Proving the finger trick for multiplying by 9

Here's the proof:

• $$n$$ is the finger that's down.
• $$n - 1$$ is the number of fingers up, to the left of the finger that's down.
• $$10 - n$$ is the number of fingers up, to the right of the finger that's down.

Thus, we need to prove $$9n = 10(n - 1) + (10 - n).$$

\begin{align} 9n &= 10(n - 1) + (10 - n) \\ &= 10n - 10 + 10 - n \\ &= 9n \end{align}