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curriculum/challenges/italian/10-coding-interview-prep/project-euler/problem-440-gcd-and-tiling.md
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---
id: 5900f5241000cf542c510037
title: 'Problem 440: GCD and Tiling'
title: 'Problema 440: GCD e Tiling'
challengeType: 5
forumTopicId: 302112
dashedName: problem-440-gcd-and-tiling
@ -8,24 +8,32 @@ dashedName: problem-440-gcd-and-tiling
# --description--
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:
Vogliamo ricoprire completamente una tavola di lunghezza $n$ e altezza 1, con blocchi 1 × 2 o 1 × 1 con una singola cifra decimale in alto:
For example, here are some of the ways to tile a board of length n = 8:
<img class="img-responsive center-block" alt="dieci blocchi 1x1 con una cifra decimale in alto, e blocco 1x2" src="https://cdn.freecodecamp.org/curriculum/project-euler/gcd-and-tiling-1.png" style="background-color: white; padding: 10px;" />
Let T(n) be the number of ways to tile a board of length n as described above.
Per esempio, ecco alcuni dei modi per piastrellare una tavola di lunghezza $n = 8$:
For example, T(1) = 10 and T(2) = 101.
<img class="img-responsive center-block" alt="esempi di modi per piastrellare una tavola di lunghezza n = 8" src="https://cdn.freecodecamp.org/curriculum/project-euler/gcd-and-tiling-2.png" style="background-color: white; padding: 10px;" />
Let S(L) be the triple sum ∑a,b,c gcd(T(ca), T(cb)) for 1 ≤ a, b, c ≤ L. For example: S(2) = 10444 S(3) = 1292115238446807016106539989 S(4) mod 987 898 789 = 670616280.
Sia $T(n)$ il numero di modi per piastrellare una tavola di lunghezza $n$ come descritto sopra.
Find S(2000) mod 987 898 789.
Per esempio, $T(1) = 10$ e $T(2) = 101$.
Sia $S(L)$ la tripla somma $\sum_{a, b, c} gcd(T(c^a), T(c^b)$ per $1 ≤ a, b, c ≤ L$.
Per esempio:
$$\begin{align} & S(2) = 10\\,444 \\\\ & S(3) = 1\\,292\\,115\\,238\\,446\\,807\\,016\\,106\\,539\\,989 \\\\ & S(4)\bmod 987\\,898\\,789 = 670\\,616\\,280. \end{align}$$
Trova $S(2000)\bmod 987\\,898\\,789$.
# --hints--
`euler440()` should return 970746056.
`gcdAndTiling()` dovrebbe restituire `970746056`.
```js
assert.strictEqual(euler440(), 970746056);
assert.strictEqual(gcdAndTiling(), 970746056);
```
# --seed--
@ -33,12 +41,12 @@ assert.strictEqual(euler440(), 970746056);
## --seed-contents--
```js
function euler440() {
function gcdAndTiling() {
return true;
}
euler440();
gcdAndTiling();
```
# --solutions--