100 Essential Things You Didn't Know You Didn't Know Page 8
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 least8 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 peoplefn1 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 your 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 arefn2 (23 × 22)/2 = 253 pairings.
The American mathematician Paul Halmos found a handy approximation for this problem in a slightly different form. He showed that if we have a gathering of a large number of people, call the number N again, then we need to take a random selection of at least 1.18 × N½ of them in order to have a greater than evens chance that two of them will share the same birthday. If you put N = 365 then the formula gives 22.544, so we need 23 people.
One of the assumptions built into this analysis is that there is an equal chance of birthdays falling on any day of the year. In practice, this is probably not quite true. Conception may be more likely at holiday times, and planned birth deliveries by Caesarean section will be unlikely to be planned for Christmas Day or New Year’s Eve. Successful sportsmen and sportswomen are another interesting case – you might investigate the birthdays of premier league footballers in a match or members of the UK athletics team. Here, I suspect you would find a bias towards birthdays falling in the autumn. The reason is nothing to do with astrological star signs. The British school year starts at the beginning of September, and so children who have birthdays in September, October and November are significantly older than those in the same school year with June, July and August birthdays at stages in life when 6 to 9 months makes a big difference in physical strength and speed. The children with autumn birthdays therefore are rather more likely to make it into sports teams and get all the impetus, opportunity and extra coaching needed to propel them along the road to success as young adults. The same may well be true of other activities that call for different types of maturity as well.
There is a range of activities where we have to give our birth dates as part of a security check. Banks, on-line shopping and airline sites make use of birth dates for part of their passenger security checking. We have seen that taking a birthday alone is not a very good idea. The chance of two customers sharing a date is rather high. You can cut the odds by requiring the year as well and adding a password. You can see that it’s essentially the same problem, but instead of having 365 dates that might be shared by two people at your party you are asking for the chance that they share a date and a choice of, say, a string of 10 letters for their password. This reduces the odds on random matches dramatically. If a password was eight alphabetical characters long then there would be 2610 choices that could be made, and Halmos’ formula would lead us to expect that we need to have 1.28 × 265, or about 15,208,161, customers before there was a 50 per cent chance of two having the same. As of July 2007 the population of the world was estimated to be 6,602,224,175, so having a password sequence that uses just 14 letters makes the number of people required for a 50 per cent chance of a random match larger than the population of the world.
fn1 Taking leap years into account makes a small change but does not change this number.
fn2 There are 23 for the first choice and then 22 for each of them, so 23 × 22, but then divide by 2 because we don’t care about the order in which we find a pair (i.e. ‘you and me’ is the same as ‘me and you’).
35
Tilting at Windmills
The answer, my friend, is blowin’ in the wind.
Bob Dylan
If you travel about the UK you increasingly encounter modern-day windmills dotting parts of the countryside like alien spaceships. Their presence is controversial. They are there to counter atmospheric pollution by offering a cleaner source of power, yet they introduce a new form of visual pollution when inappropriately located in pristine (but no doubt windswept) countryside or seascapes.
There are some interesting questions to ask about the windmills, or ‘wind turbines’ as we call them today. Old style windmills had four sails crossed at the centre like an X. Modern windmills look like aircraft propellers and generally have three arms. There are several factors that have led to the three-armed (or Danish style) windmill becoming so common. Three arms are cheaper than four. So why not two arms? Four-armed windmills have an awkward property that makes them less stable than three-armed ones. Windmills with four (or any even number) of arms have the property that when one of the arms is in its highest vertical position, extracting maximum power from the wind, the other end of the sail will be pointing vertically downwards and will be shielded from the wind by the windmill support. This leads to stresses across the sail and a tendency for the windmill to wobble, which can be dangerous in strong winds. Three-armed windmills (and indeed windmills with any odd number of arms) don’t suffer from this problem. The three arms are each 120 degrees apart, and when any one is vertical neither of the other two arms can be vertical as well. Of course, thre
e arms will catch less wind than four and so will have to rotate faster in order to generate the same amount of power.
