---
id: 5900f49f1000cf542c50ffb1
challengeType: 5
title: 'Problem 306: Paper-strip Game'
forumTopicId: 301960
---

## Description
<section id='description'>
The following game is a classic example of Combinatorial Game Theory:

Two players start with a strip of n white squares and they take alternate turns.
On each turn, a player picks two contiguous white squares and paints them black.
The first player who cannot make a move loses.

If n = 1, there are no valid moves, so the first player loses automatically.
If n = 2, there is only one valid move, after which the second player loses.
If n = 3, there are two valid moves, but both leave a situation where the second player loses.
If n = 4, there are three valid moves for the first player; she can win the game by painting the two middle squares.
If n = 5, there are four valid moves for the first player (shown below in red); but no matter what she does, the second player (blue) wins.



So, for 1 ≤ n ≤ 5, there are 3 values of n for which the first player can force a win.
Similarly, for 1 ≤ n ≤ 50, there are 40 values of n for which the first player can force a win.

For 1 ≤ n ≤ 1 000 000, how many values of n are there for which the first player can force a win?
</section>

## Instructions
<section id='instructions'>

</section>

## Tests
<section id='tests'>

```yml
tests:
  - text: <code>euler306()</code> should return 852938.
    testString: assert.strictEqual(euler306(), 852938);

```

</section>

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

<div id='js-seed'>

```js
function euler306() {

  return true;
}

euler306();
```

</div>



</section>

## Solution
<section id='solution'>

```js
// solution required
```

</section>