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:

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 4^{th} **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.