When I tutored elementary school math, one of my favorite things to show the kids was how to multiply by 9 using their fingers. (I apologize, but this trick assumes that you have five fingers on each hand. Later you'll see how to adapt it if you have nonstandard hands.)
Here's how the finger rule works. Let's say you want to multiply 4 by 9. First, put your hands in front of you with your palms away from you.
Count four fingers from the left, and bend that finger down. (This should be your left index finger.)
There are 3 fingers to the left of that finger and 6 to the right, and 4×9=36. To multiply 8 by 9, count 8 fingers from the left (this should be over to your right middle finger). There are 7 fingers to the left of it and 2 to the right, and 8×9=72.
Recently, Numberphile posted a video called Pi and Four Fingers, which includes a portion of an interview with Simon Singh, author of the recently published book The Simpsons and Their Mathematical Secrets. In the video, Singh notes that even though almost all of the characters on the Simpsons have eight fingers, they use base* ten anyway. Sadly, the Simpsons can't use the finger rule I use to multiply by nine. But if they were willing to think in base eight—as eight-fingered creatures, this would be the "natural" base for them—they could use the finger rule to multiply by seven.
The finger rule works for decimal thinkers with ten fingers because 9 is one less than the base of our number system. Each time you add 9, you are adding 10 and subtracting 1. In terms of finger multiplication, this means that by moving one finger to the right (going up to the next multiple of 9), you are putting one more finger in the left column and subtracting one from the right column.
If we're willing to think in different bases, we can use the finger rule with a different number of fingers and remove the trick's insidious bias against people and cartoon characters with nonstandard numbers of fingers on their hands. Anne Boleyn** could have used it to multiply by 10 as long as she was willing to think in base 11, and the Simpsons could use it to multiply by seven as long as they interpreted the result in base eight.
It's easier for me to pretend I have fewer fingers than I do than it is to add an extra one, so I'll show the finger rule in base eight. (Plus, it's already easy to multiply by ten, but seven is tougher, so the sevens trick might be more useful.) We'll pretend I don't have thumbs.
To multiply three by seven, count over three fingers from the left. You should be folding down the middle finger on your left hand.
There are two fingers to the left and five to the right. But 3×7 isn't 25, is it? Yes it is! We're counting in base eight, so the two represents two eights, or sixteen, not twenty. Sixteen plus five is twenty-one, so we got the right answer. Hurrah!
If you're already good at multiplying by seven, this trick might help your facility with base eight. If you're already good at thinking in base eight, maybe this trick will help you learn your seven times table. Either way, it's definitely a good way to impress your date!***
*If you'd like a refresher on what the "base" of a numbering system is, here you go. Each place in a base 10 number represents a power of 10. The furthest right place is the 100, or ones, place. (Any number other than 0 raised to the zeroth power is 1.) The next place to the left is the 101, or tens, place, the next place is the 102, or hundreds place, and so on. Other bases work the same way. For example, if we use the digits 1 and 0 to write numbers in binary, we would write the number twenty-two as 10110. There is one sixteen (24), one four(22), and one two (21). In base eight, we have a ones place, an eights place, a sixty-fours (82) place, and so on. The number one hundred written in base eight is 144 (sixty-four+four×eight+four), and sixty-four written in base eight is 100.
**Anne Boleyn probably didn't have eleven fingers. The first record of this abnormality came about fifty years after her death from someone who blamed her for Henry VIII's renunciation of Catholicism, so it is not given much credence.
***Roots of Unity accepts no responsibility for any negative experiences you may have as a result of using the sevens trick to impress a date.
It's hard to take photographs of your own hands! Pictures in this post were taken by Jon Chaika.