 1 ≤ i ≤ k,Ci is tangent to Ci+1 for 1 ≤ i k, andCk is tangent to C1.", "For example, (5, 5, 5) and (4, 3, 21) are necklace triplets, while it can be shown that (2, 2, 5) is not.", "Let T(n) be the number of necklace triplets (a, b, c) such that a, b and c are positive integers, and b ≤ n.For example, T(1) = 9, T(20) = 732 and T(3000) = 438106.", "Find T(1 000 000 000)." ], "solution": "747215561862", "name": "PE428" }, { "source": "Project Euler", "directions": [ "A unitary divisor d of a number n is a divisor of n that has the property gcd(d, n/d) = 1.The unitary divisors of 4! = 24 are 1, 3, 8 and 24.The sum of their squares is 12 + 32 + 82 + 242 = 650.", "Let S(n) represent the sum of the squares of the unitary divisors of n. Thus S(4!)=650.", "Find S(100 000 000!) modulo 1 000 000 009." ], "solution": "98792821", "name": "PE429" }, { "source": "Project Euler", "directions": [ "N disks are placed in a row, indexed 1 to N from left to right.Each disk has a black side and white side. Initially all disks show their white side.", "At each turn, two, not necessarily distinct, integers A and B between 1 and N (inclusive) are chosen uniformly at random.All disks with an index from A to B (inclusive) are flipped.", "The following example shows the case N = 8. At the first turn A = 5 and B = 2, and at the second turn A = 4 and B = 6.", "Let E(N, M) be the expected number of disks that show their white side after M turns.We can verify that E(3, 1) = 10/9, E(3, 2) = 5/3, E(10, 4) ≈ 5.157 and E(100, 10) ≈ 51.893.", "Find E(1010, 4000).Give your answer rounded to 2 decimal places behind the decimal point." ], "solution": "5000624921.38", "name": "PE430" }, { "source": "Project Euler", "directions": [ "Fred the farmer arranges to have a new storage silo installed on his farm and having an obsession for all things square he is absolutely devastated when he discovers that it is circular. Quentin, the representative from the company that installed the silo, explains that they only manufacture cylindrical silos, but he points out that it is resting on a square base. Fred is not amused and insists that it is removed from his property.", "Quick thinking Quentin explains that when granular materials are delivered from above a conical slope is formed and the natural angle made with the horizontal is called the angle of repose. For example if the angle of repose, $\\alpha = 30$ degrees, and grain is delivered at the centre of the silo then a perfect cone will form towards the top of the cylinder. In the case of this silo, which has a diameter of 6m, the amount of space wasted would be approximately 32.648388556 m3. However, if grain is delivered at a point on the top which has a horizontal distance of $x$ metres from the centre then a cone with a strangely curved and sloping base is formed. He shows Fred a picture.", " ", "We shall let the amount of space wasted in cubic metres be given by $V(x)$. If $x = 1.114785284$, which happens to have three squared decimal places, then the amount of space wasted, $V(1.114785284) \\approx 36$. Given the range of possible solutions to this problem there is exactly one other option: $V(2.511167869) \\approx 49$. It would be like knowing that the square is king of the silo, sitting in splendid glory on top of your grain.", "Fred's eyes light up with delight at this elegant resolution, but on closer inspection of Quentin's drawings and calculations his happiness turns to despondency once more. Fred points out to Quentin that it's the radius of the silo that is 6 metres, not the diameter, and the angle of repose for his grain is 40 degrees. However, if Quentin can find a set of solutions for this particular silo then he will be more than happy to keep it.", "If Quick thinking Quentin is to satisfy frustratingly fussy Fred the farmer's appetite for all things square then determine the values of $x$ for all possible square space wastage options and calculate $\\sum x$ correct to 9 decimal places." ], "solution": "23.386029052", "name": "PE431" }, { "source": "Project Euler", "directions": [ "Let S(n,m) = ∑φ(n × i) for 1 ≤ i ≤ m. (φ is Euler's totient function)You are given that S(510510,106 )= 45480596821125120. ", "Find S(510510,1011).Give the last 9 digits of your answer." ], "solution": "754862080", "name": "PE432" }, { "source": "Project