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

  It is easy to get trapped into thinking about a problem in one set way. Breaking out and being ‘imaginative’ or original in solving a problem can require a different way of thinking about it rather than just a correct implementation of principles already learned. Simple problems that involve the application of fixed rules in a faultless way can usually be conquered by the second approach. For example, if someone challenges you to a game of noughts and crosses you should never lose, regardless of whether you move first or second. There is a strategy whose worst outcome is a draw, but it will give you a win only if your opponent deviates from that optimal strategy. Alas, not all problems are as easy as finding the best move in noughts and crosses. Here is an example of a simple problem whose solution will almost certainly take you by surprise.

  Write down a 3×3 square of nine dots. Now pick up your pencil and without lifting the pencil point from the paper or retracing your path, draw four straight lines that pass through all the dots.

  Here’s one failed attempt. It misses out one of the points in the middle on the left-hand edge:

  Here’s another. It’s also one point short because it doesn’t go through the central point:

  It looks impossible doesn’t it. I can do it with four straight lines only if I retrace the pencil path, going down the diagonal and then going back and forth along the intersecting lines. But that requires the drawing of far more than four lines, even though only four appear to be present when you have finished:

  There is a way to draw four lines through all the points without lifting the pencil or retracing its path, but it requires breaking a rule that you imposed on yourself for no reason at all. It wasn’t one of the restrictions imposed at the outset. You were just so used to playing by a certain type of rule that you didn’t think to step outside the box and break it. The solution that you want simply requires that you draw straight lines that (literally) extend beyond the box of nine points before they turn back in a different direction.

  Think outside the box!

  80

  Googling in the Caribbean – The Power of the Matrix

  Cricket is organised loafing.

  William Temple

  Most sports create league tables to see who is the best team after all the participants play each other. How the scores are assigned for wins, losses and draws can be crucial for determining who comes out on top. Some years ago, football leagues decided to give three points for a win rather than two, in the hope that it would encourage more attacking play. A team would get far more credit for winning than for playing out a draw – in which each team earns only one point. But somehow this simple system seems to be rather crude. After all, should you not get more credit for beating a top team than one down at the bottom of the league?

  The 2007 Cricket World Cup in the Caribbean gives a nice example. In the second phase of the competition, the top eight teams played each other (actually each had played one of the others already in the first stage, and that result was carried forward so they only had to play six more games). They were given two points for a win, one for a tie and zero for a defeat. The top four teams in the table went on to qualify for the two semi-final knockout games. In the event of teams being level on points they were separated by their overall run-scoring rate. Here is the table:

  But, let’s think about another way of determining the team rankings that gives more credit for beating a good team than a bad one. We give each team a score that is equal to sum of the scores of the teams that they beat. Since there were no tied games we don’t have to worry about them. The overall scores look like a list of eight equations:

  A = SL + N +SA + E +W +B + I

  SL = N + W + E + B + I

  N = W+ E + B + I + SA

  SA = W + E + SL + I

  E = W + B + I

  W = B + I

  B = SA

  I = B

  This list can be expressed as a matrix equation for the list x = (A, N, W, E, B, SL, I, SA) with the form Ax = K x, where K is a constant and A is an 8×8 matrix of 0’s and 1’s denoting defeats and wins, respectively, and is given by:

  In order to solve the equations and find the total scores of each team, and hence their league ranking under this different point-scoring system, we have to find the eigenvector of the matrix A with all its entries positive or zero. Each of these solutions will require K to take a specific value. This corresponds to a solution for the list x in which all have positive (or zero – if they lost every game) scores, as is obviously required for the situation being described here. Solving the matrix for this, so-called ‘first-rank’ eigenvector, we find that it is given by

  x = (A, N, W, E, B, SL, I, SA) = (0.729, 0.375, 0.104, 0.151, 0.153, 0.394, 0.071, 0.332)

  The ranking of the teams is given by the magnitudes of their scores here, with Australia (A) at the top with 0.729 and Ireland (I) at the bottom with 0.071. If we compare this ranking with the original table we have:

  Super Eight standings

  My Ranking

  A

  A 0.729

  SL

  SL 0.394

  N

  N 0.375

  SA

  SA 0.332

  E

  B 0.153

  W

  E 0.151

  B

  W 0.104

  I

  I 0.071

  The top four teams qualifying for the semi-finals finish in exactly the same order under both systems, but three of the bottom four are quite different. Bangladesh won only one game, so scored a mere two points and finished second from bottom of the World Cup League. Under our system they jump up to fifth because their one win was against the higher ranked South Africans. England actually won two games but only against the bottom two teams, and end up ranked just behind Bangladesh (although it takes the third decimal place to separate them – 0.153 vs. 0.151). The poor West Indies finished sixth under the straightforward league system but drop a position under the rank system.

  This system of ranking is what lies at the root of the Google search engine. The matrix of results when team i plays team j corresponds to the number of web links that exist between topic i and topic j. When you search for a term, a matrix of ‘scores‘ is created by the massive computing power at Google’s disposal, which solves the matrix equation to find the eigenvector, and hence the ranked list of ‘hits’ to the word that you were searching for. It still seems like magic though.

