## Alex through the looking glass

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. I count all the things that need to be counted.
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### 3 Responses to Alex through the looking glass

1. Gordon Royle says:

Is the 69.3 rather than 72 due to continuous compounding of interest rather than, say, annual?

• Peter Cameron says:

Yes, but the convergence is quite fast. And, as Alex points out, lenders in the UK are required to give the effective rate of interest assuming continuous compounding, and 69.3 would be the correct figure to use with this.

2. Mike Osborne says:

It is not true that lenders are required to calculate the effective rate assuming continuous compounding. Consumer credit legislation in most countries is similar to the US Truth in Lending Act. Lenders are required to calculate the periodic rate of interest on the same basis as the repayments. For example, if the repayments on a car loan are monthly then the lender is required to calculate the monthly rate of interest and annualise it to the annual percentage rate (APR). The lender must quote the APR to the consumer. The convention about annualisation varies from country to country, e.g. in the US the monthly rate is multiplied by 12 and in the UK it is compounded. On this basis, when rates are low (say, 1%, as now), 69 is a better figure for ‘the rule.’ But when rates are 5% or 10%, 72 works better.

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