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

  1 + ½ + ⅓ + ¼ + . . . + 1/n

  This is a famous series that mathematicians call the ‘harmonic’ series. Let’s label as H(n) its sum after n terms are totalled; so we see that H(1) = 1, H(2) = 1.5, H(3) = 1.833, H(4) = 2.083, and so on. The most interesting thing about the sum of this series is that it grows so very slowly as the number of terms increases,fn1 so H(256) = 6.12 but H(1,000) is only 7.49 and H(1,000,000) = 14.39.

  What does this tell us? Suppose that we were to apply our formula to the rainfall records for some place in the UK from 1748 to 2004 – a period of 256 years. Then we predict that we should find only H(256) = 6.12, or about 6 record years of high (or low) rainfall. If we look at the rainfall records kept by Kew Gardens for this period then this is the number of record years there have been. We would have to wait for more than a thousand years to have a good chance of finding even 8 record years. Records are very rare if events occur at random.

  In the recent past there has been growing concern around the world about the evidence for systematic changes in climate, so called ‘global warming’, and we have noticed an uncomfortably large number of local climatic records in different places. If new records become far commoner than the harmonic series predicts, then this is telling us that annual climatic events are no longer independent annual events but are beginning to form part of a systematic non-random trend.

  fn1 In fact, when n gets very large H(n) increases only as fast as the logarithm of n and is very well approximated by 0.58 + ln(n).

  29

  A Do-It-Yourself Lottery

  The Strong Law of Small Numbers: There are not enough small numbers to satisfy all the demands placed upon them.

  Richard Guy

  If you are in need of a simple but thought-provoking parlour game to keep guests amused for a while, then one you might like to try is something that I call the Do-It-Yourself Lottery. You ask everyone to pick a positive whole number, and write it on a card along with their name. The aim is to pick the smallest number that is not chosen by anyone else. Is there a winning strategy? You might think you should go for the smallest numbers, like 1 or 2. But won’t other people think the same, and so you won’t end up with a number that is not chosen by someone else. Pick a very large number – and there are an infinite number of them to choose from – and you will surely lose. It’s just too easy for someone else to pick a smaller number. This suggests that the best numbers are somewhere in between. But where? What about 7 or 11? Surely no one else will think of picking 7?

  I don’t know if there is a winning strategy, but what the game picks up on is our reluctance to think of ourselves as ‘typical’. We are tempted to think that we could pick a low number for some reason that no one else will think of. Of course, the reason why opinion polls can predict how we will vote, what we will buy, where we will go on holiday, and how we will respond to an increase in interest rates is precisely because we are all so similar.

  I have another suspicion about this game. Although there is an infinite collection of numbers to choose from, we forget about most of them. We set a horizon somewhere around 20, or about twice the number of people in the game, if it is larger, and don’t think anyone will pick anything bigger than this. We then exclude the first few numbers up to about 5 on the grounds that they are too obvious for no one else to choose as well and choose from those that remain with roughly equal probability.

  A systematic study of preferences would involve playing this game many times over with a large sample of players (say 100 in each trial) to look at the pattern of numbers chosen and the winning choices. It would also be interesting to see how players changed their strategy if they played the game over and over again. A computer simulation of the game is not necessarily of any use because it needs to be told a strategy to adopt. Clearly the numbers are not chosen at random (in which case all numbers would get chosen with equal likelihood). Psychology is important. You try to imagine what others will choose. But the temptation to think that you don’t think like anyone else is so strong that almost all of us fall for it. Of course, if there really was a definite strategy for picking the lowest number, everyone would be acting logically to adopt it but that would prevent them choosing a number that no one else chose, and they could never win with that strategy.

  30

  I Do Not Believe It!

  It is quite a three-pipe problem and I beg that you won’t speak to me for fifty minutes.

  Sherlock Holmes

  You are appearing live in a TV game show. The manic presenter shows you three boxes, labelled A, B and C. One of them contains a cheque made out to you for £1 million. The other two contain photos of the presenter. He knows which box contains the cheque, and that cheque will be yours if you pick the right box. You go for Box A. The presenter reaches for Box C and shows everyone that it contains one of the pictures of himself. The cheque must be in Box A or Box B. You picked Box A. The presenter now asks you if you want to stick with your original choice of Box A or switch to Box B. What should you do? Maybe you have an impulse that urges you to switch your choice to Box B, while another voice is saying, Stick with Box A; he’s just trying to make you switch to a less expensive option for his employers.’ Or perhaps a more rational voice is telling you that it can’t possibly make any difference because the cheque is still where it always was and you either guessed it right first time or you didn’t.

  The answer is remarkable. You should switch your choice to Box B! If you do this you will double your chance of picking the box that contains the cheque. Stick with the choice of Box A and you have a 1/3 chance of winning the cheque; switch to Box B and the chance increases to 2/3.

