100 Essential Things You Didn't Know You Didn't Know (11 page)

If we 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 success
^fn1
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 strategy
⁶
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 dividing 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.

Steven Brams, Michael Jones and Christian Klamler have suggested a better way to divide the spoils between two parties that both feel is fair. Each party is asked to tell the arbiter how they would divide the assets equally. If they both make an identical choice then there is no problem and they immediately agree what to do. If they don’t agree, then the arbiter has to intervene.

_________A_____B_________________

Suppose the assets are put along a line and my choice of a fair division divides the line at A but your choice divides it at B. The fair division then gives me the part to the left of A and gives you the part to the right of B. In between, there is a left-over portion, which the arbiter divides in half and then gives one part to each of us. In this process we have both ended up with more than the ‘half’ we expected. Both are happy.

We could do a bit better perhaps than Brams and Co. suggest by not having the arbiter simply divide the remainder in half. We
could
repeat the whole fair division process on that piece, each choosing where we thought it is equally divided, taking our two non-overlapping pieces so that a (now smaller) piece remains, then divide that, and so on, until we are left with a last piece that (by prior agreement) is negligibly small, or until the choices of how to divide the remnant made by each of us become the same.

If there are three or more parties wishing to share the proceeds fairly then the process becomes much more complicated but is in essence the same. The solutions to these problems have been patented by New York University so that they can be employed commercially in cases where disputes have to be resolved and a fair division of assets arrived at. Applications have ranged from the US divorce courts to the Middle East peace process.

34

Many Happy Returns

You’re still so young that you think a month is a long time.

Henning Mankell

If you invite lots of people to your birthday party you might be interested to know how many you need so that there is a better than 50% chance of one of them sharing your birthday. Suppose you know nothing about your guests’ birthdays ahead of time, then, forgetting about leap years and so assuming 365 days in a year, you will need at least
⁸
253 guests to have a better than evens chance of sharing your birthday with one of them. It’s much bigger than 365 divided by two because many of the guests are likely to have the same birthdays as guests other than you. This looks like an expensive birthday surprise to cater for.

A more striking parlour game to plan is simply to look for people who share the same birthday with each other, not necessarily with you. How many guests would you need before there is a better than evens chance of two of them sharing the same birthday? If you try this question on people who haven’t worked it out in detail, then they will generally overestimate the number of people required by a huge margin. The answer is striking. With just 23 people
^fn1
there is a 50.7 per cent of two of them sharing a birthday, with 22 the chance is 47.6 per cent and with 24 it is 53.8 per cent.
⁹
Take
any two football teams, throw in the referee and among them you have an odds-on chance of the same birthday being shared by two of them. There is a simple connection with the first problem we considered of having a match with
you
r birthday, which required 253 guests for a greater than 50 per cent chance. The reason that any pairing requires just 23 people to be present is because there are so many possible pairings of 23 people – in fact, there are
^fn2
(23 × 22)/2 = 253 pairings.