---
id: 5900f48a1000cf542c50ff9c
title: 'Problem 285: Pythagorean odds'
challengeType: 5
forumTopicId: 301936
dashedName: problem-285-pythagorean-odds
---

# --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?

# --hints--

`euler285()` should return 157055.80999.

```js
assert.strictEqual(euler285(), 157055.80999);
```

# --seed--

## --seed-contents--

```js
function euler285() {

  return true;
}

euler285();
```

# --solutions--

```js
// solution required
```