64 lines
2.4 KiB
Markdown
64 lines
2.4 KiB
Markdown
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---
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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<sub>3</sub>,<var><sub>n</sub></var>=<var>n</var>(<var>n</var>+1)/2 | 1, 3, 6, 10, 15, ... |
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| Square | P<sub>4</sub>,<var><sub>n</sub></var>=<var>n</var><sup>2</sup> | 1, 4, 9, 16, 25, ... |
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| Pentagonal | P<sub>5</sub>,<var><sub>n</sub></var>=<var>n</var>(3<var>n</var>−1)/2 | 1, 5, 12, 22, 35, ... |
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| Hexagonal | P<sub>6</sub>,<var><sub>n</sub></var>=<var>n</var>(2<var>n</var>−1) | 1, 6, 15, 28, 45, ... |
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| Heptagonal | P<sub>7</sub>,<var><sub>n</sub></var>=<var>n</var>(5<var>n</var>−3)/2 | 1, 7, 18, 34, 55, ... |
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| 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>
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<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>
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<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>
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<li>This is the only set of 4-digit numbers with this property.</li>
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</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
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assert(typeof cyclicalFigurateNums() === 'number');
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```
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`cyclicalFigurateNums()` should return 28684.
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```js
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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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}
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cyclicalFigurateNums();
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```
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# --solutions--
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```js
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// solution required
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```
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