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

by John D. Barrow

  Suppose that you wanted to conceal your identity in a similar way today, how could you use simple maths to do it? Pick a couple of very large prime numbers, for example 104729 and 105037 (actually you want to pick much larger ones, with hundreds of digits, but these are big enough to get the general idea). Multiply them together to get the product: 11000419973. Incidentally, don’t trust your calculator, it probably can’t cope with such a large number and will round the answer off at some stage – my calculator gave the wrong answer 11000419970.

  Now, let’s go back to our secret publishing challenge. You want to publish your discovery and not reveal your identity publicly, but include a hidden ‘signature’ so that at some future date you can show that you wrote it. You could publish the book with that huge product of two prime numbers, 11000419973, printed in it at the back. You know the two factors (104729 and 105037) and can easily multiply them together to show that the answer is your product ‘code’. However, if other people start with 11000419973, they will not find it so easy to find the two factors. If you had chosen to multiply together two very large prime numbers, each with 400 digits, then the task of finding the two factors could take a lifetime, even if assisted by a powerful computer. It is not impossible to break our ‘code’, but using much longer numbers is an unnecessary level of security – it just needs to take a very long time.

  This operation of multiplying and factoring numbers is an example of a so-called ‘trapdoor’ operation (see Chapter 27). It’s quick and easy to go in one direction (like falling through a trapdoor), but longwinded and slow to go in the opposite direction (like climbing back up through the trapdoor). A more complicated version of multiplying two prime numbers is used as the basis for most of the world’s commercial and military codes today. For example, when you buy anything online and enter your credit card details into a secure website, the details are compounded with large prime numbers, transmitted to the company and then decrypted by prime number factorisation.

  fn1 In Italian: ‘Il numero selezionato da lei è inesistente.’

  91

  The Ice Skating Paradox

  After finishing dinner, Sidney Morgenbesser decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie, at which point Morgenbesser says, ‘In that case I’ll have the blueberry pie!’

  Academic legend

  When we make choices or cast votes, it seems rational to expect that, if we first chose K as the best among all the alternatives on offer, and then someone comes along and tells us that there is another alternative, Z, which they forgot to include, our new preferred choice will be to stick with K or to choose Z. Any other choice seems irrational because we would be choosing one of the options we rejected first time around in favour of K. How can the addition of the new option change the ranking of the others?

  The requirement that this should not be allowed to happen is so engrained in the minds of most economists and mathematicians that it is generally excluded by fiat in the design of voting systems. Yet, we know that human psychology is rarely entirely rational and there are situations where the irrelevant alternative changes the order of our preferences, as it did with Sidney Morgenbesser’s pie order (of course, he could have seen one of the pies in question between orders).

  A notorious example was a transport system that offered a red bus service as an alternative to the car. Approximately half of all travellers were soon found to use the red bus; half still used a car. A second bus, blue in colour, was introduced. We would expect one quarter of travellers to use the red bus, one quarter to use the blue bus, and one half to continue travelling by car. Why should they care about the colour of the bus? In fact, what happened was that one third used the red bus, one third the blue bus, and one third their car!

  There is one infamous situation where the effect of irrelevant alternatives was actually built into a judging procedure, with results so bizarre that they eventually led to the abandonment of that judging process. The situation in question was the judging of ice skating performances at the Winter Olympics in 2002, which saw the young American Sarah Hughes defeat favourites Michelle Kwan and Irina Slutskaya. When you watch skating on the television the scores for an individual performance (6.0, 5.9, etc.) are announced with a great fanfare. However, curiously, those marks do not determine who wins. They are just used to order the skaters. You might have thought that the judges would just add up all the marks from the two programmes (short and long) performed by each individual skater, and the one with the highest score wins the gold medal. Unfortunately, it wasn’t like that in 2002 in Salt Lake City. At the end of the short programme the order of the first four skaters was:

  Kwan (0.5), Slutskaya (1.0), Cohen (1.5), Hughes (2.0).

  They are automatically given the marks 0.5, 1.0, 1.5 and 2.0 because they have taken the first four positions (the lowest score is best). Notice that all those wonderful 6.0s are just forgotten. It doesn’t matter by how much the leader beats the second place skater, she only gets a half-point advantage. Then, for the performance of the long programme, the same type of scoring system operates, with the only difference being that the marks are doubled, so the first placed skater is given score 1, second is given 2, third 3 and so on. The score from the two performances are then added together to give each skater’s total score. The lowest total wins the gold medal.

  After Hughes, Kwan and Cohen had skated their long programmes Hughes was leading, and so had a long-programme score of 1, Kwan was second so had a score of 2, and Cohen lay third with a 3. Adding these together, we see that before Slutskaya skates the overall marks are:

  1st: Kwan (2.5), 2nd: Hughes (3.0), 3rd: Cohen (4.5)

  Finally, Slutskaya skates and is placed second in the long programme, so now the final scores awarded for the long programme are:

  Hughes (1.0), Slutskaya (2.0), Kwan (3.0), Cohen (4.0).