  How can this be? At first, there is a 1 in 3 chance that the cheque is in any box. That means a 1/3 chance that it’s in A and a 2/3 chance that it’s in B or C. When the presenter intervenes and picks a box, it doesn’t change these odds because he always picks a box that doesn’t contain the cheque. So after he opens Box C there is still a 1/3 chance that the cheque is in A but now there is a 2/3 chance that it is in B because it definitely isn’t in C. You should switch.

  Still not convinced? Look at it another way. After the presenter opens Box C you have two options. You can stick with your choice of Box A and this will ensure you will win if your original choice was right. Or you can switch boxes, to B, in which case you will be a winner only if your original choice was wrong. Your first choice of Box A will be right 1/3 of the time and wrong 2/3 of the time. So changing your box will get the cheque 2/3 of the time while staying with your first choice will be a winning strategy only 1/3 of the time.

  You should by now be convinced by this mind-changing experience.

  31

  Flash Fires

  I will show you fear in a handful of dust

  T.S. Eliot, The Waste Land

  One of the lessons that we have learned from a number of catastrophic fires is that dust is lethal. A small fire in an old warehouse can be fanned into an explosive inferno by efforts to extinguish it if those efforts blow large amounts of dust into the air where it combusts and spreads the fire through the air in a flash. Anywhere dark and unvisited – under escalators or tiers of seats, or in neglected storage depositories – where dust can build up unnoticed in significant quantities is a huge fire risk.

  Why is this? We don’t normally think of dust as being particularly inflammable stuff. What is it that transforms it into such a deadly presence? The answer is a matter of geometry. Start with a square of material and cut it up into 16 separate smaller squares. If the original square was of size 4 cm × 4 cm, then each of the 16 smaller squares will be of size 1 cm × 1 cm. The total surface area of material is the same – 16 square centimetres. Nothing has been lost. However, the big change is in the length of the exposed edges. The original square had a perimeter of length 16 cm, but the separate smaller squares each have a perimeter of 4 cm and there are 16 of them, so the total perimeter has grown four times bigger, to 4 × 16 cm = 64 cm.

  If w
e do this with a cube, then it would have 6 faces (like a die) of size 4 cm × 4 cm, and each would have an area of 16 sq cm, so the total surface area of the big cube would be 6 × 16 sq cm = 96 sq cm. But if we chopped up the big cube into 64 separate little cubes, each of size 1 cm × 1 cm × 1 cm, the total volume of material would stay the same, but the total surface area of all the little cubes (each with six faces of area 1 cm × 1 cm would have grown to be 64 × 6 × 1 sq cm = 384 sq cm.

  What these simple examples show is that if something breaks up into small pieces then the total surface that the fragments possess grows enormously as they get smaller. Fire feeds on surfaces because this is where combustible material can make contact with the oxygen in the air that it needs to sustain itself. That’s why we tear up pieces of paper when we are setting a camp fire. A single block of material burns fairly slowly because so little of it is in direct contact with the surrounding air and it is at that boundary with the air that the combustion happens. If it crumbles into a dust of fragments, then there is vastly more surface area of material in contact with the air and combustion occurs everywhere, spreading quickly from one dust fragment to another. The result can be a flash fire or a catastrophic firestorm when the density of dust motes in the air is so great that all the air is caught up in a self-sustaining inferno.

  In general, many small things are a bigger fire hazard than one large thing of the same volume and material composition. Careless logging in the forest, which takes all the biggest trees and leaves acres of splintered debris and sawdust all around the forest floor is a topical present-day example.

  Powders are dangerous in large quantities. A major disaster happened in Britain in the 1980s when a large Midlands factory making custard powder caught fire. Just a small sprinkling of powdered milk, flour or sawdust over a small flame will produce a dramatic flame bursting several metres into the air. (Don’t try it! Just watch the film.fn1)

  fn1 For a sequence of pictures of a demonstration by Neil Dixon for his school chemistry class see http://observer.guardian.co.uk/flash/page/0,,1927850,00.html

  32

  The Secretary Problem

  The chief cause of problems is solutions.

  Sevareid’s Rule

  There is a classic problem about how to make a choice from a large number of candidates; perhaps a manager is faced with 500 applicants for the post of company secretary, or a king must choose a wife from all the young women in his kingdom, or a university must choose the best student to admit from a long list of applicants. When the number of candidates is moderate you can interview them all, weigh each against the others, re-interview any you are unsure about and pick the best one for the job. If the number of applicants is huge this may not be a practical proposition. You could just pick one at random, but if there are N applicants the chance of picking the best one at random is only 1/N, and with N large this is a very small chance – less than 1% when there are more than 100 applicants. As a route to the best candidate, the first method of interviewing everyone was time-consuming but reliable; the random alternative was quick but quite unreliable. Is there a ‘best’ method, somewhere in between these two extremes, which gives a pretty good chance of finding the best candidate without spending exorbitant amounts of time in so doing?