  81

  Loss Aversion

  In theory there is no difference between theory and practice. In practice there is.

  Yogi Berra

  People seem to react very differently to the possibility of gains and losses. Economists took a long time to recognise that human behaviour is not symmetrical in this respect when it comes to decision making. We tend to be naturally risk averse and work much harder to avoid a small loss than to secure a larger gain. Being ‘loss averse’ means that losing a £50 note in the street gives you more unhappiness than the happiness you enjoy if you find a £50 note. You feel better about avoiding a 10 per cent surcharge than taking advantage of a 10 per cent reduction in train ticket prices.

  Imagine that you are a market trader who sells goods from a roadside stall. You decide that you want to obtain a certain income each day and you will carry on working until you achieve that level of sales. What happens? When trade is good you quickly reach the sales target and go home early. When trade is bad you carry on working longer and longer hours in order to meet your target. This seems irrational. You work far longer in order to avoid a shortfall in your target, but you don’t seize the opportunity to work longer when the demand is high. You are a classic example of the psychology of risk aversion.

  Some people would argue that this type of behaviour is just irrational. There is no good reason for it. On the other hand, gains and losses are not necessarily symmetrical with respect to the amount of money that you currently have. If your total wealth is £100,000, then a gain of £100,000 is to be welcomed, but
a loss of £100,000 is to be avoided much more because it will bankrupt you. The potential loss is much greater than the possible gain.

  Sometimes the taking of decisions does rest upon a purely psychological perception of apparent differences that do not truly exist. As an example, suppose the Environment Agency has to draw up plans to counter the effects on coastal homes of an anomalously high tide and expected storm surge that is expected to wreck 1,000 homes. It asks people to choose between two plans. Plan A uses all resources to build a wall in one location and will save 200 homes. Plan B uses the resources more diversely and will save all 1,000 homes from destruction with a probabilityfn1 of 1/5. Faced with this choice, most people pick the sure and positive sounding Plan A.

  Imagine, now, that the Environment Agency has a different Public Relations Officer who wants to present these two plans differently. The choice is now going to be between Plan C, which allows 800 homes to be destroyed, and Plan D, which leads to no homes being destroyed with a probability of 1/5 and all 1,000 homes being destroyed with a probabilityfn2 of 4/5. Most people choose Plan D. This is strange because Plan D is the same as Plan B, and Plan A is the same as Plan C. Our innate risk aversion makes us pick D over C, but not B over A, because we are more sensitive to losses. The sure loss of 800 homes seems worse to us than the 4/5 chance of losing 1,000. But when it comes to the saving of homes, we don’t respond so strongly to the chance of saving 1,000 as we do to the surety of saving 200. Odd.

  fn1 This means that the expected number of homes that will be saved is 1,000 × 1/5 = 200, the same number saved in Plan A.

  fn2 The expected number of homes destroyed is 800 in both Plans C and D, i.e. the expected number saved is 200, as in Plans A and B.

  82

  The Lead in Your Pencil

  We are all pencils in the hand of God.

  Mother Teresa

  The modern pencil was invented in 1795 by Nicholas-Jacques Conte, a scientist serving in the army of Napoleon Bonaparte. The magic material that was so appropriate for the purpose was the form of pure carbon that we call graphite. It was first discovered in Europe, in Bavaria at the start of the fifteenth century, although the Aztecs had used it as a marker several hundred years earlier. Initially it was believed to be a form of lead and was called ‘plumbago’ or black lead (hence the ‘plumbers’ who mend our lead water-carrying pipes), a misnomer that still echoes in our talk of pencil ‘leads’. It was called graphite only in 1789, using the Greek word ‘graphein’ meaning ‘to write’. Pencil is an older word, derived from the Latin ‘pencillus’, meaning ‘little tail’, to describe the small ink brushes used for writing in the Middle Ages.

  The purest deposits of lump graphite were found in Borrowdale near Keswick in the Lake District in 1564 and spawned quite a smuggling industry and associated black economy in the area. During the nineteenth century a major pencil manufacturing industry developed around Keswick in order to exploit the high quality of the graphite. The first factory opened in 1832, and the Cumberland Pencil Company has just celebrated its 175th anniversary, although the local mines have long been closed and supplies of the graphite used now come from Sri Lanka and other far away places. Cumberland pencils were those of the highest quality because the graphite used shed no dust and marked the paper very well. Conte’s original process for manufacturing pencils involved roasting a mixture of water, clay and graphite in a kiln at 1,9000 Fahrenheit before encasing the resulting soft solid in a wooden surround. The shape of that surround can be square, polygonal or round, depending on the pencil’s intended use – carpenters don’t want round pencils that are going to roll off the workbench. The hardness or softness of the final pencil ‘lead’ can be determined by adjusting the relative fractions of clay and graphite in the roasting mixture. Commercial pencil manufacturers typically market 20 grades of pencil, from the softest, 9B, to the hardest 9H, with the most popular intermediate value, HB, lying midway between H and B. ‘H’ means hard and ‘B’ means black. The higher the B number, the more graphite gets left on the paper. There is also an ‘F’, or Fine point, which is a hard pencil for writing rather than drawing.

