65 lines
2.5 KiB
Markdown
65 lines
2.5 KiB
Markdown
---
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id: 5900f5171000cf542c510029
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title: 'Problem 426: Box-ball system'
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challengeType: 5
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forumTopicId: 302096
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dashedName: problem-426-box-ball-system
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---
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# --description--
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Consider an infinite row of boxes. Some of the boxes contain a ball. For example, an initial configuration of 2 consecutive occupied boxes followed by 2 empty boxes, 2 occupied boxes, 1 empty box, and 2 occupied boxes can be denoted by the sequence (2, 2, 2, 1, 2), in which the number of consecutive occupied and empty boxes appear alternately.
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A turn consists of moving each ball exactly once according to the following rule: Transfer the leftmost ball which has not been moved to the nearest empty box to its right.
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After one turn the sequence (2, 2, 2, 1, 2) becomes (2, 2, 1, 2, 3) as can be seen below; note that we begin the new sequence starting at the first occupied box.
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<img class="img-responsive center-block" alt="animation showing one complete turn from (2, 2, 2, 1, 2) to (2, 2, 1, 2, 3)" src="https://cdn.freecodecamp.org/curriculum/project-euler/box-ball-system-1.gif" style="background-color: white; padding: 10px;">
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A system like this is called a Box-Ball System or BBS for short.
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It can be shown that after a sufficient number of turns, the system evolves to a state where the consecutive numbers of occupied boxes is invariant. In the example below, the consecutive numbers of occupied boxes evolves to [1, 2, 3]; we shall call this the final state.
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<img class="img-responsive center-block" alt="four turns from occupied boxes [2, 2, 2] to final state [1, 2, 3]" src="https://cdn.freecodecamp.org/curriculum/project-euler/box-ball-system-2.gif" style="background-color: white; padding: 10px;">
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We define the sequence $\\{t_i\\}$:
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$$\begin{align}
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& s_0 = 290\\,797 \\\\
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& s_{k + 1} = {s_k}^2\bmod 50\\,515\\,093 \\\\
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& t_k = (s_k\bmod 64) + 1
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\end{align}$$
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Starting from the initial configuration $(t_0, t_1, \ldots, t_{10})$, the final state becomes [1, 3, 10, 24, 51, 75].
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Starting from the initial configuration $(t_0, t_1, \ldots, t_{10\\,000\\,000})$, find the final state.
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Give as your answer the sum of the squares of the elements of the final state. For example, if the final state is [1, 2, 3] then $14 (= 1^2 + 2^2 + 3^2)$ is your answer.
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# --hints--
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`boxBallSystem()` should return `31591886008`.
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```js
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assert.strictEqual(boxBallSystem(), 31591886008);
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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 boxBallSystem() {
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return true;
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}
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boxBallSystem();
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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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