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### Number and algebra

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### For younger learners

# Diagonal Sums

Here is a $100$ square with some of the numbers shaded:

Look at the green square which contains the numbers $2, 3, 12$ and $13$.

Do you notice anything about the sum of the numbers that are diagonally opposite each other?

Look at the pink square.

What happens this time when you look at the numbers diagonally opposite each other?

What about the yellow square?

You could try with other squares which have four numbers in them.

Can you find a reason why what you notice, happens?

Look at the squares shaded red. They form the corners of a large $3$ by $3$ square.

If you add the numbers diagonally opposite each other, what do you notice with this larger square?

Can you find a reason why what you notice, happens?

What happens for squares of different sizes?

You may like to print off this 100 square to try out some different squares of numbers.

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Age 7 to 14

Challenge Level

Here is a $100$ square with some of the numbers shaded:

Look at the green square which contains the numbers $2, 3, 12$ and $13$.

Do you notice anything about the sum of the numbers that are diagonally opposite each other?

Look at the pink square.

What happens this time when you look at the numbers diagonally opposite each other?

What about the yellow square?

You could try with other squares which have four numbers in them.

Can you find a reason why what you notice, happens?

Look at the squares shaded red. They form the corners of a large $3$ by $3$ square.

If you add the numbers diagonally opposite each other, what do you notice with this larger square?

Can you find a reason why what you notice, happens?

What happens for squares of different sizes?

You may like to print off this 100 square to try out some different squares of numbers.

Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. Try lots of examples. What happens? Can you explain it?

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?