Euler", "directions": [ "Let E(x0, y0) be the number of steps it takes to determine the greatest common divisor of x0 and y0 with Euclid's algorithm. More formally:x1 = y0, y1 = x0 mod y0xn = yn-1, yn = xn-1 mod yn-1E(x0, y0) is the smallest n such that yn = 0.", "We have E(1,1) = 1, E(10,6) = 3 and E(6,10) = 4.", "Define S(N) as the sum of E(x,y) for 1 ≤ x,y ≤ N.We have S(1) = 1, S(10) = 221 and S(100) = 39826.", "Find S(5·106)." ], "solution": "326624372659664", "name": "PE433" }, { "source": "Project Euler", "directions": [ "Recall that a graph is a collection of vertices and edges connecting the vertices, and that two vertices connected by an edge are called adjacent.Graphs can be embedded in Euclidean space by associating each vertex with a point in the Euclidean space.A flexible graph is an embedding of a graph where it is possible to move one or more vertices continuously so that the distance between at least two nonadjacent vertices is altered while the distances between each pair of adjacent vertices is kept constant.A rigid graph is an embedding of a graph which is not flexible.Informally, a graph is rigid if by replacing the vertices with fully rotating hinges and the edges with rods that are unbending and inelastic, no parts of the graph can be moved independently from the rest of the graph.", "The grid graphs embedded in the Euclidean plane are not rigid, as the following animation demonstrates:", "However, one can make them rigid by adding diagonal edges to the cells. For example, for the 2x3 grid graph, there are 19 ways to make the graph rigid:", "Note that for the purposes of this problem, we do not consider changing the orientation of a diagonal edge or adding both diagonal edges to a cell as a different way of making a grid graph rigid.", "Let R(m,n) be the number of ways to make the m × n grid graph rigid. E.g. R(2,3) = 19 and R(5,5) = 23679901", "Define S(N) as ∑R(i,j) for 1 ≤ i, j ≤ N.E.g. S(5) = 25021721.Find S(100), give your answer modulo 1000000033" ], "solution": "863253606", "name": "PE434" }, { "source": "Project Euler", "directions": [ "The Fibonacci numbers {fn, n ≥ 0} are defined recursively as fn = fn-1 + fn-2 with base cases f0 = 0 and f1 = 1.", "Define the polynomials {Fn, n ≥ 0} as Fn(x) = ∑fixi for 0 ≤ i ≤ n.", "For example, F7(x) = x + x2 + 2x3 + 3x4 + 5x5 + 8x6 + 13x7, and F7(11) = 268357683.", "Let n = 1015. Find the sum [∑0≤x≤100 Fn(x)] mod 1307674368000 (= 15!)." ], "solution": "252541322550", "name": "PE435" }, { "source": "Project Euler", "directions": [ "Julie proposes the following wager to her sister Louise.She suggests they play a game of chance to determine who will wash the dishes.For this game, they shall use a generator of independent random numbers uniformly distributed between 0 and 1.The game starts with S = 0.The first player, Louise, adds to S different random numbers from the generator until S > 1 and records her last random number 'x'.The second player, Julie, continues adding to S different random numbers from the generator until S > 2 and records her last random number 'y'.The player with the highest number wins and the loser washes the dishes, i.e. if y > x the second player wins.", "For example, if the first player draws 0.62 and 0.44, the first player turn ends since 0.62+0.44 > 1 and x = 0.44.If the second players draws 0.1, 0.27 and 0.91, the second player turn ends since 0.62+0.44+0.1+0.27+0.91 > 2 and y = 0.91.Since y > x, the second player wins.", "Louise thinks about it for a second, and objects: \"That's not fair\".What is the probability that the second player wins?Give your answer rounded to 10 places behind the decimal point in the form 0.abcdefghij" ], "solution": "0.5276662759", "name": "PE436" }, { "source": "Project Euler", "directions": [ "When we calculate 8n modulo 11 for n=0 to 9 we get: 1, 8, 9, 6, 4, 10, 3, 2, 5, 7.As we see all possible values from 1 to 10 occur. So 8 is a primitive root of 11.But there is more:If we take a closer look we see:1+8=98+9=17≡6 mod 119+6=15≡4 mod 116+4=104+10=14≡3 mod 1110+3=13≡2 mod 113+2=52+5=75+7=12≡1 mod 11.", "So the powers of 8 mod 11 are cyclic with period 10, and 8", "n", " + 8", "n+1", " ≡ 8", "n+2", " (mod 11).", "8 is called a ", "Fibonacci primitive root", " of 11.", "Not every prime has a Fibonacci primitive