This is an introductory article to the Y combinator from `lambda-calculus`. But we won’t mention the `Y combinator` in this article.

This article is the first one of a serie about the Y combinator in `ruby`:

In this article, we are going to show how to write recursive functions in `ruby` without giving names to any function.

(If you are curious to see it in other languages, there is a version of the code in `clojure` and `javascript`.)

At first glance, it seems impossible: how could we refer to something that we are currently defining without using its name?

1. Live: The code is executed in your browser
2. Interactive: You can modify the code and it is evaluated as you type

## Begin with the end in mind

The end result of this article is the recursive implementation of the `factorial` function without using neither names nor loops.

Here is the code:

``````->(f){
f[f]
}[->(func){
->(n) { n == 0 ? 1 : n * func[func][n-1]}
}][19]
``````

As you can check, no mention of any names.

At first, it feels like magic.

Now, we are going to show the 4 step process that leads to this wonderful piece of code.

(We were inspired by this long but awesome article by Mike Vanier.)

# Step 0: recursive function

Let’s start with the recursive implementation of `factorial`:

``````\$factorial = ->(n) {
(n === 0) ? 1 : n * \$factorial[n - 1]
}
\$factorial[10]
``````

# Step 1: simple generator

Let’s write a function that is a generator of the `factorial` function:

``````\$factorial_gen = ->(f) {
->(n) {(n === 0) ? 1 : n * f[n - 1]}
}
nil
``````

One one hand, `factorial_gen` is not recursive.

On the other hand, `factorial_gen` is not the `factorial` function.

But the interesting thing is that if we pass `factorial` to `factorial_gen` it returns the `factorial` function:

``````\$factorial_gen[\$factorial][19]
``````

Before going on reading make sure you understand why it is true that:

`\$factorial_gen[factorial]` is equivalent to `\$factorial`

# Step 2: weird generator

Now, we are going to do something very weird: instead of using `func`, we are going to use `(func func)`. Like this:

``````\$factorial_weird = ->(f) {
->(n) {(n === 0) ? 1 : n * f[f][n - 1]}
}
nil
``````

The funny thing now is that if we apply `factorial_weird` to itself we get the `factorial` function:

``````\$factorial_weird[\$factorial_weird][19]
``````

Before going on reading make sure you understand why it is true that:

`\$factorial_weird[factorial_weird]` is equivalent to `factorial`

# Step 3: Recursion without names

Now, let’s write down the application of `factorial_weird` to itself, using the body of `factorial_weird` instead of its name:

``````\$factorial_no_names = ->(f) {
->(n) {(n === 0) ? 1 : n * f[f][n - 1]}
}[->(f) {
->(n) {(n === 0) ? 1 : n * f[f][n - 1]}
}]
\$factorial_no_names[19]
``````

And we got a recursive implementation of `factorial` without using any names!

We gave it a name just for the convenience of using it.

As you can check, this is a completely valid implementation of `factorial`:

``````(1..11).map(&\$factorial_no_names).to_a
``````

Do you understand why this is equivalent to the code we shown in the beginning of the article?

``````->(f){
f[f]
}[->(func){
->(n) { n == 0 ? 1 : n * func[func][n-1]}
}][19]
``````

Can you write your own implementation of other recursive functions without names?

In our next article, we are going to show the Y combinator in action in `ruby`.