65 lines
1.5 KiB
Markdown
65 lines
1.5 KiB
Markdown
---
|
||
id: 5900f48a1000cf542c50ff9c
|
||
challengeType: 5
|
||
title: 'Problem 285: Pythagorean odds'
|
||
---
|
||
|
||
## Description
|
||
<section id='description'>
|
||
Albert chooses a positive integer k, then two real numbers a, b are randomly chosen in the interval [0,1] with uniform distribution.
|
||
The square root of the sum (k·a+1)2 + (k·b+1)2 is then computed and rounded to the nearest integer. If the result is equal to k, he scores k points; otherwise he scores nothing.
|
||
|
||
For example, if k = 6, a = 0.2 and b = 0.85, then (k·a+1)2 + (k·b+1)2 = 42.05.
|
||
The square root of 42.05 is 6.484... and when rounded to the nearest integer, it becomes 6.
|
||
This is equal to k, so he scores 6 points.
|
||
|
||
It can be shown that if he plays 10 turns with k = 1, k = 2, ..., k = 10, the expected value of his total score, rounded to five decimal places, is 10.20914.
|
||
|
||
If he plays 105 turns with k = 1, k = 2, k = 3, ..., k = 105, what is the expected value of his total score, rounded to five decimal places?
|
||
</section>
|
||
|
||
## Instructions
|
||
<section id='instructions'>
|
||
|
||
</section>
|
||
|
||
## Tests
|
||
<section id='tests'>
|
||
|
||
```yml
|
||
tests:
|
||
- text: <code>euler285()</code> should return 157055.80999.
|
||
testString: assert.strictEqual(euler285(), 157055.80999, '<code>euler285()</code> should return 157055.80999.');
|
||
|
||
```
|
||
|
||
</section>
|
||
|
||
## Challenge Seed
|
||
<section id='challengeSeed'>
|
||
|
||
<div id='js-seed'>
|
||
|
||
```js
|
||
function euler285() {
|
||
// Good luck!
|
||
return true;
|
||
}
|
||
|
||
euler285();
|
||
```
|
||
|
||
</div>
|
||
|
||
|
||
|
||
</section>
|
||
|
||
## Solution
|
||
<section id='solution'>
|
||
|
||
```js
|
||
// solution required
|
||
```
|
||
</section>
|