100 Essential Things You Didn't Know You Didn't Know Read online

  The origin of all these tens in the counting system is not hard to find. It is at our fingertips. Most ancient human cultures used their fingers in some way for counting. As a result you find counting systems based on groups of five (fingers of one hand), ten (fingers of both hands), groups of twenty (fingers plus toes), or mixtures of all or some of these systems. Our own counting system betrays a complicated history in which different counting systems merged to form new ones by the presence of old words that reflect the previous base. Thus we have a word like ‘dozen’, for 12, or ‘score’ (derived from the old Saxon word sceran, meaning to shear or to cut) for 20, with its interesting triple meaning of 20, to make a mark or to keep count. All three meanings reflect the time when tallies were kept on pieces of wood by marking (scoring) them in groups of 20.

  Despite the ubiquity of the base 10 counting system in early culture, there is one unusual case where a Central American Indian society used a base 8 counting system. Can you think why this might be? I used to ask mathematicians if they could think of a good reason, and they usually responded by saying that 8 was a good number, to use because it has lots of factors, it divides exactly by 2 and 4, so you can divide portions into quarters without creating a new type of quantity that we call a fraction. The only time I got the right answer though was when I asked a large group of 8-yearold children and one girl immediately produced the answer: they were counting the gaps between their fingers. If you hold things between your fingers, strings or pieces of material, this is a natural way to count. The base eighters were finger counters too.

  59

  Getting a Mandate

  Democracy used to be a good thing, but now it has gotten into the wrong hands.

  Jesse Helms

  Politicians have a habit of presuming that they have a much greater mandate than they really do. If you have a roster of policies on offer to the electorate, the fact that you gained the most voters overall does not mean that a majority of voters favour each one of your policies over your opponent’s alternative. And if you win the election by a narrow margin, what sort of mandate do you have?

  For simplicity, assume there are just two candidates (or parties) in the election. Suppose the winner gets W votes and the loser gets L votes, so the total number of valid votes cast was W+L. In any number of events of this size, the random statistical ‘error’ that you expect to occur is given by the square root of W+L, so if W+L = 100 there will be a statistical uncertainty of 10 in either direction. In order for the winner of the election to be confident that they haven’t won because of a significant random variation in the whole voting process – counting, casting and sorting votes – we need to have the winning margin greater than the random variation:

  W−L > √(W+L)

  If 100 votes were cast, then the winning margin needs to exceed 10 votes in order to be convincing. As an example, in the 2000 US Presidential electionfn1 Bush received 271 electoral college votes and Gore received 266. The difference was just 5, far less than the square root of 271 + 266, which is about 23.

  More amusingly, it is told that Enrico Fermi, the great Italian high-energy physicist who was a key player in the creation of the first atomic bomb – and a very competitive tennis player – once responded to being beaten at tennis by 6 games to 4 by remarking that the defeat was not statistically significant because the margin was less than the square root of the number of games they played!

  Let’s suppose you have won the election and have a margin of victory large enough to quell concerns about purely random errors being responsible, how large a majority in your favour do you think you need in order to claim that you have a ‘mandate’ from the electorate for your policies? One interesting suggestion is to require that the fraction of all the votes that the winner receives, W/(W+L), exceeds the ratio of the loser’s votes to the winner’s votes, L/W. This ‘golden’ mandate condition is therefore that

  W/(W+L) > L/W

  This requires that W/L > (1+√5)/2 = 1.61, which is the famous ‘golden ratio’. This means that you would require a fraction W/(W+L) greater than 8/13 or 61.5% of all the votes cast for the two parties. In the last general election in the UK, Labour won 412 seats and the Conservatives 166, so Labour had 71.2% of these 578 seats, enough for a ‘golden’ mandate. By contrast, in the 2004 US Election, Bush won 286 electoral votes, Kerry won 251, and so Bush received only 53.3% of the total, less than required for a ‘golden’ mandate.

  fn1 Some aspects of this election remain deeply suspicious from a purely statistical point of view. In the crucial Florida vote the result of the recount was very mysterious. Just re-examining ballot papers produced a gain of 2,200 votes for Gore and 700 for Bush. Since one would expect there to be an equal chance of an ambiguous ballot being cast for either candidate, this huge asymmetry in the destination of the new ballots accepted in the recount suggests that something else of a non-random nature was going on in either the first count or the recount.

  60

  The Two-headed League

  But many that are first shall be last; and the last shall be first.

  Gospel of St. Matthew

  In 1981 the Football Association in England made a radical change to the way its leagues operated in an attempt to reward more attacking play. They proposed that 3 points be awarded for a win rather than the 2 points that had traditionally been the victor’s reward. A draw still received just 1 point. Soon other countries followed suit, and this is now the universal system of point scoring in football league competitions all over the world. It is interesting to look at the effect this has had on the degree of success that a dull, non-winning team can have. In the era of 2 points for a win it was easily possible to win the league with 60 points from 42 games, and so a team that gained 42 points from drawing all its games could finish in the top half of the league – indeed, Chelsea won the old First Division Championship in 1955 with the lowest ever points total of 52. Today, with 3 points for a win, the champion side needs over 90 points from its 38 games and an all-drawing side will find its 42 points will put it three or four from the bottom, fighting against relegation.

