47 lines
1.5 KiB
Markdown
47 lines
1.5 KiB
Markdown
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---
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id: 5900f49f1000cf542c50ffb1
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title: 'Problem 306: Paper-strip Game'
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challengeType: 5
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forumTopicId: 301960
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dashedName: problem-306-paper-strip-game
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---
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# --description--
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The following game is a classic example of Combinatorial Game Theory:
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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.
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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.
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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.
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For 1 ≤ n ≤ 1 000 000, how many values of n are there for which the first player can force a win?
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# --hints--
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`euler306()` should return 852938.
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```js
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assert.strictEqual(euler306(), 852938);
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```
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# --seed--
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## --seed-contents--
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```js
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function euler306() {
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return true;
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}
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euler306();
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```
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# --solutions--
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```js
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// solution required
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```
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