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---
id: 5900f3a91000cf542c50febc
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 |
| -------------- | --------------------------------------------------------------------- | --------------------- |
| Triangle | P< sub > 3< / sub > ,< var > < sub > n< / sub > < / var > =< var > n< / var > (< var > n< / var > +1)/2 | 1, 3, 6, 10, 15, ... |
| Square | P< sub > 4< / sub > ,< var > < sub > n< / sub > < / var > =< var > n< / var > < sup > 2< / sup > | 1, 4, 9, 16, 25, ... |
| Pentagonal | P< sub > 5< / sub > ,< var > < sub > n< / sub > < / var > =< var > n< / var > (3< var > n< / var > − 1)/2 | 1, 5, 12, 22, 35, ... |
| Hexagonal | P< sub > 6< / sub > ,< var > < sub > n< / sub > < / var > =< var > n< / var > (2< var > n< / var > − 1) | 1, 6, 15, 28, 45, ... |
| Heptagonal | P< sub > 7< / sub > ,< var > < sub > n< / sub > < / var > =< var > n< / var > (5< var > n< / var > − 3)/2 | 1, 7, 18, 34, 55, ... |
| Octagonal | P< sub > 8< / sub > ,< var > < sub > n< / sub > < / var > =< var > n< / var > (3< var > n< / var > − 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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< ol >
< li > 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).< / li >
< li > Each polygonal type: triangle (P< sub > 3,127< / sub > = 8128), square (P< sub > 4,91< / sub > = 8281), and pentagonal (P< sub > 5,44< / sub > = 2882), is represented by a different number in the set.< / li >
< li > This is the only set of 4-digit numbers with this property.< / li >
< / ol >
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Find the sum of the only ordered set of six cyclic 4-digit numbers for which each polygonal type: triangle, square, pentagonal, hexagonal, heptagonal, and octagonal, is represented by a different number in the set.
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# --hints--
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`cyclicalFigurateNums()` should return a number.
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```js
assert(typeof cyclicalFigurateNums() === 'number');
```
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`cyclicalFigurateNums()` should return 28684.
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```js
assert.strictEqual(cyclicalFigurateNums(), 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() {
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return true;
}
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cyclicalFigurateNums();
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```
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# --solutions--
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```js
// solution required
```