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 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?
Let's generate a sequence where the next number is the mean of the previous two numbers.
Now start with three numbers.
This time, we can generate a sequence where the next number is the mean of the last three numbers.