Happy Numbers

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

Zooming in on the Squares

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

Litov's Mean Value Theorem

Stage: 3 Challenge Level:
What happens if you reverse the two original numbers?
Try starting with $2, 8$ instead of $8, 2$.

What happens if you keep one of the starting numbers constant and only change the other?

It might be faster to use a spreadsheet or to write a short program to help you identify the limiting value (the value you get if you continue this process indefinitely).

Keep a record of your results.