root.", "There are 323 primes less than 10000 with one or more Fibonacci primitive roots and the sum of these primes is 1480491.", "Find the sum of the primes less than 100,000,000 with at least one Fibonacci primitive root." ], "solution": "74204709657207", "name": "PE437" }, { "source": "Project Euler", "directions": [ "For an n-tuple of integers t = (a1, ..., an), let (x1, ..., xn) be the solutions of the polynomial equation xn + a1xn-1 + a2xn-2 + ... + an-1x + an = 0.", "Consider the following two conditions:", "x1, ..., xn are all real.If x1, ..., xn are sorted, ⌊xi⌋ = i for 1 ≤ i ≤ n. (⌊·⌋: floor function.)", "In the case of n = 4, there are 12 n-tuples of integers which satisfy both conditions.We define S(t) as the sum of the absolute values of the integers in t.For n = 4 we can verify that ∑S(t) = 2087 for all n-tuples t which satisfy both conditions.", "Find ∑S(t) for n = 7." ], "solution": "2046409616809", "name": "PE438" }, { "source": "Project Euler", "directions": [ "Let d(k) be the sum of all divisors of k.We define the function S(N) = ∑1≤i≤N ∑1≤j≤Nd(i·j).For example, S(3) = d(1) + d(2) + d(3) + d(2) + d(4) + d(6) + d(3) + d(6) + d(9) = 59.", "You are given that S(103) = 563576517282 and S(105) mod 109 = 215766508.Find S(1011) mod 109." ], "solution": "968697378", "name": "PE439" }, { "source": "Project Euler", "directions": [ "We want to tile a board of length n and height 1 completely, with either 1 × 2 blocks or 1 × 1 blocks with a single decimal digit on top:", "For example, here are some of the ways to tile a board of length n = 8:", "Let T(n) be the number of ways to tile a board of length n as described above.", "For example, T(1) = 10 and T(2) = 101.", "Let S(L) be the triple sum ∑a,b,c gcd(T(ca), T(cb)) for 1 ≤ a, b, c ≤ L.For example:S(2) = 10444S(3) = 1292115238446807016106539989S(4) mod 987 898 789 = 670616280.", "Find S(2000) mod 987 898 789." ], "solution": "970746056", "name": "PE440" }, { "source": "Project Euler", "directions": [ "For an integer M, we define R(M) as the sum of 1/(p·q) for all the integer pairs p and q which satisfy all of these conditions:", " 1 ≤ p < q ≤ M p + q ≥ M p and q are coprime.", "We also define S(N) as the sum of R(i) for 2 ≤ i ≤ N.We can verify that S(2) = R(2) = 1/2, S(10) ≈ 6.9147 and S(100) ≈ 58.2962.", "Find S(107). Give your answer rounded to four decimal places." ], "solution": "5000088.8395", "name": "PE441" }, { "source": "Project Euler", "directions": [ "An integer is called eleven-free if its decimal expansion does not contain any substring representing a power of 11 except 1.", "For example, 2404 and 13431 are eleven-free, while 911 and 4121331 are not.", "Let E(n) be the nth positive eleven-free integer. For example, E(3) = 3, E(200) = 213 and E(500 000) = 531563.", "Find E(1018)." ], "solution": "1295552661530920149", "name": "PE442" }, { "source": "Project Euler", "directions": [ "Let g(n) be a sequence defined as follows:g(4) = 13,g(n) = g(n-1) + gcd(n, g(n-1)) for n > 4.", "The first few values are:", " n4567891011121314151617181920... g(n)1314161718272829303132333451545560... ", "You are given that g(1 000) = 2524 and g(1 000 000) = 2624152.", "Find g(1015)." ], "solution": "2744233049300770", "name": "PE443" }, { "source": "Project Euler", "directions": [ "A group of p people decide to sit down at a round table and play a lottery-ticket trading game. Each person starts off with a randomly-assigned, unscratched lottery ticket. Each ticket, when scratched, reveals a whole-pound prize ranging anywhere from £1 to £p, with no two tickets alike. The goal of the game is for each person to maximize his ticket winnings upon leaving the game.", "An arbitrary person is chosen to be the first player. Going around the table, each player has only one of two options:", "1. The player can scratch his ticket and reveal its worth to everyone at the table.2. The player can trade his unscratched ticket for a previous player's scratched ticket, and then leave the game with that ticket. The previous player then scratches his newly-acquired ticket and reveals its worth to everyone at the table.", "The game ends once all tickets have been scratched. All players still remaining at the table must leave with their currently-held tickets.", "Assume that each player uses the optimal strategy for maximizing