  The strange thing about graphite is that it is a form of pure carbon that is one of the softest solids known, and one of the best lubricants because the six carbon atoms that link to form a ring can slide easily over adjacent rings. Yet, if the atomic structure is changed, there is another crystalline form of pure carbon, diamond, that is one of the hardest solids known.

  An interesting question is to ask how long a straight line could be drawn with a typical HB pencil before the lead was exhausted. The thickness of graphite left on a sheet of paper by a soft 2B pencil is about 20 nanometres and a carbon atom has a diameter of 0.14 nanometres, so the pencil line is only about 143 atoms thick. The pencil lead is about 1 mm in radius and therefore square mm in area. If the length of the pencil is 15 cm, then the volume of graphite to be spread out on a straight line is 150 cubic mm. If we draw a line of thickness 20 nanometres and width 2 mm, then there will be enough lead to continue for a distance L = 150π/4×10-7 mm = 1,178 kilometres. But I haven’t tested this prediction!

  83

  Testing Spaghetti to Destruction

  Every time I see a Parceline van I shall remember Miles Kington. Because it was Miles who had decided that it was the name of an Italian pasta dish.

  Richard Ingrams

  Hold both ends of a long, brittle, rod of dry spaghetti. Flex it and gradually move the ends together so that the rod snaps. You might have expected that eventually the rod would snap into two pieces, leaving you holding one in each hand. Strangely, this never happens. The spaghetti always breaks into more than two pieces. This is odd. If you had snapped a thin rod of wood or plastic it would have broken into two pieces. Why does the spaghetti behave differently? Richard Feynman was puzzled by this question as well and a story appears in his biography, told by Daniel Hillis:

  Once we were making spaghetti . . . If you get a spaghetti stick and you break it, it will almost always break into three pieces. Why is this true – why does it break into three pieces? We spent the next two hours coming up with crazy theories. We thought up experiments, like breaking it underwater because we thought that might dampen the sound, the vibrations. Well, we ended up at the end of a couple of hours with broken spaghetti all over the kitchen and no real good theory about why spaghetti breaks in three.

  More recently some light has been shed on this problem, which turned out to be unexpectedly difficult. A brittle rod of anything, not just spaghetti, will break when it gets curved by more than a critical amount, called its ‘rupture curvature’. There is no mystery about that, but what happens next is interesting. When the break first occurs, one end of each part will be left free while the other end is held in your hand. The free end that has suddenly been released tries to straighten itself and sends waves of curvature back along its length towards your hand where it is held fixed. These waves reflect and meet others arriving at different places along the spaghetti rod. When they meet, a sudden jump in curvature occurs, sufficient to break the flexed spaghetti again. New waves of curvature get produced by this new breaking and can lead to more local increases in curvature beyond the critical value at different points in the spaghetti. As a result, the spaghetti will break in one or more other places after it first fractures. The breaking stops when there is no longer enough energy left to allow the waves to travel along the pasta rod you are left holding. Any fragments that find themselves free at both ends just fall to the ground.

  84

  The Gherkin

  Think cool; think cucumber.

  Stephen Moss

  The most dramatic modern construction in the City of London is 30 St Mary Axe, more commonly known as the Swiss Re building, the Pine Cone or simply the Gherkin. Prince Charles sees it as symptomatic of a rash of carbuncular towers on the face of London. The architects, Norman Foster and Partners, heralded it as a signature building for the modern age and received the 2004 RI
BA Stirling Prize for their creation. It has succeeded in putting the Swiss Re insurance company in the public eye and has stimulated a wide-ranging debate about the desirability of towers on the traditional horizons and sight-lines of the City of London. Alas, while there is an ongoing debate about the aesthetic success of the Gherkin, there is not much doubt that it has been a bit of a commercial disappointment for Swiss Re. The company occupies just the first 15 of the 34 floors, but has never succeeded in renting the other half of the building to another, single, organisation. This is not entirely surprising: the type of high-profile commercial enterprise able to afford such space would recognise that the building has become so totally associated with the name of Swiss Re that it would be forever playing second fiddle and would gain no kudos at all by its presence there. As a result the space has been parcelled up into smaller lets.

  The most obvious feature of the Gherkin is that it’s big – 180 metres high – and the creation of a tower on such a scale creates structural and environmental problems. Today, engineers can create sophisticated computer models of a big building that enable them to study its response to wind and heat, its take-up of fresh air from the outside, and its effect on passers-by at ground level. Tinkering with one aspect of the design, like the reflectivity of its surface, will have effects in many other areas – changing the internal temperature and air-conditioning requirements, for instance – and all the consequences can be seen at once using sophisticated computer simulations of the building. It is no good following a ‘one thing at a time’ approach to designing a complicated structure like a modern building, you have to do a lot of things all at once.

  The Gherkin’s elegant curved profile is not just driven by aesthetics or some mad designer’s desire to be spectacular and controversial. The tapering shape, starting narrowest at street level and bulging most at floor 16, before narrowing again steadily towards the top, was chosen in response to the computer models.