110 lines
2.7 KiB
Markdown
110 lines
2.7 KiB
Markdown
|
---
|
||
|
|
id: 59c3ec9f15068017c96eb8a3
|
||
|
|
title: Farey sequence
|
||
|
|
challengeType: 5
|
||
|
|
forumTopicId: 302266
|
||
|
|
dashedName: farey-sequence
|
||
|
|
---
|
||
|
|
|
||
|
|
# --description--
|
||
|
|
|
||
|
|
The [Farey sequence](https://en.wikipedia.org/wiki/Farey sequence "wp: 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.
|
||
|
|
|
||
|
|
The *Farey sequence* is sometimes incorrectly called a *Farey series*.
|
||
|
|
|
||
|
|
Each Farey sequence:
|
||
|
|
|
||
|
|
<ul>
|
||
|
|
<li>starts with the value 0, denoted by the fraction $ \frac{0}{1} $</li>
|
||
|
|
<li>ends with the value 1, denoted by the fraction $ \frac{1}{1}$.</li>
|
||
|
|
</ul>
|
||
|
|
|
||
|
|
The Farey sequences of orders `1` to `5` are:
|
||
|
|
|
||
|
|
<ul>
|
||
|
|
<li style='list-style: none;'>${\bf\it{F}}_1 = \frac{0}{1}, \frac{1}{1}$</li>
|
||
|
|
<li style='list-style: none;'>${\bf\it{F}}_2 = \frac{0}{1}, \frac{1}{2}, \frac{1}{1}$</li>
|
||
|
|
<li style='list-style: none;'>${\bf\it{F}}_3 = \frac{0}{1}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{1}{1}$</li>
|
||
|
|
<li style='list-style: none;'>${\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}$</li>
|
||
|
|
<li style='list-style: none;'>${\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>
|
||
|
|
</ul>
|
||
|
|
|
||
|
|
# --instructions--
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
# --hints--
|
||
|
|
|
||
|
|
`farey` should be a function.
|
||
|
|
|
||
|
|
```js
|
||
|
|
assert(typeof farey === 'function');
|
||
|
|
```
|
||
|
|
|
||
|
|
`farey(3)` should return an array
|
||
|
|
|
||
|
|
```js
|
||
|
|
assert(Array.isArray(farey(3)));
|
||
|
|
```
|
||
|
|
|
||
|
|
`farey(3)` should return `["1/3","1/2","2/3"]`
|
||
|
|
|
||
|
|
```js
|
||
|
|
assert.deepEqual(farey(3), ['1/3', '1/2', '2/3']);
|
||
|
|
```
|
||
|
|
|
||
|
|
`farey(4)` should return `["1/4","1/3","1/2","2/4","2/3","3/4"]`
|
||
|
|
|
||
|
|
```js
|
||
|
|
assert.deepEqual(farey(4), ['1/4', '1/3', '1/2', '2/4', '2/3', '3/4']);
|
||
|
|
```
|
||
|
|
|
||
|
|
`farey(5)` should return `["1/5","1/4","1/3","2/5","1/2","2/4","3/5","2/3","3/4","4/5"]`
|
||
|
|
|
||
|
|
```js
|
||
|
|
assert.deepEqual(farey(5), [
|
||
|
|
'1/5',
|
||
|
|
'1/4',
|
||
|
|
'1/3',
|
||
|
|
'2/5',
|
||
|
|
'1/2',
|
||
|
|
'2/4',
|
||
|
|
'3/5',
|
||
|
|
'2/3',
|
||
|
|
'3/4',
|
||
|
|
'4/5'
|
||
|
|
]);
|
||
|
|
```
|
||
|
|
|
||
|
|
# --seed--
|
||
|
|
|
||
|
|
## --seed-contents--
|
||
|
|
|
||
|
|
```js
|
||
|
|
function farey(n) {
|
||
|
|
|
||
|
|
}
|
||
|
|
```
|
||
|
|
|
||
|
|
# --solutions--
|
||
|
|
|
||
|
|
```js
|
||
|
|
function farey(n){
|
||
|
|
let farSeq=[];
|
||
|
|
for(let den = 1; den <= n; den++){
|
||
|
|
for(let num = 1; num < den; num++){
|
||
|
|
farSeq.push({
|
||
|
|
str:num+"/"+den,
|
||
|
|
val:num/den});
|
||
|
|
}
|
||
|
|
}
|
||
|
|
farSeq.sort(function(a,b){
|
||
|
|
return a.val-b.val;
|
||
|
|
});
|
||
|
|
farSeq=farSeq.map(function(a){
|
||
|
|
return a.str;
|
||
|
|
});
|
||
|
|
return farSeq;
|
||
|
|
}
|
||
|
|
```
|