53 lines
1.5 KiB
Markdown
53 lines
1.5 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 sk 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 s5 = 7.
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The sequence S = {sk : k ≥ 0} starts {0, 1, 2, 4, 5, 7, 9, 12, ...}.
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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: 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.
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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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Given Σ(M(n))3 = 8150 for 1 ≤ n ≤ 20.
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Find Σ(M(n))3 for 1 ≤ n ≤ 1000.
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# --hints--
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`euler391()` should return 61029882288.
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```js
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assert.strictEqual(euler391(), 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 euler391() {
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return true;
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}
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euler391();
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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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