253 lines
6.8 KiB
Markdown
253 lines
6.8 KiB
Markdown
---
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id: 5900f3a91000cf542c50febc
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title: 'Problem 61: Cyclical figurate numbers'
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challengeType: 5
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forumTopicId: 302173
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dashedName: problem-61-cyclical-figurate-numbers
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---
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# --description--
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Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:
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| Type of Number | Formula | Sequence |
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| -------------- | ----------------------------- | --------------------- |
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| Triangle | $P_3(n) = \frac{n(n+1)}{2}$ | 1, 3, 6, 10, 15, ... |
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| Square | $P_4(n) = n^2$ | 1, 4, 9, 16, 25, ... |
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| Pentagonal | $P_5(n) = \frac{n(3n−1)}2$ | 1, 5, 12, 22, 35, ... |
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| Hexagonal | $P_6(n) = n(2n−1)$ | 1, 6, 15, 28, 45, ... |
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| Heptagonal | $P_7(n) = \frac{n(5n−3)}{2}$ | 1, 7, 18, 34, 55, ... |
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| Octagonal | $P_8(n) = n(3n−2)$ | 1, 8, 21, 40, 65, ... |
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The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three interesting properties.
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1. The set is cyclic, in that the last two digits of each number is the first two digits of the next number (including the last number with the first).
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2. Each polygonal type: triangle ($P_3(127) = 8128$), square ($P_4(91) = 8281$), and pentagonal ($P_5(44) = 2882$), is represented by a different number in the set.
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3. This is the only set of 4-digit numbers with this property.
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Find the sum of all numbers in ordered sets of `n` cyclic 4-digit numbers for which each of the $P_3$ to $P_{n + 2}$ polygonal types, is represented by a different number in the set.
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# --hints--
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`cyclicalFigurateNums(3)` should return a number.
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```js
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assert(typeof cyclicalFigurateNums(3) === 'number');
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```
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`cyclicalFigurateNums(3)` should return `19291`.
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```js
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assert.strictEqual(cyclicalFigurateNums(3), 19291);
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```
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`cyclicalFigurateNums(4)` should return `28684`.
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```js
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assert.strictEqual(cyclicalFigurateNums(4), 28684);
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```
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`cyclicalFigurateNums(5)` should return `76255`.
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```js
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assert.strictEqual(cyclicalFigurateNums(5), 76255);
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```
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`cyclicalFigurateNums(6)` should return `28684`.
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```js
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assert.strictEqual(cyclicalFigurateNums(6), 28684);
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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 cyclicalFigurateNums(n) {
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return true;
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}
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cyclicalFigurateNums(3);
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```
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# --solutions--
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```js
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function cyclicalFigurateNums(n) {
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function getChains(chain, n, numberTypes, numsExcludingLastNeededType) {
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if (chain.length === n) {
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return [chain];
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}
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const nextNumbers = getNextNumbersInChain(
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chain[chain.length - 1],
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numsExcludingLastNeededType
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);
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const chains = [];
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for (let j = 0; j < nextNumbers.length; j++) {
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const nextNumber = nextNumbers[j];
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if (chain.indexOf(nextNumber) === -1) {
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const nextChain = [...chain, nextNumber];
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chains.push(
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...getChains(nextChain, n, numberTypes, numsExcludingLastNeededType)
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);
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}
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}
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return chains;
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}
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function getNextNumbersInChain(num, numsExcludingLastNeededType) {
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const results = [];
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const beginning = num % 100;
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numsExcludingLastNeededType.forEach(number => {
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if (Math.floor(number / 100) === beginning) {
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results.push(number);
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}
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});
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return results;
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}
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function fillNumberTypes(n, numberTypes, numsExcludingLastNeededType) {
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const [, lastTypeCheck, lastTypeArr] = numberTypes[n - 1];
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for (let i = 1000; i <= 9999; i++) {
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for (let j = 0; j < n - 1; j++) {
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const [, typeCheck, typeArr] = numberTypes[j];
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if (typeCheck(i)) {
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typeArr.push(i);
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numsExcludingLastNeededType.add(i);
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}
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}
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if (lastTypeCheck(i)) {
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lastTypeArr.push(i);
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}
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}
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}