  With these changes in mind, let’s imagine a league where the football authorities decide to change the scoring system just after the final whistles blow on the last day of the season. Throughout the season they have been playing 2 points for a win and 1 point for a draw. There are 13 teams in the league and they play each other once, so every team plays 12 games. The All Stars win 5 of their games and lose 7. Remarkably, every other game played in the league is drawn. The All Stars therefore score a total of 10 points. All the other teams score 11 points from their 11 drawn games, and 7 of them score another 2 points when they beat the All Stars, while 5 of them lose to the All Stars and score no more points. So 7 of the other teams end up with 13 points and 5 of them end up with 12 points. All of them have scored more than the All Stars, who therefore find themselves languishing at the bottom of the league table.

  Just as the despondent All Stars have got back to the dressing room after their final game and realise they are bottom of the league, facing certain relegation and probable financial ruin, the news filters through that the league authorities have voted to introduce a new points scoring system and apply it retrospectively to all the matches played in the league that season. In order to reward attacking play they will award 3 points for a win and 1 for a draw. The All Stars quickly do a little recalculating. They now get 15 points from their 5 wins. The other teams get 11 points from their 11 drawn games still. But now the 7 that beat the All Stars only get another 3 points each, while the 5 that lost to them get nothing. Either way, all the other teams score only 11 points or 14 points and the All Stars are now the champions!

  61

  Creating Something out of Nothing

  Mistakes are good. The more mistakes, the better. People who make mistakes get promoted. They can be trusted. Why? They’re not dangerous. They can’t be too serious. People who don’t make mistakes eventually fall off cliffs,
a bad thing because everyone in free fall is considered a liability. They might land on you.

  James Church, A Corpse in the Koryo

  If you are one of those people who have to give lectures or ‘presentations’ using a computer package like PowerPoint, then you have probably also discovered one of its weaknesses – especially if you are an educator. When your presentation ends it is typically time for questions from the audience about what you have said. One of the facts of life about such questions is that they are very often most easily and effectively answered by drawing or writing something. If you are at a blackboard or have an overhead projector with acetate sheets and a pen in front of you, then any picture you need is easily drawn. But armed just with your standard laptop computer you are a bit stuck. You can’t easily ‘draw’ anything over your presentation unless you have a ‘tablet PC’. Which all goes to show how much we rely on pictures to explain what we mean. They are more direct than words. They are analog, not digital.

  Some mathematicians are suspicious of drawings. They like proofs that make no reference to a picture you might have drawn in a way that biases what you think might be true. But most mathematicians are quite the opposite. They like pictures and see them as a vital guide to seeing what might be true and how to go about showing it. Since that opinion is in the majority, let’s show something that will make the minority happy. Suppose you have 8 × 8 square metres of expensive flooring that is made up of four pieces – two triangles and two quadrilaterals – as shown in the floor plan here.

  It is easy to see that the total area of the square is 8 × 8 = 64 square metres. Now let’s take our four pieces of flooring with the given dimensions and lay them down in a different way. This time to create a rectangle, like this:

  Something strange has happened though. What is the area of the new rectangular carpet? It is 13 × 5 = 65 square metres.18 We have created one square metre of carpet out of nothing! What has happened? Cut out the pieces and try it for yourself.

  62

  How to Rig An Election

  I will serve as a consultant for your group for your next election. Tell me who you want to win. After talking to the members I will design a ‘democratic procedure’ which ensures the election of your candidate.

  Donald Saari19

  As we have already seen in Chapter 14, elections can be tricky things. There are many ways to count votes, and if you do it unwisely you can find that candidate A beats B who beats C who loses to A! This is undesirable. Sometimes we find ourselves voting several times on a collection of candidates with the weakest being dropped at each stage, so that the votes for that candidate can be transferred to other candidates in the next round of voting.

  Even if you are not spending your days casting formal votes for candidates, you will be surprised how often you are engaged in voting. What film shall we go to see? What TV channel shall we watch? Where shall we go on holiday? What’s the best make of fridge to buy? If you enter into discussion with others about questions that different possible answers then you are really engaged in casting a ‘vote’ – your preference – and the successful ‘candidate’ is the choice that is finally made. These decisions are not reached by all parties casting a vote. The process is usually much more haphazard. Someone suggests one film. Then someone suggests another because it is newer. Then someone says the newer one is too violent and we should pick a third one. Someone turns out to have seen that one already so it’s back to the first choice. Someone realises it’s no good for children so they suggest another. People are wearying by now and agree on that proposal. What is happening here is interesting. One possibility at a time is being considered against another one, and this process is repeated like rounds in a tournament. You never consider all the attributes of all the possible films together and vote. The outcome of the deliberations therefore depend very strongly on the order in which you consider one film versus another. Change the order in which you consider the films and the attributes you use for comparing them and you can end up with a very different winner.

