### Calendar Capers

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?

### Reverse to Order

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?

### Card Trick 2

Can you explain how this card trick works?

# 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.