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function isCyclicalChain(chain, n, numberTypes) {
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const numberTypesInChain = getNumberTypesInChain(chain, numberTypes);
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if (!isChainAllowed(numberTypesInChain, n)) {
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return false;
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}
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const isChainCyclic =
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Math.floor(chain[0] / 100) === chain[chain.length - 1] % 100;
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return isChainCyclic;
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}
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function getNumberTypesInChain(chain, numberTypes) {
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const numbersInChain = {};
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for (let i = 0; i < numberTypes.length; i++) {
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const numberTypeName = numberTypes[i][0];
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numbersInChain[numberTypeName] = [];
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}
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for (let i = 0; i < chain.length; i++) {
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for (let j = 0; j < n; j++) {
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const [typeName, , typeNumbers] = numberTypes[j];
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const typeNumbersInChain = numbersInChain[typeName];
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if (typeNumbers.indexOf(chain[i]) !== -1) {
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typeNumbersInChain.push(chain[i]);
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}
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}
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}
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return numbersInChain;
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}
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function isChainAllowed(numberTypesInChain, n) {
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for (let i = 0; i < n; i++) {
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const typeName = numberTypes[i][0];
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const isNumberWithTypeInChain = numberTypesInChain[typeName].length > 0;
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if (!isNumberWithTypeInChain) {
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return false;
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}
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for (let j = i + 1; j < n; j++) {
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const otherTypeName = numberTypes[j][0];
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if (
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isNumberRepeatedAsOnlyNumberInTwoTypes(
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numberTypesInChain[typeName],
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numberTypesInChain[otherTypeName]
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)
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) {
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return false;
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}
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}
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}
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return true;
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}
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function isNumberRepeatedAsOnlyNumberInTwoTypes(
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typeNumbers,
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otherTypeNumbers
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) {
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return (
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typeNumbers.length === 1 &&
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otherTypeNumbers.length === 1 &&
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typeNumbers[0] === otherTypeNumbers[0]
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);
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}
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function isTriangle(num) {
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return ((8 * num + 1) ** 0.5 - 1) % 2 === 0;
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}
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function isSquare(num) {
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return num ** 0.5 === parseInt(num ** 0.5, 10);
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}
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function isPentagonal(num) {
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return ((24 * num + 1) ** 0.5 + 1) % 6 === 0;
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}
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function isHexagonal(num) {
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return ((8 * num + 1) ** 0.5 + 1) % 4 === 0;
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}
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function isHeptagonal(num) {
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return ((40 * num + 9) ** 0.5 + 3) % 10 === 0;
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}
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function isOctagonal(num) {
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return ((3 * num + 1) ** 0.5 + 1) % 3 === 0;
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}
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const numberTypes = [
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['triangle', isTriangle, []],
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['square', isSquare, []],
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['pentagonal', isPentagonal, []],
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['hexagonal', isHexagonal, []],
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['heptagonal', isHeptagonal, []],
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['octagonal', isOctagonal, []]
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];
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const numsExcludingLastNeededType = new Set();
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fillNumberTypes(n, numberTypes, numsExcludingLastNeededType);
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const nNumberChains = [];
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const [, , lastType] = numberTypes[n - 1];
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for (let i = 0; i < lastType.length; i++) {
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const startOfChain = lastType[i];
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nNumberChains.push(
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...getChains([startOfChain], n, numberTypes, numsExcludingLastNeededType)
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);
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}
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const cyclicalChains = nNumberChains.filter(chain =>
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isCyclicalChain(chain, n, numberTypes)
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);
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let sum = 0;
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for (let i = 0; i < cyclicalChains.length; i++) {
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for (let j = 0; j < cyclicalChains[0].length; j++) {
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sum += cyclicalChains[i][j];
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}
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}
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return sum;
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}
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```
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