---
title: Y combinator
id: 594810f028c0303b75339ad5
challengeType: 5
---

## Description
<section id='description'>
<p>
In strict
<a href="https://en.wikipedia.org/wiki/Functional programming" title="wp: functional programming">functional programming</a> and
the <a href="https://en.wikipedia.org/wiki/lambda calculus" title="wp: lambda calculus">lambda calculus</a>,
functions (lambda expressions) don't have state and are only allowed to refer to arguments of enclosing functions.
This rules out the usual definition of a recursive function wherein a function is associated with the state of a variable and this variable's state is used in the body of the function.
</p>
<p>
The <a href="http://mvanier.livejournal.com/2897.html">Y combinator</a> is itself a stateless function that,
when applied to another stateless function, returns a recursive version of the function. The Y combinator is
the simplest of the class of such functions, called
<a href="https://en.wikipedia.org/wiki/Fixed-point combinator" title="wp: fixed-point combinator">fixed-point combinators</a>.
</p>
Task:

    Define the stateless Y combinator function and use it to compute
    <a href="https://en.wikipedia.org/wiki/Factorial" title="wp: factorial">factorial</a>.

<code>factorial(N)</code> function is already given to you.
See also <a href="http://vimeo.com/45140590">Jim Weirich: Adventures in Functional Programming</a>.
</section>

## Instructions
<section id='instructions'>

</section>

## Tests
<section id='tests'>

```yml
tests:
  - text: Y must return a function
    testString: assert.equal(typeof Y(f => n => n), 'function', 'Y must return a function');
  - text: factorial(1) must return 1.
    testString: assert.equal(factorial(1), 1, 'factorial(1) must return 1.');
  - text: factorial(2) must return 2.
    testString: assert.equal(factorial(2), 2, 'factorial(2) must return 2.');
  - text: factorial(3) must return 6.
    testString: assert.equal(factorial(3), 6, 'factorial(3) must return 6.');
  - text: factorial(4) must return 24.
    testString: assert.equal(factorial(4), 24, 'factorial(4) must return 24.');
  - text: factorial(10) must return 3628800.
    testString: assert.equal(factorial(10), 3628800, 'factorial(10) must return 3628800.');

```

</section>

## Challenge Seed
<section id='challengeSeed'>

<div id='js-seed'>

```js
function Y(f) {
  return function() {
  // Good luck!
  };
}

var factorial = Y(function(f) {
  return function (n) {
    return n > 1 ? n * f(n - 1) : 1;
  };
});
```

</div>


### After Test
<div id='js-teardown'>

```js
var factorial = Y(f => n => (n > 1 ? n * f(n - 1) : 1));
```

</div>

</section>

## Solution
<section id='solution'>


```js
var Y = f => (x => x(x))(y => f(x => y(y)(x)));

```

</section>