100 Essential Things You Didn't Know You Didn't Know Page 12
fn1 The ‘Little Boy’ bomb dropped on Hiroshima was a fission device with 60kg of uranium-235 and created a blast equivalent to 13 kilotons of TNT, killing approximately 80,000 people immediately; the ‘Fat Man’ bomb dropped on Nagasaki was a fission bomb with 6.4kg of plutonium-239, equivalent to the blast from 21 kilotons of TNT. About 70,000 people were killed instantly.
fn2 The name ‘mushroom cloud’ became commonplace in the early 1950s, but the comparison between bomb debris patterns and ‘mushrooms’ dates back at least to newspaper headlines in 1937.
52
Walk Please, Don’t Run!
You can spot the northern Europeans because they walk faster than the leisurely art of the paseo strictly requires.
Benidorm travel guide
Walk along a busy city high street and most of the people around you will be walking at about the same speed. A few people are in a bit of a hurry and a few others move very slowly, perhaps because of age, infirmity or totally impractical footwear. When you walk, you keep a part of one foot in contact with the ground all the time and you straighten your leg as you push off from the ground. Indeed, the rules of race walking make these the defining features of walking, which distinguish it from running: a failure to adhere to them results in warnings and ultimately disqualification from a race. As you walk, your hips will rise and fall as your centre moves in a gentle circular arc each time you make a complete stride. So if the length of your leg from the ground to your hips is L, you will be creating an acceleration equal to v2/L upwards towards the centre of that circular arc of movement. This cannot become greater than the acceleration due to gravity, g, that pushes us down to the ground (or we would take off!), and so g > v2/L and we deduce that, roughly, the top speed for normal walking is about √(gL). Since g = 10 ms-2 and a typical leg length is 0.9 metre, the top speed for ordinary walkers is about 3 metres per second – a fairly good estimate – and the taller you are, the larger L will be and the faster you will walk, although because of the square root there really isn’t much difference between the walking speeds of people with the usual range of heights.
Another way to interpret this result is to look at people (or other two-legged creatures) trying to get from one place to another as quickly as possible and to ask at what speed they stop walking and break into a run. The critical speed √(gL) is the fastest rate of progress you can expect to make without breaking contact with the ground (‘lifting’ as the race walkers say). Once you start breaking contact you can go much faster, with a maximum speed of about where S ~ 0.3 m is the difference in length between your straight leg and bent leg when you push off the ground and n ~ 10 is the number of strides you use to accelerate up to full speed.
Race walkers walk much faster than 3 metres per second. The world record for walking 1,500 metres, set by American Tim Lewis in 1988, is 5 minutes 13.53 seconds, an average speed of 4.78 ms-1. This event is rarely walked, and so it is interesting to look at the highly competitive world record for 20 kilometres, the shorter of the two championship events. This was reduced to 1 hr 17 mins and 16 secs by the Russian walker, Vladimir Kanaykin, on 29 September 2007, an average speed of 4.3 ms-1 over more than 12.5 miles! These speeds manage to exceed our estimate of √(gL) because race walkers use a much more efficient style of walking than we do when we stroll down the road. They do not rock their centres up and down and are able to roll their hips in a very flexible fashion to produce a longer stride length and higher frequency of striding. This highly efficient movement, coupled with very high levels of fitness, enables them to sustain impressive speeds over long periods of time. The world record holder over 50 km, more than 31 miles, averages more than 3.8 ms-1 and will cover the marathon distance (42.2 km) in 3 hours 6 minutes en route.
53
Mind-reading Tricks
Every positive integer is one of Ramanujan’s personal friends.
John E. Littlewood
Think of a number between 1 and 9. Multiply it by 9 and add the digits of this new number together. Subtract 4 from your answer and you will be left with a single-digit number. Next, convert this number to a letter: if your number is 1 it becomes A, 2 becomes B, 3 becomes C, 4 becomes D, 5 becomes E, 6 becomes F and so on. Now think of a type of animal that begins with your chosen letter and imagine that animal as strongly as you can. Hold it vividly in the forefront of your mind. If you look at the note 14 at the back of the book, you will see that I have read your mind and discovered the animal that you were imagining.
