53 lines
1.7 KiB
Markdown
53 lines
1.7 KiB
Markdown
---
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id: 5900f4b71000cf542c50ffca
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title: 'Problem 331: Cross flips'
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challengeType: 5
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forumTopicId: 301989
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dashedName: problem-331-cross-flips
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---
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# --description--
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N×N disks are placed on a square game board. Each disk has a black side and white side.
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At each turn, you may choose a disk and flip all the disks in the same row and the same column as this disk: thus $2 × N - 1$ disks are flipped. The game ends when all disks show their white side. The following example shows a game on a 5×5 board.
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<img class="img-responsive center-block" alt="animation showing game on 5x5 board" src="https://cdn.freecodecamp.org/curriculum/project-euler/cross-flips.gif" style="background-color: white; padding: 10px;">
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It can be proven that 3 is the minimal number of turns to finish this game.
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The bottom left disk on the $N×N$ board has coordinates (0, 0); the bottom right disk has coordinates ($N - 1$,$0$) and the top left disk has coordinates ($0$,$N - 1$).
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Let $C_N$ be the following configuration of a board with $N × N$ disks: A disk at ($x$, $y$) satisfying $N - 1 \le \sqrt{x^2 + y^2} \lt N$, shows its black side; otherwise, it shows its white side. $C_5$ is shown above.
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Let $T(N)$ be the minimal number of turns to finish a game starting from configuration $C_N$ or 0 if configuration $C_N$ is unsolvable. We have shown that $T(5) = 3$. You are also given that $T(10) = 29$ and $T(1\\,000) = 395\\,253$.
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Find $\displaystyle \sum_{i = 3}^{31} T(2^i - i)$.
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# --hints--
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`crossFlips()` should return `467178235146843500`.
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```js
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assert.strictEqual(crossFlips(), 467178235146843500);
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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 crossFlips() {
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return true;
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}
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crossFlips();
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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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