Choose any three by three square of dates on a calendar page...
Can you create a Latin Square from multiples of a six digit number?
Can you explain how this card trick works?
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.