The efficiency of windmills is an interesting question, which was first solved back in 1919 by the German engineer Albert Betz. A windmill’s sails are like a rotor that sweeps out an area of air, A, which approaches it at a speed U and then passes on at another, lower, speed V. The loss in air speed as a result of the sails’ action is what enables power to be extracted from the moving air by the windmill. The average speed of the air at the sails is 1/2 (U + V). The mass of air per unit time passing through the rotating arms is F = DA × 1/2 (U + V), where D is the density of air. The power generated by the windmill is just the difference in the rate of change of kinetic energy of the air before and after passing the rotors. This is P = 1/2 FU2 − 1/2 FV2. If we use the formula for F in this equation, then we see that the power generated is
P = 1/4 DA(U2 – V2 )(U + V)
But if the windmill had not been there the total power in the undisturbed wind would have just been P0 = 1/2 DAU3. So, the efficiency of the windmill in extracting power from the moving mass of air is going to be P/P0 , which equals 1, indicating 100 per cent efficiency as a power generator, when P = P0. Dividing the last two formulae, we have
P/P0 = 1/2 {1 – (V/U)2} × {1 + (V/U)}
This is an interesting formula. When V/U is small, P/P0 is close to 1/2; as V/U approaches its maximum value of 1, so P/P0 tends to zero because no wind speed has been extracted. Somewhere in between, when V/U = 1/3, P/P0 takes its maximum value. In that case, the power efficiency has its maximum value, and it is equal to 16/27, or about 59.26%. This is Betz’s Law for the maximum possible efficiency of a windmill or rotor extracting energy from moving air. The reason that the efficiency is less than 100% is easy to understand. If it had been 100% then all the energy of motion of the incoming wind would have to be removed by the windmill sails – for example, this could be achieved by making the windmill a solid disc so it stopped all the wind – but then the rotors would not turn. The downstream wind speed (our quantity V) would be zero because no more wind would flow through the windmill sails.
In practice, good wind turbines manage to achieve efficiencies of about 40%. By the time the power is converted to usable electricity there are further losses in the bearings and transmission lines, and only about 20% of the available wind power will ultimately get turned into usable energy.
The maximum power that can be extracted from the wind by any sail, rotor, or turbine, occurs when V/U = 1/3, so Pmax = (8/27) × D × A × U3. If the diameter of the rotor circle is d then the area is A= πd2/4 and if the windmill operates at about 50% efficiency, then the power output is about 1.0 × (d/2 m)2 × (U/1 ms-1)3 watts.
36
Verbal Conjuring
Posh and Becks failed to appear from their hotel room, having been confused by the ‘Do Not Disturb’ sign hanging on the inside of the door.
Angus Deayton
Skilful conjuring, viewed close up, can be very perplexing, and it becomes astonishing if the conjuror shows you how it was done. How easily he misled you; how blind you were to what was going on right under your nose; how simple it was. You soon realise how incompetent a judge you would be in the face of claims of spoon bending or levitation. Scientists are the easiest to fool: they are not used to Nature conspiring to trick them. They believe almost everything they see is true. Magicians believe nothing.
In this spirit, I want to relate to you a little mathematical story, adapted from a version by Frank Morgan, that is an example of verbal conjuring. You keep track of everything in the story but something seems to disappear in the telling – a sum of money no less – and you have to figure out where it went; or, indeed, whether it was ever there.
Three travellers arrive at a cheap hotel late at night, each with just £10 in his wallet. They decide to share one large room and the hotel charges them £30 for a one-night stay, so they each put in £10. After they have gone up to their room with the bell boy carrying their bags, the hotel receptionist receives an email from the head office of the hotel chain saying that they are running a special offer and the room rate is reduced to £25 for guests staying tonight. The receptionist, being scrupulously honest about such things, immediately sends her bell boy back to the room of the three new guests with a £5 note for their rebate. The bell boy is less scrupulous. He hadn’t received a tip for carrying their bags and he can’t see how to split £5 into three, so he decides to keep £2 for himself, as a ‘tip’, and give the three guests a rebate of £1 each. Each of the three guests has therefore spent £9 on the room and the bell boy has £2 in his pocket. That makes a total of £29. But they paid £30 – what happened to the other £1?fn1
fn1 By following the argument closely, you will see that there is no missing pound. In the end the three guests have a total of £3, the bell boy has £2, and the hotel has £25.