This is a very simple trick, and you ought to be able to work out how I was able to guess the animal of your choice with such a high likelihood of success. There is a little mathematics involved, in that some simple properties of numbers are exploited, but there is also a psychological – and even zoological – ingredient as well.
There is another trick of this general sort that involves only the properties of numbers. It uses the number 1089, which you may well already have listed among your favourites. It was the year in which there was an earthquake in England; it is also a perfect square (33×33); but its most striking property is the following.
Pick any three-digit number in which the digits are all different (like 153). Make a second number by reversing the order of the three digits (so 351). Now take the smaller of the two numbers away from the larger (so 351 − 153 = 198; if your number has only two digits, like 23, then put a 0 in front, so 023). Now add this to the number you get by writing it backwards (so 198 + 891 = 1089). Whatever number you chose at the outset, you will end up with 1089 after this sequence of operations!15
54
The Planet of the Deceivers
You can fool some of the people some of the time, and some of the people all the time, but you cannot fool all the people all of the time.
Abraham Lincoln
One of the human intuitions that has been honed by countless generations of social interaction is trust. It is founded upon an ability to assess how likely it is that someone is telling the truth. One of the sharp distinctions between different environments is whether we assume people are honest until we have reason to think otherwise, or whether we assume them to be dishonest until we have reason to think otherwise. One encounters this distinction in the bureaucracies of different countries. In Britain officialdom is based upon the premise that people are assumed to be honest, but I have noticed that in some other countries the opposite is the default assumption and rules and regulations are created under a presumption of dishonesty. When you make an insurance claim, you will discover which option your company takes in its dealings with its customers.
Imagine that our quest to explore the Universe has made contact with a civilisation on the strange world of Janus. The monitoring of their political and commercial activity over a long period has shown us that on the average their citizens tell the truth ¼ of the time and lie 3/4 of the time. Despite this worrying appraisal, we decide to go ahead with a visit and are welcomed by the leader of their majority party who makes a grand statement about his benevolent intentions. This is followed by the leader of the opposition party getting up and saying that the Leader’s statement was a true one. What is the likelihood that the Leader’s statement was indeed a true one?
We need to know the probability that the Leader’s statement was true given that the opposition head said it was. This is equal to the probability that the Leader’s statement was true and the opposition head’s claim was true divided by the probability that the opposition head’s statement was true. Well, the first of these – the probability that they were both telling the truth is just ¼ × ¼ = 1/16. The probability that the opponent spoke the truth is the sum of two probabilities: the first is the probability that the Leader told the truth and his opponent did too, which is ¼ × ¼ = 1/16, and the probability that the Leader lied and his opponent lied too, which is 3/4 × 3/4 = 9/16. So, the probability16 that the Leader’s statement was really true was just 1/16 ÷ 10/16, that is 1/10.
55
How to Win the Lo
ttery
A lottery is a taxation – upon all the fools in creation. And heaven be praised, it is easily raised, credulity’s always in fashion.
Henry Fielding
The UK National Lottery has a simple structure. You pay £1 to select six different numbers from the list 1,2,3,. . ., 48,49. You win a prize if at least three of the numbers on your ticket match those on six different balls selected by a machine that is designed to make random choices from the 49 numbered balls. Once drawn, the balls are not returned to the machine. The more numbers you match, the bigger the prize you win. Match all six and you will share the jackpot with any others who also share the same six matching numbers. In addition to the six drawn balls, an extra one is drawn and called the ‘Bonus Ball’. This affects only those players who have matched five of the six numbers already drawn. If they also match the Bonus Ball then they get a larger prize than those who matched only the other five numbers.
