Algorithms and Data Structures implemented in JavaScript for beginners, following best practices.
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/**
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* Problem 44 - Pentagon numbers
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*
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* @see {@link https://projecteuler.net/problem=44}
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*
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* Pentagonal numbers are generated by the formula, Pn=n(3n−1)/2. The first ten pentagonal numbers are:
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* 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...
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* It can be seen that P4 + P7 = 22 + 70 = 92 = P8. However, their difference, 70 − 22 = 48, is not pentagonal.
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* Find the pair of pentagonal numbers, Pj and Pk, for which their sum and difference are pentagonal and D = |Pk − Pj| is minimised; what is the value of D?
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*
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* @author ddaniel27
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*/
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function problem44 (k) {
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if (k < 1) {
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throw new Error('Invalid Input')
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}
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while (true) {
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k++
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const n = k * (3 * k - 1) / 2 // calculate Pk
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for (let j = k - 1; j > 0; j--) {
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const m = j * (3 * j - 1) / 2 // calculate all Pj < Pk
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if (isPentagonal(n - m) && isPentagonal(n + m)) { // Check sum and difference
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return n - m // return D
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}
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}
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}
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}
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/**
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* Function to check if a number is pentagonal or not
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* This function solves n
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* applying the solution for a quadratic function
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* @see {@link https://en.wikipedia.org/wiki/Quadratic_function}
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*/
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function isPentagonal (n) {
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const pent = (Math.sqrt(24 * n + 1) + 1) / 6
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return pent === Math.floor(pent)
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}
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export { problem44 }
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