HOW MANY SQUARES ARE THERE IN THIS GRID?

Before we see the quickest way to answer the question (scroll down if you are an expert!), let’s see some different possible approaches:

GCSE APPROACH:
There are 16 little squares.
There is the one large square (the entire grid).
But there are also some “2 by 2” squares, like the red one – I count 9 of these:

2 by 2 and 2 by 3 squares

And there are some “3 by 3” squares, like the blue one – I count 4 of these:

So the total number of squares is 16+9+4+1 = 30.

A-LEVEL APPROACH:
Hey – I recognise those: the number of each size of square is itself a square number!
There are $4^2 = 16$ little squares.
There are $3^2=9$ “2 by 2” squares.
There are $2^2=4$ “3 by 3” squares.
There is $1^2=1$ big square (the whole grid).

So the total number of squares is $1^2 + 2^2 + 3^2 + 4^2 = 30$

Now we have noticed this, it’s easy to see that a slightly bigger “5 by 5” grid would have a total of $1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55$ squares – and so on for larger grids still.

FURTHER-MATHS A-LEVEL APPROACH:
$S_n = \sum_1^n r^2 = \frac{1}{6}n(n+1)(2n+1)$
The total number of squares is:
$\frac{1}{6} \times 4 \times (4+1) \times (8+1) = 30$


BEYOND A-LEVEL:
It’s amusing to notice that the answer is the 4th square-based pyramid number.

These numbers count the total number of oranges needed to build a pyramid of them with n layers.

Finding relations between number, shape and algebra is a central theme of higher maths as this gives us a multi-pronged approach to solving puzzles.

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