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?

- You take the first digit of the number to be squared
- make another digit by adding one to the first digit
- multiply the two digits
- multiply the result by 11 and append a 0
- 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!