curriculum/challenges/english/blocks/project-euler-problems-101-to-200/5900f3ec1000cf542c50feff.md
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.
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.
By finding the difference between tile $n$ and each of its six neighbors we shall define $PD(n)$ to be the number of those differences which are prime.
For example, working clockwise around tile 8 the differences are 12, 29, 11, 6, 1, and 13. So $PD(8) = 3$.
In the same way, the differences around tile 17 are 1, 17, 16, 1, 11, and 10, hence $PD(17) = 2$.
It can be shown that the maximum value of $PD(n)$ is $3$.
If all of the tiles for which $PD(n) = 3$ are listed in ascending order to form a sequence, the 10th tile would be 271.
Find the 2000th tile in this sequence.
hexagonalTile(10) should return 271.
assert.strictEqual(hexagonalTile(10), 271);
hexagonalTile(2000) should return 14516824220.
assert.strictEqual(hexagonalTile(2000), 14516824220);
function hexagonalTile(tileIndex) {
return true;
}
hexagonalTile(10);
class PrimeSeive {
constructor(num) {
const seive = Array(Math.floor((num - 1) / 2)).fill(true);
const upper = Math.floor((num - 1) / 2);
const sqrtUpper = Math.floor((Math.sqrt(num) - 1) / 2);
for (let i = 0; i <= sqrtUpper; i++) {
if (seive[i]) {
// Mark value in seive array
const prime = 2 * i + 3;
// Mark all multiples of this number as false (not prime)
const primeSquaredIndex = 2 * i ** 2 + 6 * i + 3;
for (let j = primeSquaredIndex; j < upper; j += prime) {
seive[j] = false;
}
}
}
this._seive = seive;
}
isPrime(num) {
return num === 2
? true
: num % 2 === 0
? false
: this.isOddPrime(num);
}
isOddPrime(num) {
return this._seive[(num - 3) / 2];
}
};
function hexagonalTile(tileIndex) {
const primeSeive = new PrimeSeive(tileIndex * 420);
let count = 1;
let n = 1;
let number = 0;
while (count < tileIndex) {
if (primeSeive.isPrime(6*n - 1) &&
primeSeive.isPrime(6*n + 1) &&
primeSeive.isPrime(12*n + 5)) {
number = 3*n*n - 3*n + 2;
count++;
if (count >= tileIndex) break;
}
if (primeSeive.isPrime(6*n + 5) &&
primeSeive.isPrime(6*n - 1) &&
primeSeive.isPrime(12*n - 7) && n != 1) {
number = 3*n*n + 3*n + 1;
count++;
}
n++;
}
return number;
}