---
id: 5900f4f31000cf542c510006
challengeType: 5
title: 'Problem 391: Hopping Game'
---

## Description
<section id='description'>
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.
</section>

## Instructions
<section id='instructions'>

</section>

## Tests
<section id='tests'>

```yml
tests:
  - text: <code>euler391()</code> should return 61029882288.
    testString: assert.strictEqual(euler391(), 61029882288, '<code>euler391()</code> should return 61029882288.');

```

</section>

## Challenge Seed
<section id='challengeSeed'>

<div id='js-seed'>

```js
function euler391() {
  // Good luck!
  return true;
}

euler391();
```

</div>



</section>

## Solution
<section id='solution'>

```js
// solution required
```
</section>
