61 lines
1.5 KiB
Markdown
61 lines
1.5 KiB
Markdown
---
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id: 5900f5241000cf542c510037
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title: 'Problem 440: GCD and Tiling'
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challengeType: 5
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forumTopicId: 302112
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dashedName: problem-440-gcd-and-tiling
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---
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# --description--
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We want to tile a board of length $n$ and height 1 completely, with either 1 × 2 blocks or 1 × 1 blocks with a single decimal digit on top:
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<img class="img-responsive center-block" alt="ten blocks 1x1 with single decimal digit on top, and 1x2 block" src="https://cdn.freecodecamp.org/curriculum/project-euler/gcd-and-tiling-1.png" style="background-color: white; padding: 10px;">
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For example, here are some of the ways to tile a board of length $n = 8$:
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<img class="img-responsive center-block" alt="examples of ways to tile a board of length n = 8" src="https://cdn.freecodecamp.org/curriculum/project-euler/gcd-and-tiling-2.png" style="background-color: white; padding: 10px;">
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Let $T(n)$ be the number of ways to tile a board of length $n$ as described above.
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For example, $T(1) = 10$ and $T(2) = 101$.
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Let $S(L)$ be the triple sum $\sum_{a, b, c} gcd(T(c^a), T(c^b))$ for $1 ≤ a, b, c ≤ L$.
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For example:
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$$\begin{align}
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& S(2) = 10\\,444 \\\\
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& S(3) = 1\\,292\\,115\\,238\\,446\\,807\\,016\\,106\\,539\\,989 \\\\
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& S(4)\bmod 987\\,898\\,789 = 670\\,616\\,280.
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\end{align}$$
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Find $S(2000)\bmod 987\\,898\\,789$.
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# --hints--
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`gcdAndTiling()` should return `970746056`.
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```js
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assert.strictEqual(gcdAndTiling(), 970746056);
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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 gcdAndTiling() {
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return true;
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}
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gcdAndTiling();
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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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