Recently we looked at how to square two-digit numbers ending in 5 in your head. What about the other square numbers? Well, there’s a quick way to do that in your head as well.

Let’s look at 37^{2}.

First of all round the 37 up or down to the nearest 10.

You added 3 to get 40, so you also need to subtract 3 to get 34 (37-3=34)

Multiply these to numbers together 40×34 = 4 x 34 x 10 = 1360 (4×30=120, 4×4=16, 120 +16 = 136, 136×10=1360)

Finally, add on the square of the number you added and subtracted (here it was 3 and 3×3=9 so add 9) 1360+9=1369.

So 37^{2} = 1369

Here’s another example: 62^{2}

Round this down to 60, and then because you subtracted 2, you also have to add 2 so you get 60×64 (6 x 64 x 10). 3 x 60 = 360, 6 x 4 = 24, 360 + 24 = 384, 384 x 10 = 3840

Remember to add the square of the number you rounded down by. 2^{2} = 4 so add 4.

3840 + 4 = 3844

So 62^{2} = 3844

If you’ve enjoyed these tricks then you might enjoy the Secrets of Mental Maths course from The Great Courses. It’s fascinating!

Related post: How to multiply any 2-digit number by 11 (in your head!)