# Why does lattice multiplication work?

My brother (who is currently in 4th grade) is learning a technique called Lattice multiplication in school. It is a technique that involves taking two numbers, placing them on the outside of a square, and then computing the product. However, he said that when he asked his teacher how it worked, she said "Magic". After reviewing it, I'm puzzled as well. Can anyone explain why this technique works mathematically? Here is an example:

Lattice Multiplication

• And proof-verification isn't an appropriate tag for this anyway. May I suggest arithmetic? Anyway, this description doesn't contain enough information to reproduce the technique, except maybe via some lucky guessing. It would be useful if you included a worked example here. – Travis Willse Feb 10 '16 at 23:33
• Will do. Just added a link. – Chandler Freeman Feb 10 '16 at 23:36

This is really just ordinary long multiplication, but with lazy evaluation of carries, and the rows are inverted. That is, imagine multiplying $255 \times 25$ the usual way. The first row would have the result of multiplying $255 \times 5$, which is $1275$, and the second row would have the result of multiplying $255 \times 20$, which is $5100$. We then add $1275+5100 = 6375$.
Well, this is the same thing, except that first of all, we reverse the rows, so that we multiply $255 \times 20$ first. Secondly, we leave all the carries unevaluated at first, so that instead of writing $5100$, we write ${}^04^10^10^00$, where the $1$'s are the carry results of multiplying $5 \times 2$. The first $1$ will carry leftward to join the $4$, and the second $1$ will carry leftward to join the $0$, but unlike traditional multiplication, we leave that undone for the time being.
The second row is then the result of $255 \times 5$, except that instead of $1275$, we have $^10^25^25$. Now, finally, we add ${}^04^10^10^00+{}^10^25^25$:
where now all the carries are done at once: $4+1+1 = 6, 1+2 = 3, 5+2 = 7, 5 = 5$. The lattice lines are drawn so as to visually facilitate this, but otherwise, there's nothing terribly magical going on here. Notice that there are no carries in the addition; otherwise, this method would not appear to as great an advantage. I consider that a bit of intellectual dishonesty, really.