Ben working at home with his Dad sent us a solution to this problem:

Ben reckons for the four x four square, there should be 16 little squares inside. He says the next one should be five x five containing 25 little squares.

Alistair from Histon and Impington Infants School agreed with Ben's answers, and went a little further explaining how he worked out the last part:

I didn't know the answer to a square with 81 little squares, so I guessed 20, but 20 times 20 is 400 which is too big. I then thought of 10, but I knew that that would be 100 squares which is also too big, so I tried 9. 9 times 9 is 81, so the square would be 9 squares by 9 squares.

This is very well reasoned through Alistair.

Ryo who is 7 and goes to Moorfield Junior School adds:

I also know that 12 on each side will have 144 little squares. All these numbers are called square numbers because they always make squares.

Thank you to all three of you for sending in your contributions.