d269909faa
* fix: clean-up Project Euler 381-400 * fix: missing image extension * fix: missing subscripts Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> Co-authored-by: Tom <20648924+moT01@users.noreply.github.com>
60 lines
1.6 KiB
Markdown
60 lines
1.6 KiB
Markdown
---
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id: 5900f4f31000cf542c510006
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title: 'Problem 391: Hopping Game'
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challengeType: 5
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forumTopicId: 302056
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dashedName: problem-391-hopping-game
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---
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# --description--
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Let $s_k$ be the number of 1’s when writing the numbers from 0 to $k$ in binary.
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For example, writing 0 to 5 in binary, we have 0, 1, 10, 11, 100, 101. There are seven 1’s, so $s_5 = 7$.
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The sequence $S = \\{s_k : k ≥ 0\\}$ starts $\\{0, 1, 2, 4, 5, 7, 9, 12, \ldots\\}$.
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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.
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For example, with $n = 5$ and starting with $c = 0$:
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- Player 1 chooses 4, so $c$ becomes $0 + 4 = 4$.
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- Player 2 chooses 5, so $c$ becomes $4 + 5 = 9$.
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- Player 1 chooses 3, so $c$ becomes $9 + 3 = 12$.
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- etc.
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Note that $c$ must always belong to $S$, and each player can increase $c$ by at most $n$.
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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 $M(20) = 4$.
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It can be verified $\sum M{(n)}^3 = 8150$ for $1 ≤ n ≤ 20$.
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Find $\sum M{(n)}^3$ for $1 ≤ n ≤ 1000$.
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# --hints--
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`hoppingGame()` should return `61029882288`.
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```js
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assert.strictEqual(hoppingGame(), 61029882288);
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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 hoppingGame() {
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return true;
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}
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hoppingGame();
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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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