The happy numbers kata

This weekend my 9 year old son was given the following homework:

Choose a two-digit number (eg. 23), square each digit and add them together. Keep repeating this until you reach 1 or the cycle carries on in a continuous loop. If you reach 1 then the number you started with is a “happy number”.

Can you find all the happy numbers between 1 and 100?

For example, 31 is happy because

31 => 32 + 12 => 10 => 12 + 02 => 1

But 4 is not happy because

4 => 16 => 37 => 58 => 89 => 145 => 42 => 20 => 4

and we have a cycle.

I thought this might make an interesting code kata, so while he beavered away with pencil and paper I set to writing a little program to do the job. Why not give it a try yourself?

What tests, if any, did you write? How many times did your stack overflow? What data structure(s) do you need?
Did you use recursion? Or loops? Or filters? Or something else?
Did you manage to eliminate all of the duplication? Can you eliminate all of the conditionals? Is any of this altered by your choice of programming language?
How long does it take to run? At what range size (eg. 1-100, 1-1000, 1-10000) does it take more than, say, 5 seconds to run? Does it run in linear time? Polynomial time? What mathematical tricks can you find to speed it up?
What will you do differently next time?

First time through, I wrote no tests. My stack overflowed three times. I used a dictionary to hold the results, with a simple array for cycle detection and a recursive algorithm. I made no attempt to eliminate either duplication or conditionals, and I used none of the arithmetic tricks that my son found while doing the exercise by hand.

Next time, however, will be different…

Supplementary (for the mathematicians among you):

Are there any numbers for which the series neither terminates at 1 nor eventually repeats forever?