# Proving the finger trick for multiplying 6-10

Let the thumb be $$1,$$ the index finger $$2,$$ the middle finger $$3,$$ etc. Finger $$x,$$ on the left hand, touches finger $$y$$ on the right hand. That is, if $$x = 2$$ and $$y = 3,$$ then the index finger on the left hand is touching the middle finger on the right hand. There are $$x - 1$$ fingers to the right of finger $$x$$ on the left hand, and $$5 - x$$ fingers to the left. Similarly, there are $$y - 1$$ fingers to the left of finger $$y$$ on the right hand, and $$5 - y$$ fingers to the right. The number of ones in the final product, is thus

$$(5 - x)(5 - y) = xy - 5x - 5y + 25$$

The number of tens in the final product, is equal to the fingers touching, plus the number of fingers below the touching fingers, which is

$$2 + (x - 1) + (y - 1) = x + y$$

So the number of ones it contributes to the final product, is

$$10(x + y) = 10x + 10y$$

If we want to multiply $$6$$ by $$9,$$ we would put fingers $$x = 1$$ and $$y = 4$$ together. In general, if we want to multiply $$a$$ and $$b,$$ we're putting together $$x = a - 5$$ and $$y = b - 5.$$ Thus $$a = x + 5$$ and $$b = y + 5,$$ are the two numbers we're multiplying. From all of this, we can see the equation we want to prove is

$$(x + 5)(y + 5) = (10x + 10y) + (xy - 5x - 5y + 25)$$

Expanding the left-hand side gives us

$$xy + 5x + 5y + 25$$

But simplifying the right-hand side, by combining like terms, gives us the same thing. Thus, our equation is true, and the trick has been proven to always work.