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107 lines
2.6 KiB
Markdown
107 lines
2.6 KiB
Markdown
---
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id: 59c3ec9f15068017c96eb8a3
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title: Farey sequence
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challengeType: 1
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forumTopicId: 302266
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dashedName: farey-sequence
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---
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# --description--
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The Farey sequence <code>F<sub>n</sub></code> of order `n` is the sequence of completely reduced fractions between `0` and `1` which, when in lowest terms, have denominators less than or equal to `n`, arranged in order of increasing size.
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The *Farey sequence* is sometimes incorrectly called a *Farey series*.
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Each Farey sequence:
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<ul>
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<li>starts with the value 0, denoted by the fraction $ \frac{0}{1} $</li>
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<li>ends with the value 1, denoted by the fraction $ \frac{1}{1}$.</li>
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</ul>
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The Farey sequences of orders `1` to `5` are:
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<ul>
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<li style='list-style: none;'>${\bf\it{F}}_1 = \frac{0}{1}, \frac{1}{1}$</li>
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<li style='list-style: none;'>${\bf\it{F}}_2 = \frac{0}{1}, \frac{1}{2}, \frac{1}{1}$</li>
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<li style='list-style: none;'>${\bf\it{F}}_3 = \frac{0}{1}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{1}{1}$</li>
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<li style='list-style: none;'>${\bf\it{F}}_4 = \frac{0}{1}, \frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{1}{1}$</li>
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<li style='list-style: none;'>${\bf\it{F}}_5 = \frac{0}{1}, \frac{1}{5}, \frac{1}{4}, \frac{1}{3}, \frac{2}{5}, \frac{1}{2}, \frac{3}{5}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{1}{1}$</li>
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</ul>
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# --instructions--
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Write a function that returns the Farey sequence of order `n`. The function should have one parameter that is `n`. It should return the sequence as an array.
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# --hints--
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`farey` should be a function.
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```js
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assert(typeof farey === 'function');
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```
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`farey(3)` should return an array
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```js
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assert(Array.isArray(farey(3)));
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```
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`farey(3)` should return `['0/1','1/3','1/2','2/3','1/1']`
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```js
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assert.deepEqual(farey(3),['0/1', '1/3', '1/2', '2/3', '1/1']);
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```
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`farey(4)` should return `['0/1','1/4','1/3','1/2','2/3','3/4','1/1']`
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```js
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assert.deepEqual(farey(4), ['0/1', '1/4', '1/3', '1/2', '2/3', '3/4', '1/1']);
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```
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`farey(5)` should return `['0/1','1/5','1/4','1/3','2/5','1/2','3/5','2/3','3/4','4/5','1/1']`
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```js
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assert.deepEqual(farey(5), [
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'0/1',
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'1/5',
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'1/4',
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'1/3',
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'2/5',
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'1/2',
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'3/5',
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'2/3',
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'3/4',
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'4/5',
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'1/1'
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]);
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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 farey(n) {
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}
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```
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# --solutions--
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```js
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function farey(n) {
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const sequence = [{ string: "0/1", float: 0.0 }];
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for (let i = 1; i < n; i++) {
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for (let j = n; j >= i; j--) {
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if (i === 1 || j % i > 0) {
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sequence.push({ string: `${i}/${j}`, float: i / j });
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}
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}
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}
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return sequence
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.sort((a, b) => a.float - b.float)
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.map(e => e.string)
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}
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```
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