What are your chances of picking six numbers from the 49 possibilities correctly, assuming that the machinefn1 picks winning numbers at random? The drawing of each ball is an independent event that has no effect on the next drawing, aside from reducing the number of balls to be chosen from. The chance of getting the first of the 6 winning numbers from the 49 is therefore just the fraction 6/49. The chance of picking the next of the remaining 5 from the 48 balls that remain is 5/48. The chance of picking the next of the remaining 4 from the 47 balls that remain is 4/47. And so on, the remaining three probabilities being 3/46, 2/45 and 1/44. So the probability that you pick them all independently and share the jackpot is
6/49 × 5/48 × 4/47 × 3/46 × 2/45 × 1/44 = 720/10068347520
If you divide this out you get the odds as 1 in 13,983,816 – that’s about one chance in 13.9 million. If you want to match 5 numbers plus the Bonus Ball, then the odds are 6 times smaller, and your chance of sharing the prize is 1 in 13,983,816/6 or 1 in 2,330,636.
Let’s take the collection of all the possible draws – all 13,983,816 of them – and ask how many of them will result in 5, or 4, or 3, or 2, or 1, or zero numbers being chosen correctly.17 There are just 258 of them that get 5 numbers correct, but 6 of them win the Bonus Ball prize, so that leaves 252; 13,545 of them get 4 balls correct,246,820 of them that get 3 balls correct, 1,851,150 of them that get 2 balls correct, 5,775,588 of them get just 1 ball correct, and 6,096,454 of them get none of them correct. So to get the odds for you to get, say, 5 numbers correct you just divide the number of ways it can happen by the total number of possible combinations, i.e. 252/13,983,816, which means odds of 1 in 55,491 if you buy one lottery ticket. For matching 4 balls the odds are 1 in 1,032; for matching 3 balls they are 1 in 57. The number of the 13,983,816 outcomes that win a prize is 1 + 258 + 13,545 + 246,820 = 260,624 and so the odds of winning any prize when you buy a single ticket are 1 in 13,983,816/260,624, that is about 1 in 54. Buy a ticket a week with an extra one on your birthday and at Christmas and you have an evens chance of winning something.
This arithmetic is not very encouraging. Statistician John Haigh points out that the average person is more likely to drop dead within one hour of purchasing a ticket than to win the jackpot. Although it is true that if you don’t buy a ticket you will certainly not win, what if you buy lots of tickets?
The only way to be sure of winning a lottery is to buy all the tickets. There have been several attempts to use such a strategy in different lotteries around the world. If no jackpot is won in the draw, then usually the unwon prize is rolled over to the following week to create a super-jackpot. In such situations it might be attractive to try to buy almost all the tickets. This is quite legal! The Virginia State Lottery in the USA is like the UK Lottery except the six winning numbers are chosen from only 44 balls, so there are 7,059,052 possible outcomes. When the jackpot had rolled over to $27 million, Australian gambler Peter Mandral set in operation a well-oiled ticket buying and printing operation that managed to buy 90% of the tickets (a failure by some of his team was responsible for the worrying gap of 10%). He won the rollover jackpot and went home with a healthy profit on his $10 million outlay on tickets and payments to his ticket-buying ‘workers’.
fn1 Strictly speaking there are 12 machines (which each have names) and 8 sets of balls that can be used for the public draw of the winning numbers on television. The machine and the set of balls to be used at any draw are chosen at random from these candidates. This point is usually missed by those who carry out statistical analyses of the results of the draw since the Lottery began. Since the most likely source of a non-random element favouring a particular group of numbers over others would be associated with features of a particular machine or ball, it is important to do statistical studies for each machine and set of balls separately. Such biases would be evened out by averaging the results over all the sets of balls and machines.
56
A Truly Weird Football Match
Own goal: Own goals tend, like deflections, to be described with sympathy for those who fall victim to them. Often therefore preceded by the adjectives freak or bizarre even when ‘incompetent’ or ‘stupid’ might come more readily to mind.