  The result is extraordinary: the overall winner is Hughes because the final scores are:

  1st: Hughes (3.0), 2nd: Slutskaya (3.0), 3rd: Kwan (3.5), 4th: Cohen (5.5)

  Hughes has been placed ahead of Slutskaya because, when the total scores are tied, the superior performance in the long programme is used to break the tie. But the effect of the poorly constructed rules is clear. The performance of Slutskaya results in the positions of Kwan and Hughes being changed. Kwan is ahead of Hughes after both of them have skated, but after Slutskaya skates Kwan finds herself behind Hughes! How can the relative merits of Kwan and Hughes depend on the performance of Slutskaya? The paradox of irrelevant alternatives rules ok.

  92

  The Rule of Two

  History is just one damn thing after another.

  Henry Ford

  Infinities are tricky things and have perplexed mathematicians and philosophers for thousands of years. Sometimes the sum of a never-ending list of numbers will become infinitely large; sometimes it will get closer and closer to a definite number; sometimes it will defy having any type of definite sum at all. A little while ago I was giving a talk about ‘infinity’ that included a look at the simple geometric series

  S = ½ + ¼ + ⅛ + + + + . . .

  And so on, forever: every term in the sum exactly half the size of its predecessor. The sum of this series is actually equal to 1, but someone in the audience who wasn’t a mathematician wanted to know if there was any way to show him that this is true.

  Fortunately, there is a simple demonstration that just uses a picture. Draw a square of size 1 × 1, so its area is 1. Now let’s divide the square in half by dividing it from top to bottom into two rectangles. Each of them must have an area equal to ½. Now divide one of these rectangles in two to make two smaller rectangles, each with area equal to one-quarter. Now divide one of these smaller rectangles in half to make two more rectangles, each of area equal to one-eighth. Keep on going like this, making a rectangle of half the area of the previous one
, and look at the picture. The original square has just had its whole area subdivided into a never-ending sequence of regions that fill it completely. The total area of the square is equal to the sum of the areas of the pieces that I have left intact at each stage of the cutting process, and the areas of these pieces is just equal to our series S. So the sum of the series S must be equal to 1, the total area of the square.

  Usually when we encounter a series like S for the first time, we work out its sum in another way. We notice that each successive term is one half of the previous one and then multiply the whole series by 1/2 so we have

  ½ × S = ¼ + ⅛ + + + + . . .

  But we notice that the series on the right is just the original series, S, minus the first term, which is ½. So we have that ½× S = S – ½, and S = 1 again.

  93

  Segregation and Micromotives

  The world is full of obvious things which nobody by any chance ever observes.

  Sherlock Holmes in The Hound of the Baskervilles

  In many societies there is a significant segregation between communities of different types – racial, ethnic, religious, cultural and economic. In some cases there is loudly stated dislike of one community by another, but in others there doesn’t seem to be any overt attempt to be separate and the different communities get on well as individuals in their spheres of activity. However, the tendencies of individuals may not be a good guide to the behaviour of a group because of the interplay between many individual choices. When some of the statistical methods used by scientists to study the collective behaviour of large numbers of things are applied to populations of people, some very simple but unexpected truths emerge.

  In 1978 Thomas Schelling, an American political scientist, decided to investigate how racial segregation came about in American cities. Many people assumed that it was simply a result of racial intolerance. Some thought that it might be overcome by throwing different communities together in a random mix, but were surprised to find that the result was always segregation into different racial sub-communities again, even though the residents seemed to be quite tolerant of other ethnic groups when questioned in surveys. What emerged from a mathematical study of virtual societies using computer simulations was that very slight imbalances result in complete segregation despite an average tolerant outlook. Suppose that a family will move, because of intolerance or to avoid it, if more that one in three of its neighbours are different from them, but will stay put if fewer than one in five are different. In this situation a random mix of two sorts of families (‘blue’ and ‘red’) that differ in some way (race, religion or class, say) will gradually become more and more polarised, until eventually it is completely segregated into a totally blue and a totally red community, just like a mixture of oil and water,fn1 with empty ‘buffer’ regions between them. In a region with above average reds the blues move, leading to above average blues elsewhere, so the reds in that new above average blue neighbourhood move out, and so on. The moves all tend to be towards regions where there is an above average concentration of the mover’s type. The boundary regions between different regions are always very sensitive since single movers can tip the balance one way or the other. It is more stable to evolve towards these boundary regions being empty to create a buffer between the segregated communities.

  These simple insights were very important. They showed that very strong segregation was virtually inevitable in mixed communities and didn’t imply that there was serious intolerance. Segregation doesn’t necessarily mean prejudice – although it certainly can, as the examples of the United States, Rhodesia, South Africa and Yugoslavia show. Better to foster close links between the separate communities than to try to prevent them forming. Macrobehaviour is fashioned by micromotives that need not be part of any organised policy.

  fn1 Actually, this familiar example should not be examined too closely. If the dissolved air is removed from water, say by repeated freezing and thawing, it will mix with oil quite smoothly.

