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118 lines
3.0 KiB
Markdown
118 lines
3.0 KiB
Markdown
---
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id: 5900f3ec1000cf542c50feff
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title: 'Problem 128: Hexagonal tile differences'
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challengeType: 1
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forumTopicId: 301755
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dashedName: problem-128-hexagonal-tile-differences
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---
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# --description--
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A hexagonal tile with number 1 is surrounded by a ring of six hexagonal tiles, starting at "12 o'clock" and numbering the tiles 2 to 7 in an anti-clockwise direction.
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New rings are added in the same fashion, with the next rings being numbered 8 to 19, 20 to 37, 38 to 61, and so on. The diagram below shows the first three rings.
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<img class="img-responsive center-block" alt="前三圈排列好的六角砖，数字编号为 1 到 37，其中砖 8 和砖 17高亮" src="https://cdn.freecodecamp.org/curriculum/project-euler/hexagonal-tile-differences.png" style="background-color: white; padding: 10px;" />
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通过计算砖 $n$ 和它周围 6 块砖的数字差，我们定位 $PD(n)$ 为数字差中素数的个数。
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例如，围绕砖 8 顺时针方向的差额分别为 12、29、11、6、1 和 13。 则 $PD(8) = 3$。
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同理，围绕砖 17 的差额为 1、17、16、1、11 和 10，所以 $PD(17) = 2$。
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可以发现 $PD(n)$ 的最大值是 $3$。
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如果 $PD(n) = 3$ 的砖按升序排列，那么第 10 块砖将会是 271。
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求序列中的第 2000 块砖。
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# --hints--
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`hexagonalTile(10)` should return `271`.
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```js
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assert.strictEqual(hexagonalTile(10), 271);
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```
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`hexagonalTile(2000)` should return `14516824220`.
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```js
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assert.strictEqual(hexagonalTile(2000), 14516824220);
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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 hexagonalTile(tileIndex) {
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return true;
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}
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hexagonalTile(10);
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```
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# --solutions--
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```js
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class PrimeSeive {
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constructor(num) {
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const seive = Array(Math.floor((num - 1) / 2)).fill(true);
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const upper = Math.floor((num - 1) / 2);
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const sqrtUpper = Math.floor((Math.sqrt(num) - 1) / 2);
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for (let i = 0; i <= sqrtUpper; i++) {
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if (seive[i]) {
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// Mark value in seive array
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const prime = 2 * i + 3;
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// Mark all multiples of this number as false (not prime)
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const primeSqaredIndex = 2 * i ** 2 + 6 * i + 3;
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for (let j = primeSqaredIndex; j < upper; j += prime) {
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seive[j] = false;
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}
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}
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}
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this._seive = seive;
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}
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isPrime(num) {
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return num === 2
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? true
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: num % 2 === 0
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? false
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: this.isOddPrime(num);
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}
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isOddPrime(num) {
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return this._seive[(num - 3) / 2];
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}
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};
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function hexagonalTile(tileIndex) {
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const primeSeive = new PrimeSeive(tileIndex * 420);
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let count = 1;
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let n = 1;
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let number = 0;
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while (count < tileIndex) {
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if (primeSeive.isPrime(6*n - 1) &&
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primeSeive.isPrime(6*n + 1) &&
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primeSeive.isPrime(12*n + 5)) {
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number = 3*n*n - 3*n + 2;
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count++;
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if (count >= tileIndex) break;
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}
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if (primeSeive.isPrime(6*n + 5) &&
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primeSeive.isPrime(6*n - 1) &&
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primeSeive.isPrime(12*n - 7) && n != 1) {
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number = 3*n*n + 3*n + 1;
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count++;
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}
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n++;
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}
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return number;
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}
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```
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