the expected value of his ticket winnings. ", "Let E(p) represent the expected number of players left at the table when the game ends in a game consisting of p players (e.g. E(111) = 5.2912 when rounded to 5 significant digits).", "Let S1(N) = E(p)Let Sk(N) = Sk-1(p) for k > 1", "Find S20(1014) and write the answer in scientific notation rounded to 10 significant digits. Use a lowercase e to separate mantissa and exponent (e.g. S3(100) = 5.983679014e5)." ], "solution": "1.200856722e263", "name": "PE444" }, { "source": "Project Euler", "directions": [ "For every integer n>1, the family of functions fn,a,b is defined by fn,a,b(x)≡ax+b mod n for a,b,x integer and 01, the family of functions fn,a,b is defined by fn,a,b(x)≡ax+b mod n for a,b,x integer and 01, the family of functions fn,a,b is defined by fn,a,b(x)≡ax+b mod n for a,b,x integer and 0 1).", "Here are the possible seating arrangements for N = 15:", "We see that if the first person chooses correctly, the 15 seats can seat up to 7 people.We can also see that the first person has 9 choices to maximize the number of people that may be seated.", "Let f(N) be the number of choices the first person has to maximize the number of occupants for N seats in a row. Thus, f(1) = 1, f(15) = 9, f(20) = 6, and f(500) = 16.", "Also, ∑f(N) = 83 for 1 ≤ N ≤ 20 and ∑f(N) = 13343 for 1 ≤ N ≤ 500.", "Find ∑f(N) for 1 ≤ N ≤ 1012. Give the last 8 digits of your answer." ], "solution": "73811586", "name": "PE472" }, { "source": "Project Euler", "directions": [ "Let $\\varphi$ be the golden ratio: $\\varphi=\\frac{1+\\sqrt{5}}{2}.$Remarkably it is possible to write every positive integer as a sum of powers of $\\varphi$ even if we require that every power of $\\varphi$ is used at most once in this sum.Even then this representation is not unique.We can make it unique by requiring that no powers with consecutive exponents are used and that the representation is finite.E.g: $2=\\varphi+\\varphi^{-2}$ and $3=\\varphi^{2}+\\varphi^{-2}$", "To represent this sum of powers of $\\varphi$ we use a string of 0's and 1's with a point to indicate where the negative exponents start.We call this the representation in the phigital numberbase.So $1=1_{\\varphi}$, $2=10.01_{\\varphi}$, $3=100.01_{\\varphi}$ and $14=100100.001001_{\\varphi}$. The strings representing 1, 2 and 14 in the phigital number base are palindromic, while the string representating 3 is not. (the phigital point is not the middle character).", "The sum of the positive integers not exceeding 1000 whose phigital representation is palindromic is 4345.", "Find the sum of the positive integers not exceeding $10^{10}$ whose phigital representation is palindromic." ], "solution": "35856681704365", "name": "PE473" }, { "source": "Project Euler", "directions": [ "For a positive integer n and digits d, we define F(n, d) as the number of the divisors of n whose last digits equal d.For example, F(84, 4) = 3. Among the divisors of 84 (1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84), three of them (4, 14, 84) have the last digit 4.", "We can also verify that F(12!, 12) = 11 and F(50!, 123) = 17888.", "Find F(106!, 65432) modulo (1016 + 61)." ], "solution": "9690646731515010", "name": "PE474" }, { "source": "Project Euler", "directions": [ "12n musicians participate at a music festival. On the first day, they form 3n quartets and practice all day.", "It is a disaster. At the end of the day, all musicians decide they will never again agree to play with any member of their quartet.", "On the second day, they form 4n trios, each musician avoiding his previous quartet partners.", "Let f(12n) be the number of ways to organize the trios amongst the 12n musicians.", "You are given f(12) = 576 and f(24) mod 1 000 000 007 = 509089824.", "Find f(600) mod 1 000 000 007." ], "solution": "75780067", "name": "PE475" }, { "source": "Project Euler", "directions": [ "Let R(a, b, c) be the maximum area covered by three non-overlapping circles inside a triangle with edge lengths a, b and c.", "Let S(n) be the average value of R(a, b, c) over all integer triplets (a, b, c) such that 1 ≤ a ≤ b ≤ c a + b ≤ n", "You are given S(2) = R(1, 1, 1) ≈ 0.31998, S(5) ≈ 1.25899.", "Find S(1803) rounded to 5 decimal places behind the decimal point." ], "solution": "110242.87794", "name": "PE476" } ]