### Calendar Capers

Choose any three by three square of dates on a calendar page...

### Card Trick 2

Can you explain how this card trick works?

### The Patent Solution

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?

# Happy Numbers

##### Stage: 3 Challenge Level:

Take any whole number between $1$ and $999$ inclusive and add the squares of the digits to get a new number. For example, starting from $138$ we get $1 + 3^2 + 8^2 =74.$

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.