 # Difference of squares - even faster multiplication

When multiplying numbers we have talked on this forum about cross multiplication and about Art Benjamins way.

Let’s talk about another easy way to multiply. This one is especially quick if you are good with squares.

An example.
23 x 27 = 25^2 - 2^2 or 25 squared minus 4.
Most people know how to quickly square 25 to get 625.
23 x 27 = 625 - 4 = 621.

This technique is called ‘difference of squares’.
This is based on the fact that (a-b)(a+b) = a^2 - b^2.

In short it is the average squared minus the difference squared.

In the example I used (25-2)(25+2) = 25^2 - 2^2

It works also if the numbers are further apart:
36X84 = 60^2 - 24^2 = 3600 - 576 = 3024

See how quick and easy this is?

If the average does not come out nice, just change the computation:

36X85 = 36X84 + 36 = 3024 + 36 = 3060

If you don’t know a certain square but you do know one close by, here is a 2 step process.
Assuming you don’t know 61^2 by heart:
37X85 = 61^2 - 24^2 = 60^2 +60 +61 - 24^2 = 3600 - (576-121) = 3600 - 455 = 3145

This is based on the fact that 61^2 = 60^2 + 60 +61 or 60^ + 2X60 +1
Also 62^2 = 60^2 + 4X60 + 4
63^2 = 60^2 + 6X60 + 9

Nice method, How do you know which squares to use?

You take the square of the average of the 2 numbers and subtract the square of the difference.

For example 23 X 37 = 30^2 - 7^2 = 900 - 49 = 851

and also, what do you do if the average of the 2 numbers have decimals?

You prevent that.
For example:

## 46 57 X

Instead of 51.5^2 - 5.5^2 - which is difficult - I would change this into:

46 X 56 + 46 =
51^2 - 5^2 + 46 =
2601 - 25 + 46 =
2622

Isn’t it the square of the difference from the average, not just the square of the difference? Maybe that part of the definition is implied, but I’m new to this. So it’d be:

Square of the average minus the square of the difference (from the average).

Thanks for this insight! Time to work on my squares. Yes, this is a great way of saying this.

It was kind of implied and also shown in the examples, but yes, indeed, if your definition works better for you, then great; use it.

If you want I can write a couple of pieces about how to quickly learn the squares.

I’ve been browsing this forum looking for exactly that! If you have some techniques to share I - and others - would eat it up Great stuff!

If I wasn’t already spending 3 hours a day on memory techniques I would join you all

Hey, another question: Is it possible to use this with the multiplication of 3 two digit numbers?

Like 23x35x63

Unfortunately, no.
23X35 can be calculated using 29 & 6 of course.
(However in this case 22x35+35 might be a lot quicker).

The result must be multiplied by 63.

I don’t know of a way to do this a lot quicker.

@chiguin In MCWC2014 we had to finish 10 3x3x3 tasks in 10 minutes
like 534 x 742 x 967.
Everyone uses his own techniques. I used something like criss-cross product multiplication, but because it’s three dimensions the algorithm gets a bit complicated. It’s easy to make mistakes. One mistake and the task counts as wrong and gets 0 points (even if one finds the rest 7-8 digits correctly) . I showed Mr. Bouman my algorithm for a generic 2x2x2 task like the one you mentioned (23 x 35 x 63) , and it’s of course that’s much simpler than the 3x3x3s we had to solve. It seems indimidating, but on pen and paper and the correct algorithm, it can be done, if one is very focused.

About 2x2 the difference of squares is fast yes, but not guaranteed for success. One has to be lucky to be presented with a task that can fall into the easy difference-of-squares category. Not all tasks are so easy as
56 x 64 or 27x33
Nodas

Three digit squares :
abc

a^2 + b^2 + c^2 + 2ab + 2ac + 2ac