I’ve been learning a bit about Golomb rulers recently: a ruler which has so few markings that if you can use it to measure some whole number length, then you can only measure it in one way. I first read about them on the monthly AMS feature column, about their applications inside and outside of maths (to codes, radar, sonar and suchlike), and then watched an excellent TED talk using one particularly useful two-dimensional generalisation (a Costas array) to create a piece of piano music so dissonant that no time-step or jump in pitch between any pair of notes (not necessarily adjacent) is the same.
A perfect Golomb ruler with markings at 0,1,4 and 6, can measure any whole length from 1 to 6, but each in only one way.
I started to wonder about whether Golomb rulers had anything to do with a real-life problem I’ve previously written about, that someone who owned a barge had wanted answered. He asked about how to cut ropes into different lengths so you can knot them together in combinations and get a large variety of new lengths. I had decided the link was merely thematic, until someone else asked me whether they were the same, and prompted me to have a closer look. It turns out the two are somewhat linked, and what’s more, the link can be viewed as a silly little piece of mathemagic!
I was recently travelling on a budget flight with a friend, with no assigned seating. Walking up to the queue, we wondered whether we would be able to sit together. As people joined behind us, my friend, who whole-heartedly detests maths in any disguise, pointed out that we certainly could. To be so full-up to mean that we couldn’t, each row of three would need at least two people sitting in it. Since we could see from the queue that we had more than a third of the passengers still left to board after us, then as long as we weren’t trampled in a boarding stampede, we could certainly sit together.
This was actually a subconscious application of the pigeonhole principle, one of the most intuitive theorems that mathematicians use. It states that if you have objects to put in boxes then one of those boxes must contain at least two of the objects.
Or if you put pigeons in pigeonholes (or envelopes in pigeon-holes) then, if , at least one of the holes has at least two pigeons. So far, the type of maths you wouldn’t be shocked to see explained on television by a primary-coloured puppet.
Some proverbs come in contradictory pairs, for instance “Too many cooks spoil the broth” and “Many hands make light work”. I’d like to present an example that I feel illustrates two of these simultaneously. Everyone knows “Don’t put all your eggs in one basket”, including banks, which diversify by holding many different investments of different types. But while it may be in banks’ best interests to lower their levels of risk through diversification, it may plausibly raise the risk of a system-wide failure: if all banks follow that same advice, the rest of us may be “In for a penny, in for a pound”.
This is the argument given by Beale et al. in their paper Individual versus systemic risk and the Regulator’s Dilemma. Here’s a simple example that illustrates the main idea, adapted from a related comment in the earlier paper Systemic risk: the dynamics of model banking system.
Some people are playing a dice game. Each of them rolls a single fair die once and receives £1 for rolling a 1, £2 for a 2, and so on, up to £6 for a 6. However, afterwards they each have to pay £1.50 for the privilege of playing this rewarding game. Each of them starts out with no money, and so if they roll a 1, they go bankrupt.
If you play this game, there’s a chance that you’d go bankrupt on your roll. Let’s say ten people play the game: the chance of all of them going bankrupt on their single throw is , which is about one in sixty million.