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id	title	challengeType	forumTopicId	dashedName
5900f39f1000cf542c50feb2	Problem 51: Prime digit replacements	5	302162	problem-51-prime-digit-replacements

--description--

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 n prime value family.

--hints--

primeDigitReplacements(6) should return a number.

assert(typeof primeDigitReplacements(6) === 'number');

primeDigitReplacements(6) should return 13.

assert.strictEqual(primeDigitReplacements(6), 13);

primeDigitReplacements(7) should return 56003.

assert.strictEqual(primeDigitReplacements(7), 56003);

primeDigitReplacements(8) should return 121313.

assert.strictEqual(primeDigitReplacements(8), 121313);

--seed--

--seed-contents--

function primeDigitReplacements(n) {

  return true;
}

primeDigitReplacements(6);

--solutions--

const NUM_PRIMES = 1000000;
const PRIME_SEIVE = Array(Math.floor((NUM_PRIMES-1)/2)).fill(true);
(function initPrimes(num) {
  const upper = Math.floor((num - 1) / 2);
  const sqrtUpper = Math.floor((Math.sqrt(num) - 1) / 2);
  for (let i = 0; i <= sqrtUpper; i++) {
    if (PRIME_SEIVE[i]) {
      // Mark value in PRIMES array
      const prime = 2 * i + 3;
      // Mark all multiples of this number as false (not prime)
      const primeSqaredIndex = 2 * i ** 2 + 6 * i + 3;
      for (let j = primeSqaredIndex; j < upper; j += prime) {
        PRIME_SEIVE[j] = false;
      }
    }
  }
})(NUM_PRIMES);

function isPrime(num) {
  if (num === 2) return true;
  else if (num % 2 === 0) return false
  else return PRIME_SEIVE[(num - 3) / 2];
}

function primeDigitReplacements(n) {
  function isNFamily(number, n) {
    const prime = number.toString();
    const lastDigit = prime[prime.length - 1];
    return doesReplacingMakeFamily(prime, '0', n) ||
      doesReplacingMakeFamily(prime, '2', n) ||
      (lastDigit !== '1' && doesReplacingMakeFamily(prime, '1', n));
  }

  function doesReplacingMakeFamily(prime, digitToReplace, family) {
    let miss = 0;
    const base = parseInt(
      prime
        .split('')
        .map(digit => digit == digitToReplace ? '0' : digit)
        .join('')
    );
    const replacements = parseInt(
      prime
        .split('')
        .map(digit => digit === digitToReplace ? '1' : '0')
        .join('')
    );
    const start = prime[0] === digitToReplace ? 1 : 0;
    for (let i = start; i < 10; i++) {
      const nextNumber = base + i * replacements;
      if (!isPartOfFamily(nextNumber, prime)) miss++;
      if (10 - start - miss < family) break;
    }
    return 10 - start - miss === family;
  }

  function isPartOfFamily(number, prime) {
    return (
      isPrime(number) && number.toString().length === prime.length
    );
  }

  for (let number = 1; number < 125000; number++) {
    if (isPrime(number) && isNFamily(number, n)) {
      return number;
    }
  }
  return -1;
}

3.4 KiB Raw Blame History