37
Financial Investment with Time Travellers
I’m, not a soothsayer! I’m a con man, that’s all! I can’t make any genuine predictions. If I could have foreseen how this was going to turn out I’d have stayed at home.
René Goscinny and Albert Uderzo, Asterix and the Soothsayer
Imagine that there are advanced civilisations in the Universe that have perfected the art (and science) of travelling in time. It is important to realise that travelling into the future is totally uncontroversial. It is predicted to occur in Einstein’s theories of time and motion that so accurately describe the world around us and is routinely observed in physics experiments. If two identical twins were separated, with one staying here on Earth while the other went on a round space-trip then the space-travelling twin would find himself younger than his stay-at-home brother when he returned to meet him on Earth. The returning twin has time travelled into the future of the stay-at-home twin.
So, time travelling into the future is just a matter of practicality: can you build means of transportation that can withstand the stresses and achieve the velocities close to the speed of light necessary to make it a noticeable reality? Time travel into the past is another matter entirely. This is the realm of so-called changing-the-past paradoxes, although most of them are based upon a misconception.fn1
Here is an observational proof that time travellers are not engaged in systematic economic activity in our world. The key economic fact that we notice is that interest rates are not zero. If they are positive then backward time travellers could use their knowledge of share prices gleaned from the future to travel into the past and invest in the stocks that they know will increase in value the most. They would make huge profits everywhere across the investment and futures markets. As a result, interest rates would be driven to zero. Alternatively, if interest rates were negative (so that investments are worth less in the future), time travellers could sell their investment at its current high price and repurchase it in the future at a lower price so as to travel backwards in time to resell at a high price again. Again, the only way for the markets to stop this perpetual money machine is by driving interest rates to zero. Hence, the observations that there exist interest rates that are not zero means this type of share-dealing activity is not being carried out by time travellers from the future.fn2
The same type of argument would apply to casinos and other forms of gambling. Indeed, this might be a better target for the time travellers because it is tax free! If you know the future winner of the Derby or the next number on which the roulette ball will stop, then a short trip to the past will guarantee a win. The fact that casinos and other forms of gambling still exist – and make a significant profit – is again a telling argument that time-travelling gamblers do not exist.
These examples might seem rather fanciful but, on reflection, might not the same arguments not be levelled against many forms of extrasensory perception or parapsychic forms of knowledge? Anyone who could foresee the future would have such an enormous advantage that they would be able to amass stupendous wealth quickly and ea
sily. They could win the lottery every week. If reliable intuitions about the future did exist in some human (or pre-human) minds, then they would provide the owners with an enormous evolutionary advantage. They would be able to foresee hazards and plan for the future without uncertainty. Any gene that endowed its owners with that insurance policy against all eventualities would spread far and wide and its owners would soon come to dominate the population. The fact that psychic ability is apparently so rare is a very strong argument against its existence.
fn1 See J.D. Barrow, The Infinite Book (Vintage), for details.
fn2 This argument was first made by the Californian economist Marc Reinganum. For more conservative investors, note that £1 invested in 2007 in a savings account giving 4% compound interest would have risen in value to £1× (1 + 0.04)1000= 108 million billion pounds by the year 3007. However, this will probably still be what you would have to pay for your Sunday newspaper.
38
A Thought for Your Pennies
One of the evils of money is that it tempts us to look at it rather than at the things that it buys.
E.M. Forster
Coins cause delays. Buy something that costs 79 pence (or 79 cents) and you will be rummaging in the depths of your purse for an exact combination of coins to make up the amount. If you don’t, and offer £1 (or $1) instead, then you are about to receive even more small change and it will take you even longer to find the change you need next time. The question is: what is the collection of different coin values that is best for making up your change?