  It’s just the same with elections. Suppose you have 24 people who have to choose a leader from 8 possible candidates (A, B, C, D, E, F, G and H). The ‘voters’ divide up into three groups who decide on the following orderings of their preferences among the candidates:

  1st group: A B C D E F G H

  2nd group: B C D E F G H A

  3rd group: C D E F G H A B

  At first glance it looks as if C is the preferred candidate overall, taking 1st, 2nd and 3rd places in the three ranking lists. But H’s mother is very keen for H to get the leader’s job and comes to ask if we can make sure that H wins the vote. It looks hopeless as H is last, second last and third last on the preference lists. There must be no chance of H becoming leader. We make it clear to H’s mother that everything must follow the rules and no dishonesty is allowed. So, the challenge is to find a voting system that makes H the winner.

  All we need to do in order to oblige is set up a tournament-style election and pick the winner of each 2-person contest using the preferences of the three groups listed above. First, pit G against F, and we see F wins 3–0. F then plays E, and loses 3–0. E then plays D and loses 3–0. D then plays C and loses 3–0. C then plays B and loses 2–1. B then plays A and loses 2–1. That leaves A to play H in the final match up. H beats A 2–1. So H is the winning choice in this ‘tournament’ to decide on the new leader.

  What was the trick? Simply make sure the stronger candidates eliminate each other one by one in the early stages and introduce your ‘protected’ candidate only at the last moment, ensuring that they are compared only with other candidates they can beat. So, with the right seeding, a British tennis player can win Wimbledon after all.

  63

  The Swing of the Pendulum

  England swings like a pendulum do.

  Roger Miller

  The story is told that in the sixteenth century the great Italian scientist Galileo Galilei used to amuse himself by watching the swinging of the great bronze chandelier that hung from the ceiling of the cathedral in Pisa. It may have been set in motion to spread the aroma of incense, or it may have been disturbed by the need to lower it to replenish the candles. What he saw fascinated him. The rope suspending the chandelier from the ceiling was very long and so the chandelier swung backwards and forwards like a pendulum, very slowly: slow enough to time how long it would take to make a complete journey out and back to its starting point. Galileo observed what happened on many occasions. On each occasion the chandelier swung differently; sometimes making only very small swings; sometimes somewhat larger ones – but he had noticed something very important. The period of time taken by the swinging chandelier to complete a single out and back swing was the same regardless of how far it swung. Given a large push, it went further than if it were given a small one. But if it went further, it went faster and took the same time to get back to where it started from as if it were pushed very gently.

  This discoveryfn1 has far-reaching consequences. If you have a grandfather clock then you will have to wind it up about once a week. Galileo’s discovery means that if the pendulum has stopped, it doesn’t matter how your push starts it swinging again. As long as the push is not too big, it will take the same time to swing back and forth and the resulting ‘tick’ will have the same duration. Were it not so, then pendulum clocks would be very tedious objects. You would have to set the amplitude of the swinging pendulum exactly right in order that the clock keeps the same time as it did before it stopped. Indeed, Galileo’s acute observation is what led to the idea of the pendulum clock. The first working version was made by the Dutch physicist Christiaan Huygens in the 1650s.

  Finally, there is a nice test of whether a physicist’s logic is stronger than his survival instinct that exploits the swing of a pendulum. Imagine that we have a huge, heavy pendulum, like the cathedral chandelier that Galileo watched. Stand to one side of it and draw the pendulum weight towards you until it just touches the tip of your
physicist’s nose. Now just let the pendulum weight go. Don’t give it any push. It will swing away from your nose and then return back towards the tip of your nose. Will you flinch? Should you flinch? The answers are, well, ‘Yes and no’.fn2

  fn1 Galileo thought it was true for all swings of the chandelier, no matter how far they went. In fact, that is not true. It is true to very high accuracy for oscillations of ‘small’ amplitude. This type of oscillation is known to scientists as ‘simple harmonic motion’. It describes the behaviour of almost every stable system in Nature after it is slightly perturbed from its equilibrium state.

  fn2 The pendulum cannot swing back to a greater height than where it began from (unless someone hits it to give it more energy). In practice, the pendulum always loses a little of its energy overcoming air resistance and overcoming friction at the point of its support, and so it will never quite return to the same height that it began from. The physicist is actually quite safe – but will always flinch none the less.

  64

  A Bike with Square Wheels

  The bicycle is just as good company as most husbands and, when it gets old and shabby, a woman can dispose of it and get a new one without shocking the entire community.

  Ann Strong

  If your bike’s like my bike it’s got circular wheels. You might just have one of them, but you’ve probably got two of them, but either way they are circular. However, you will probably be surprised to learn that it didn’t have to be like this. You can have a perfectly steady ride on a bike with square wheels, as long as you ride on the right type of surface.

  The important feature of a rolling wheel for the cyclist is that you don’t keep jolting up and down as the bicycle moves forward. With circular wheels on a flat surface this is the case: the centre of the cyclist’s body moves forward in a straight line when the bicycle moves forward in a straight line without any slipping. Use square wheels on a flat surface and you are going to have a very uncomfortable up-and-down ride. But could there be a differently shaped road surface that results in a smooth ride when you are using square wheels? All we need to check is whether there is a shape that leads to a straight-line motion of the cyclist when he has square wheels.