Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# 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.*
## You may also like

### Novemberish

Or search by topic

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?**

** **

**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.

"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."

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.