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# Filling the Gaps

**Will any of the numbers in the seventh column be a sum of three squares?**
**Can you prove it?**
*With thanks to Vicky Neale who created this task in collaboration with NRICH.*

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Age 14 to 16

Challenge Level

*Filling the Gaps printable sheet*

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?**

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**Can you make any conjectures about the columns in which squares, and sums of two squares, would appear if the grid continued beyond 96?**

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**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.

"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."