  94

  Not Going with the Flow

  Email is a wonderful thing for people whose role in life is to be on top of things. But not for me; my role is to be on the bottom of things.

  Don Knuth

  We have seen in the previous chapter an example of collective behaviour where no individual wants to find themselves in a significant minority. Not all situations are like this. If you are wanting to get away from it all at an idyllic island holiday destination, you want to be in the minority, not the majority, when it comes to everyone’s holiday destination of choice. The pub that ‘everyone’ chooses to go to because of the music or the food is going to turn out to be a far from ideal experience if you have to queue to get in, can’t find a chair and have to wait an hour to be served. You will do better at a less popular venue.

  This is like playing a game where you ‘win’ by being in the minority. Typically, there will be an average number of people who choose to go to each of the venues on offer, but the fluctuations around the average will be very large. In order to reduce them and converge on a strategy that is more useful, it is necessary to use past information about the attendance at the venue. If you just try to guess the psychology of fellow customers, you will end up committing the usual sin of assuming that you are not average. You think that your choice will not also be made by lots of other people acting on the same evidence – that’s why you find that everyone else has decided to go for the same stroll by the river on a sunny Sunday afternoon.

  If there are two venues to choose between, then as a result of everyone’s accumulated experience the optimal strategy gets closer and closer to half of the people going to each venue – so neither is specially popular or unpopular – on the average. At first, the fluctuations around the average are quite large, and you might turn up at one venue to find a smaller than average crowd. As time goes on, you use more and more past experience to evaluate when and if these fluctuations will occur and act accordingly, trying to go to the venue with the smaller number. If everyone acts in this way, the venue will maintain the same average number of attendees over time but the fluctuations will steadily diminish. The last ingredient of this situation is that there will be players who trust their memories and analyses of past experience, and there will be others who don’t or who only appeal to experience a fraction of the times when they have to make a choice. This tends to split the population into two groups – those who follow past experience totally and those who ignore it. Since the consequences of making the wrong choice are far more negative (no dinner, wasted evening) than the consequences of making the right choice are positive (quicker dinner, more comfortable evening), greater care is made to avoid wrong choices, and players tend to hedge their bets and go for each available choice with equal probability over the long run. Adopting a more adventurous strategy results in more extreme losses than gains. The result is a rather cautious and far from optimal pattern of group decision making and all the eating venues are less than full.

  95

  Venn Vill They Ever Learn

  There are two groups of people in the world; those who believe that the world can be divided into two groups of people, and those who don’t.

  Anon.

  John Venn came from the east of England, near the fishing port of Hull, and went – as promising mathematicians did – to Cambridge, where he entered Gonville and Caius College as a student in 1853. Graduating among the top half-dozen students in mathematics, he was elected into a college teaching fellowship. He then left the college for four years and was ordained a priest in 1859, following in the line of his distinguished father and grandfather, who were prominent figures in the evangelical wing of the Anglican Church. However, instead of following the ecclesiastical path that had been cleared for him, he returned to Caius in 1862 to teach logic and probability. Despite this nexus of chance, logic and theology, Venn was also a practical man and rather good at building machines. He constructed one for bowling cricket ball which was good enough to clean b
owl one of the members of the Australian touring cricket team on four occasions when they visited Cambridge in 1909.

  It was his college lectures in logic and probability that made Venn famous. In 1880 he introduced a handy diagram for representing logical possibilities. It soon replaced alternatives that had been tried by the great Swiss mathematician Leonard Euler and the Oxford logician and Victorian surrealist writer Lewis Carroll. It was eventually dubbed the ‘Venn diagram’ in 1918.

  Venn’s diagrams represented possibilities by regions of space. Here is a simple one that represents all the possibilities where there are two attributes.

  Suppose A is the collection of all brown animals and B is the collection of all cats. Then the hatched overlap region contains all brown cats; the region of A not intersected by B contains all the brown animals other than cats; the region of B not intersected by A represents all the cats that are not brown; and finally, the black region outside both A and B represents everything that is neither a brown animal nor a cat.

  These diagrams are widely used to display all the different sets of possibilities that can exist. Yet one must be very careful when using them. They are constrained by the ‘logic’ of the two-dimensional page they are drawn on. Suppose we represent four different sets by the circles A, B, C and D. They are going to represent the collections of friendships between three people that can exist among four people Alex, Bob, Chris and Dave. The region A represents mutual friendships between Alex, Bob and Chris; region B friendships between Alex, Bob and Dave; region C friendships between Bob, Chris and Dave; and region D friendships between Chris, Dave and Alex. The way the Venn-like diagram has been drawn displays a sub-region where A, B, C and D all intersect. This would mean that the overlap region contains someone who is a member of A, B, C and D. But there is no such person who is common to all those four sets.