The Squares Challenge

Board Size012345678910
Number of Squares001

The challenge comes in various stages:

Stage 1

Complete the table showing the number of squares in boards up to size 10 by 10.

After completing the table up to 5 by 5 by counting the squares on the pictures above, try looking for a rule for continuing the sequence. There are a couple of hints below. When you think you have the answer, you can check by clicking on the button above to show the number of squares in a 10 by 10 board

Stage 2

Find a formula for the number of squares on an n by n board.

With a bit of ingenuity, it is possible to "spot" the formula. The hints below may help.

Alternatively, you could use the "Method of finite differences" which will find a formula for any sequence provided it is a polynomial.

Stage 3

Prove that the formula found in stage 2 is correct - ie that it will work for boards of any size.

This is a real challenge! I have a way of doing it which involves looking at how many squares there are at each different angle, and combining those totals into a neat summation (Σ expression). Then that can be simplified using A Level algebra. But that is all rather pedestrian, and the formula itself is so elegant that I wonder whether there is a way of arriving at it directlyfrom the structure of the board. If you have any insight into this, please let me know:

Stage 3a

If Stage 3 is too much of a challenge, an easier "proving" stage is to prove that the formula will always give a whole number. (This is not obvious from the formula, but the proof is quite elegant.)