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## 'Filling the Gaps' printed from http://nrich.maths.org/

Charlie has been thinking about which numbers can be written
as a sum of two square numbers. He took a $10\times10$
grid, and shaded the square numbers in blue and the sums of two
squares in yellow.

He hoped to find a pattern, but couldn't see anything
obvious.

Vicky suggested changing the number of columns in the grid, so they
reduced it by one:

"There seems to be a diagonal pattern."

"If the rows were one shorter, then those diagonals would line
up into vertical columns, wouldn't they?"

"Let's try it..."

What do you notice about the
positions of the square numbers?
What do you notice about the
positions of the sums of two square numbers?
Can you make any conjectures about
the columns in which squares, and sums of two squares, would
appear if the grid continued beyond 96?
Can you prove any of your
conjectures?
You might like to look back at the nine-column grid and ask
yourself the same questions.

Charlie couldn't write every number as a sum of two squares.
He wondered what would happen if he allowed himself three squares.

Will any of the numbers in
the seventh column be a sum of three squares?

Can you prove
it?

"We must be able to
write every number if we are allowed to include sums of four
squares!"

"Yes, but it's not easy to prove. Several great
mathematicians worked on it over a long period before

Lagrange gave the first proof in 1770."

With thanks to Vicky Neale
who created this task in collaboration with
NRICH.