  There is, and its simplicity and relative effectiveness are doubly surprising, so let’s set out the ground rules. We have N known applicants for a ‘job’, and we are going to consider them in some random order. Once we have considered a candidate we can mark how they stand relative to all the others that we have seen, although we are only interested in knowing the best candidate we have seen so far. Once we have considered a candidate we cannot recall them for reconsideration. We only get credit for appointing the best candidate. All other choices are failures. So, after we have interviewed any candidate all we need to note is who is the best of all the candidates (including them) that we have seen so far. How many of the N candidates do we need to see in order to have the best chance of picking the strongest candidate and what strategy should we adopt?

  Our strategy is going to be to interview the first C of the N candidates on the list and then choose the next one of the remaining candidates we see who is better than all of them. But how should we choose the number C? That is the question.

  Imagine we have three candidates 1, 2 and 3, where 3 is actually better than 2, who is better than 1; then the six possible orders that we could see them in are

  123 132 213 231 312 321

  If we decided that we would always take the first candidate we saw, then this strategy would pick the best one (number 3) in only two of the six interview patterns so we would pick the best person with a probability of 2/6, or 1/3. If we always let the first candidate go and picked the next one we saw who had a higher rating, then we would get the best candidate in the second (132), third (213), and the fourth cases (231) only, so the chance of getting the best candidate is now 3/6, or 1/2. If we let two candidates go and picked the third one we saw with a higher rating then we would get the best candidate only in the first (123) and third (213) cases, and the chance of getting the best one is again only 1/3. So, when there are three candidates the strategy of letting one go and picking the next with a better rating gives the best chance of getting the best candidate.

  This type of analysis can be extended to the situation where the number of candidates, N, is larger than three. With 4 candidates, there are 24 different orderings in which we could see them all. It turns out that the strategy of letting one candidate go by and then taking the next one that is better still gives the best chance of finding the best candidates, and it does so with a chance of successfn1 equal to 11/24. The argument can be continued for any number of applicants and the result of seeing the first 1, or 2, or 3, or 4, and so on, candidates and then taking the next one that is better in order to see how the chance of getting the best candidate changes.

  As the number of candidates increases the strategy and outcome get closer and closer to one that is always optimal. Consider the case where we have 100 candidates. The optimal strategy6 is to see 37 of them and then pick the next one that we see who is better than any of them and then see no one else. This will result in us picking the best candidate for the job with a probability of about 37.1% – quite good compared with the 1% chance if we had picked at random.7

  Should you use this type of strategy in practice? It is all very well to say that when you are interviewing job candidates you should interview all of them, but what if you apply the same reasoning to a search process for new executives, or your ‘search’ for a wife, for the outline of the next bestseller or the perfect place to live? You can’t search for your whole lifetime. When should you call a halt and decide? Less crucially, if you are looking for a motel to stay in or a restaurant to eat at, or searching for the best holiday deal on line or the cheapest petrol station, how many options should you look at before you take a decision? These are all sequential choice problems of the sort we have been looking at in the search for an optimal strategy. Experience suggests that we do not search for long enough before making a final choice. Psychological pressures, or simple impatience (either our own or that of others), push us into making a choice long before we have seen a critical fraction, 37 per cent, of the options.

  fn1 Picking the first candidate or the last always gives a chance of 1/4, letting 2 candidates go gives a chance of 5/12. Letting one go gives a chance of 11/24, which is optimal.

  33

  Fair Divorce Settlements: the Win–Win Solution

  Conrad Hilton was very generous to me in the divorce settlement. He gave me 5,000 Gideon Bibles.

  Zsa Zsa Gabor

  ‘It’s nice to share, Dad,’ our three-year old son once remarked as he looked at my ice cream after finishing his own. But sharing is not so simple. If you need to divide something between two or more people what should you aim to do? It is easy to think that all you need is to make a division that you think is fair, and for two people this means divid
ing the asset in half. Unfortunately, although this might work when dividing something that is very simple, like a sum of money, it is not an obvious strategy when the asset to be shared means different things to different people. If we need to divide an area of land between two countries then each might prize something, like water for agriculture or mountains for tourism, differently. Alternatively, the things being divided might involve undesirables – like household chores or queuing.

  In the case of a divorce settlement there are many things that might be shared, but each person places a different value on the different parts. One might prize the house most, the other the collection of paintings or the pet dog. Although you, as a possible mediator, have a single view of the value of the different items to be shared, the two parties ascribe different values to the parts of the whole estate. The aim of a mediator must be to arrive at a division that both side are happy with. That need not mean that the halves are ‘equal’ in any simple numerical sense.

  A simple and traditional way to proceed is to ask one partner to specify a division of assets into two parts and then allow the other partner to choose which of the two parts they want. There is an incentive for the person who specifies the division to be scrupulously fair because they may be on the receiving end of any unfairness if their partner chooses the ‘better’ piece. This method should avoid any envy persisting about the division process (unless the divider knows something about the assets that the other doesn’t – for example, that there are oil reserves under one area of the land). Still, there is a potential problem. The two partners may still value the different parts differently so what seems better for one may not seem so from the other’s point of view.