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Adding All Nine

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!

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Multiplication Square

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Take a look at the multiplication square below:

times table


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?

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