Secrets of the Mathematical Ninja: Squaring three-digit numbers (Part II)

In part I, I showed you how to square a three-digit number in your head by squaring the first digit, adding double the cross-product (the first digit times by the other two digits) and adding the square of the final digits, with appropriate shifting.

Now, what about something like $798^2$? You could, of course, work out the numbers as 49, 7 × 98 × 2 = 1372, and 9604, adding them with shifts to get 6272, then 636804 (phew!), or you could split it up differently. You could do it as $(800 - 2)^2$.

That's much easier, and it works the same way. Let me walk you through it:

  • $8^2 = 64$
  • $8 \times -2 \times 2 = -32$; shifting it and adding on, I get 6368.
  • $(-2)^2 = 04$, making 636804

(Note that I write that last bit as 04 - because I'm mentally moving each sum two places to the left, I don't want to miss out the 0 in the answer!)

The advantages of this way over the way in part 1 is that you only need to know your squares up to 50, which is much less to learn. The taking away needs a bit of practice, but it's not all that bad.

But wait, there's more! Next week, I'll show you how to make your workload even less by using half-hundreds.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


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I teach in my home in Abbotsbury Road, Weymouth.

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