What are Happy Numbers?
Let us take the number 79. If we square each of the digits and add them, we get 7×7 + 9×9 = 49 + 81 = 130. Let us denote this transformation T(n). Hence, T(79) = 130. Now let us do the same for 130. We get 1 + 9 + 0 = 10. Doing again for 10, we get 1 + 0 = 1. So, if we start with a number n, and end with 1, the number n and all the numbers in the chain are called happy numbers.
79 ⟼ 130 ⟼ 10 ⟼ 1
However, if we start with the number 80, we get something interesting. The chain gets into a loop and gets stuck there.
80 ⟼ 64 ⟼ 52 ⟼ 29 ⟼ 85 ⟼ 89 ⟼ 145 ⟼ 42 ⟼ 20 ⟼ 4 ⟼ 16 ⟼ 37 ⟼ 58 ⟼ 89
Now we could think that in the former case, it got into a loop at 1 and got stuck there. It turns out there are only two loops that any natural number can get stuck –  and [89, 145, 42, 20, 4, 16, 37, 58]. If we add 0 to the mix, then also .
I was thinking wouldn’t it be cool to see all these chains visually? So I wrote a small program that would create a simple digraph with numbers as nodes and each edge corresponding to a number n and T(n).
I created a graph for 0 <= n <=200. When I rendered it with dot (Graphviz), it gave me this spectacular visual! As expected, we could see 3 clusters. The tiniest one has only one number 0. The small cluster consists of all happy numbers. The larger one is all unhappy numbers.
Zooming into the loop at the center of unhappy numbers, we see the sequence I mentioned earlier [89, 145, 42, 20, 4, 16, 37, 58].
I have used a d3 port of Graphviz to render the graph on the browser. You can find the code here: https://github.com/rmadhuram/happy-numbers
Graphviz is an open-source graph visualization software developed by AT & T Research.
If I try to render more than 200 numbers, say 250, 500, etc., the rendering is not clean. I have an idea for an algorithm that could do a better graph layout for these types of structures. I hope to implement it when I find some time!