NRICH: Multiplication Square

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:

part of times table

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:

part of times table

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:

part of times table

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


Algebra & Pre-Algebra, Linear Equations
High School, Educator

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NRICH (University of Cambridge)

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Challenge
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Click here for a poster of this problem.