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?
The rule is that the next number in
the sequence is the average of the last two numbers.
Now start with three numbers, say $4,
The new rule is that the next number
is the mean of the last three numbers.