### Finger Multiplication

“Finger multiplication” is an old trick for using the bottom-left (easier to memorize) quadrant of the times table to do multiplication in the top-right quadrant. It’s easy to work out why it works, but unsatisfying, but I found rederiving the technique gave more insight.

How might we do this? To go from one quadrant to the other, we subtract from 10, or generally from the base b: (b-x)(b-y) = b²-bx-by+xy

Well, there’s our xy term, to get it alone we want to subtract off b²-bx-by = b(b-x-y), which we can do by adding -b(b-x-y) = b(x+y-b). The factor of b is trivial to deal with, just use place value, so that leaves us with x+y-b. That’s easy enough to calculate on its own, but we can distribute the -b half to each number as (x-b/2)+(y-b/2). We have the numbers b-x and x-b/2 to do arithmetic on, these sum to b/2, the number of fingers on a hand, so if you hold up x-b/2 (x-5 for us pentadactyl folk) fingers on a hand, then the down fingers number b-x. And that’s how you do finger multiplication, hold up x-5 on one hand, y-5 on the other, put the sum of the up fingers in the tens place and the product of the down fingers in the units place *et voilà*.

Now that we’ve reverse engineered it for the existing use case, can we come up with something like it for others? How about the case where only one term is > 5 ?

We want to move this from the top-left to the bottom-left of the times table, so x∙(b-y) = bx-xy. So, put the smaller number in the tens place, and subtract off the product of it and the “down fingers” from the large number. 3×7? 7 has 3 down fingers, so 30-9 = 21. 4×8? 8 has 2 down fingers, so 40-8 = 32.