A few years ago now, I wrote about the launch of Alex’s Adventures in Numberland, a maths book for the general public by Alex Bellos.
This year, I have read the follow-up, Alex through the Looking Glass, which I got in a much more low-key fashion: in the station bookshop in Guildford.
It is a very sure-footed performance, and better focussed than its predecessor (in the same way that Lewis Carroll’s second Alice book is, with its framing device of a chess game). Most of the book is about numbers and geometry: favourite numbers, Benford’s and Zipf’s laws, triangles, conics, and circles, exponential growth and decay, negative and imaginary numbers, calculus, foundations, and a very nice final chapter on cellular automata including the description of the first self-reproducing Game of Life configuration to be discovered, Andrew Wade’s Gemini.
Something I learned that I didn’t know before. Two numbers are written on separate pieces of paper which are placed face down on a table. You are allowed to look at one, and then you must say whether it is larger or smaller than the other. There is a strategy which gives you a better than even chance of winning! Seems impossible? Answer at the end. (The small print: the two numbers are real numbers and are unequal.)
And here is something Alex missed. On page 138 he explains clearly the “rule of 72”, which says, in essence, that a sum of money at x percent compound interest will double in approximatelly 72/x years. (I learned this rule with 70 in place of 72.) Then on page 148 he remarks that because the logarithm of 2 is 0.693, we have 2x = e0.693x. But he doesn’t connect these two facts, which show that the correct rule is a “rule of 69.3”, so that the one I learned was slightly more accurate than Alex’s.
Here is another unremarked connection. The pattern produced by the cellular automaton using “rule 90” is described as the Sierpinski triangle, which in a sense is correct (it is discrete, but by re-scaling as it grows we can produce a sequence of figures which converge to Sierpinski’s triangle). But this pattern is something else too: it is Pascal’s triangle mod 2 (live and dead cells corresponding to odd and even entries).
And here is the solution to the puzzle earlier. Choose a random number. (I don’t care how you do it as long as every non-empty interval has positive probability of occurring.) If x is greater than the number on the paper you looked at, you guess that the other number is also greater; if it is less, you guess that the other number is as well. Now, if in reality x is smaller than either of the numbers, or greater than either, then you have a 50% chance of being right; but if x lies between the two, you will definitely be correct, so you have improved your odds.