**Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?**

Take a look at the multiplication square.

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?

Can you prove your prediction?

Age 14 to 16