John Leigh and David Woodhouse, The Football Lexicon
What is the most bizarre football match ever played? In that competition I think there is only one winner. It has to be the infamous encounter between Grenada and Barbados in the 1994 Shell Caribbean Cup. This tournament had a group stage before the final knockout matches. In the last of the group stage games Barbados needed to beat Grenada by at least two clear goals in order to qualify for the next stage. If they failed to do that, Grenada would qualify instead. This sounds very straightforward. What could possibly go wrong?
Alas, the law of unforeseen consequences struck with a vengeance. The tournament organisers had introduced a new rule in order to give a fairer goal difference advantage to teams that won in extra time by scoring a ‘golden goal’. Since the golden goal ended the match, you could never win by more than one goal in such a circumstance, which seems unfair. The organisers therefore decided that a golden goal would count as two goals. But look what happened.
Barbados soon took a 2–0 lead and looked to be coasting through to the next phase. Just seven minutes from full time Grenada pulled a goal back to make it 2–1. Barbados could still qualify by scoring a third goal, but that wasn’t so easy with only a few minutes left. Better to attack their own goal and score an equaliser for Grenada because they then had the chance to win by a golden goal in extra time, which would count as two goals and so Barbados would qualify at Grenada’s expense. Barbados obliged by putting the ball into their own net to make it 2–2 with three minutes left. Grenada realised that if they could score another goal (at either end!) they would go through, so they attacked their own goal to get that losing goal that would send them through on goal difference. But Barbados resolutely defended the Grenada goal to stop them scoring and sent the match into extra time. In extra time the Barbadians took their opponents by surprise by attacking the Grenada goal and scored the winning golden goal in the first five minutes. If you don’t believe me, watch it on YouTube!fn1
fn1 http://www.youtube.com/watch?v=ThpYsN-4p7w
57
An Arch Problem
Genius is four parts perspiration and one part having a focused strategic overview.
Armando Iannucci
An old arch of stones can seem a very puzzling creation. Each stone looks as if it has been put in place individually, but the whole structure looks as if it cannot be supported until the last capstone is put in place: you can’t have an ‘almost’ arch. So, how could it have been made?
The problem is an interesting one because it is reminiscent of a curious argument that is much in evidence in the United States under the name of ‘Intelligent Design’. Roughly speaking, its advocates pick on some complicated things that exist in the natural world and argue that they must have been ‘designed’ in that form rather than have evolved by a step-by-step process from simpler f
orms because there is no previous step from which they could have developed. This is a little subjective, of course – we may not be very imaginative in seeing what the previous step was – but at root the problem is just like our arch, which is a complicated construct that doesn’t seem to be one step away from a slightly simpler version of an arch with one stone missing.
Our unimaginative thinking in the case of the arch is that we have got trapped into thinking that all structures are built up by adding bits to them. But some structures can be built by subtraction. Suppose we started with a heap of stones and gradually shuffled them and removed stones from the centre of the pile until we left an arch behind. Seen in this way we can understand what the ‘almost’ arch looks like. It has part of the central hole filled in. Real sea arches are made by the gradual erosion of the hole until only the outer arch remains. Likewise, not all complexity in Nature is made by addition.
58
Counting in Eights
The Eightfold Path: Right view, right intention, right speech, right action, right livelihood, right effort, right mindfulness and right concentration.
The Noble Eightfold Way
We count in ‘tens’. Ten ones make ten, ten tens make a hundred, ten hundreds make a thousand and so on forever. This is why our counting system is called a ‘decimal’ system. There is no limit to its scope if you have enough labels to name the results. We have words like million, billion and trillion for some of the large numbers, but not for every one that you might need to write down. Instead we have a handy notation that writes 10n to denote the number which is 1 followed by n noughts, so a thousand is 103.