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 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?
Can you prove your prediction?
Click here for a poster of this problem.
Age 14 to 16
