Choose any three by three square of dates on a calendar page...
Can you explain how this card trick works?
A combination mechanism for a safe comprises thirty-two tumblers
numbered from one to thirty-two in such a way that the numbers in
each wheel total 132... Could you open the safe?
We can repeat this process using the squares of $7$ and $4$ to
get $65$ and then continue the process indefinitely.
Start with $145$ and see what happens.
Now we come to the most important step! Show that, whatever
number we start with less than $1000$, we always get another number
which is less than $1000$.
Choose some numbers less than $1000$ and, each time, repeat the
process until you notice a pattern.
Make some conjectures about what happens in general.