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.

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Investigate the successive areas of light blue in these diagrams.

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?