Dear Uncle Colin,

In Basic Maths For Dummies, you mention a method for multiplying numbers from 6 × 6 to 10 × 10 on your fingers. It’s almost magical! Why does it work?

-- Does It Guarantee Interesting Times Sums?

The method works because of algebra. If you call the number of the finger on your right hand $a$ and the number on your left-hand $b$, you’re working out $(5+a) \times (5+b)$ – for example, to get 6 × 8, you use your thumb ($a=1$) and third finger ($b=3$). When you multiply out the brackets, you get $25 + 5 × a + 5 × b + a × b$.
The method gives you ($a+b$) tens from the fingers ‘below’ the touching ones, and – a bit more complicatedly – $(5-a) × (5-b)$ units from the fingers ‘above’. (Check it with $a=1$ and $b=3$, or any other combination and convince yourself it works!) Altogether, you have $10 × (a+b) + (5 - a) × (5 - b) = 10 × a + 10 × b + 25 - 5 × a - 5 × b + a \times b$, or $25 + 5 × a + 5 × b + a \times b$, the same as before – so it works, whatever values you pick for $a$ and $b$ :o)