Choose any three by three square of dates on a calendar page.
Circle any number on the top row, put a line through the other
numbers that are in the same row and column as your circled number.
Repeat this for a number of your choice from the second row. You
should now have just one number left on the bottom row, circle it.
Find the total for the three numbers circled. Compare this total
with the number in the centre of the square. What do you find? Can
you explain why this happens?
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two 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.