A nearly useless multiplication shortcut


#1

Recently, during a short train ride, I did some squaring in my head. I took 68*68. I had just reviewed the method of subtracting and adding the same number to both factors in order to make one of the factors a round number for easier calculation. So, changing 68 to 70 (adding 2) would mean to subtract the same 2 from the other factor 68 and then doing the multiplication on those changed factors, and after this just adding the square of the changing number 2. In general:

n * n = (n+k) * (n-k) + (k*k)

My example:

68 * 68
= (68+2) * (68-2) + 2*2
= 70 * 66 + 4
= 4620 + 4
= 4624

As soon as I saw 70 * 66 I realized this would be easy, since 66 = 6 * 11. The calculation of 70 * 66 boils down to 70 * 6 * 11 = 420 * 11 = 4620 (multiplying by 11 is one of the most simple tasks in mental arithmetics, right?) or 7 * 6 * 11 * 10 = 42 * 11 * 10 = 462 * 10 = 4620.

So, I wondered what is necessary to have these simple solutions to the squaring of a larger 2-digit number?

It turns out, this is was needed: “When the difference of the digits of the number to be squared is equal to the difference of the number to the nearest next multiple of 10”!

So, these two-digit numbers qualify for the method: 26, 47, 68, 89 … thats’s not many.

And how do you do the calculation then?

  1. You take the first digit of the number to be squared
  2. make another digit by adding one to the first digit
  3. multiply the two digits
  4. multiply the result by 11 and append a 0
  5. add the square of the difference of the original number from the next multiple of 10.

So:

68 * 68 = 6*7 * 11 + 2*2 = 42*11 + 4 = 462|0 + 4 = 4624

OK, the use of this method is limited, but it was fun to find out!


#2

I love it!
Quick, simple, elegant.