When I was young, I read lots of books with titles (or at least subheadings) along the lines of, “Amaze Your Friends and Confound Your Enemies” – a lot of them were filled with tricks and oddities like the Birthday Paradox, or the old saw about the elephant from Denmark.
It turns out that if you read and continue to read enough of this kind of thing, you can continue to amaze your friends well into adulthood! A lot of times this means appearing to be good at mental arithmetic. And the trick to being good at mental arithmetic is not to be especially fast at rote calculation. It lies in a web of knowledge about numbers.
An aside about maths teaching
I often think that number theory is poorly served by the mathematical curriculum almost everywhere. Kids learn times tables and are introduced to prime numbers, and then I think the number theory track more or less stops, and a student doesn’t meet it again until university-level maths. Which is a shame, because there are many interesting and fun problems in number theory that can be easily stated and understood by a ten-year-old but which still remain unsolved. Also, a more continuous (ha!) grounding in number theory would give us a better understanding of some very important features of the modern world – an obvious example being cryptography.
I was lucky enough, as a 12-13 year old, to have a recreational maths class at school for a year where we tackled problems together. It was distinctly constructivist in nature – whether by design or not – and it was a really fun class because we were typically all trying to solve a hard problem over the course of a few weeks. “Copying” other people’s work and building on it towards a solution was par for the course – as in real life! We took inspiration from the writings of people like Martin Gardner, Sam Loyd, Raymond Smullyan, Henry Dudeney and Eric Emmet. The teacher would ramp up the difficulty of problems as we went – and in mathematics there is almost always a way, having solved one problem, to remove a constraint or make it more general in some way to provide a step up to the next level.
A typical class exchange:
Teacher: Who can tell me how many squares there are on a chessboard?
Student A: 64!
Teacher: Ah yes. Correct. But I see you’re only counting the squares that are one square in size, as it were. I think there might be more squares there…
Student A: …
Student B: Wait a minute…
Class: *realization* *time passes, working-out*
Teacher: Correct. For your chessboard there. But my chessboard has n squares on a side, not 8.
Class: *argh* *more time passes, probably a week*
Class: Um… (a few tries, and then)… n(n+1)(2n+1)/6 ?
Teacher: Right! But… hm… my chessboard got broken, now it’s not square any more, it’s n by m. Oh, and I want to know how many rectangles are on it.
Class: *mind blown*
Associations – it’s what brains do
Anyway, the human brain is fantastically good at constructing associations. And being “good at maths” is about having lots of those associations when it comes to numbers. Mathematicians often have this. Computer people have this facility, when it comes to powers of 2, and it looks astounding to muggles when it comes out in another context (eg Biology class):
Teacher (thinking he is asking a hard question): This germ divides in two every hour. If we start with just one germ here, how many will we have after a day?
Nerdette in the back row (instantly): 16,777,216
Rest of the class: How the hell…?
When you have enough associations, they start to overlap and provide multiple ways to an answer. I was recently out with a group of friends and at the end of the night we came to pay the bill. There were seven of us, and the bill was $195. I immediately knew it was about $28 each, and I didn’t have to calculate, because of some associations:
- 196 is 14 squared, which is 7 * 28. Immediate answer. (Square numbers up to 20 or so – very useful to know.)
- In the UK weight is often measured in stones. 14lbs = 1 stone, and I know that 98lbs = 7 stones or 7*14 = 98. And as 98*2 = 196, because (100-2)*2 = 200-4 = 196, so 14*2 = 28. Corroboration by overlapping association.
“Wizardry” with 1/7
Another second’s thought provides the exact amount per person, because 1/7 is a useful and interesting fraction to know. It has a recurring 6-digit pattern, and it cycles. Once you know the six digits of 1/7, it’s trivial to figure out/remember the other fractions:
- 1/7 = 0.142857142857142857…
- 2/7 = 0.285714…
- 3/7 = 0.428571…
- 4/7 = 0.571428…
- 5/7 = 0.714285…
- 6/7 = 0.857142…
This is a fun trick for kids: compute 142857 * 2, 142857 * 3, etc and see how the digits cycle. Then try 142857 * 7… and you get 999999. Neat.
Anyway, 28*7=196 but the bill is $195, which means actually everyone pays $28, less 1/7 of a dollar. So the exact figure is $27 and 85.714285… cents.
Fun for kids
When numbers are your friends, it’s easy to look like you’re a wizard. And it’s really just about forming those associations. When I see 41, I think of Euler’s famous expression x2 + x + 41 which is prime for every x from 0 to 39. When I see 153, I think, “Hello, 13 + 53 + 33!” And similarly for many other numbers and mathematical techniques, thanks to all that reading and playing with numbers as a child. These days I entertain my kids by having them do fun math tricks like:
Enter any three-digit number into your calculator (say 456)
Multiply by 7
Now multiply again by 11
Now multiply again by 13
The result is the original number “doubled up” (say 456456) – because 7 * 11 * 13 = 1001
I’m teaching them how to amaze their friends and confound their enemies!