### Calendar Capers

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

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

# Multiplication Square

##### Stage: 3 Challenge Level:

Take a look at the multiplication square below:

Pick any 2 by 2 square and add the numbers on each diagonal.
For example, if you take:

the numbers along one diagonal add up to $77$ ($32 + 45$)
and the numbers along the other diagonal add up to $76$ ($36 + 40$).

Try a few more examples.
What do you notice?
Can you show (prove) that this will always be true?

Now pick any 3 by 3 square and add the numbers on each diagonal.
For example, if you take:

the numbers along one diagonal add up to $275$ ($72 + 91 + 112$)
and the numbers along the other diagonal add up to $271$ ($84 + 91 + 96$).

Try a few more examples.
What do you notice this time?
Can you show (prove) that this will always be true?

Now pick any 4 by 4 square and add the numbers on each diagonal.
For example, if you take:

the numbers along one diagonal add up to $176$ ($24 + 36 + 50 + 66$)
and the numbers along the other diagonal add up to $166$ ($33 + 40 + 45 + 48$).

Try a few more examples.
What do you notice now?
Can you show (prove) that this will always be true?

Can you predict what will happen if you pick a 5 by 5 square, a 6 by 6 square ... an n by n square, and add the numbers on each diagonal?