Ian Stewart
Page 27
Its velocity vector is the derivative of this with respect to t, which is
(10 - 10 cos 10t, 10 sin 10t)
This vanishes when
cos 10t = 1, sin 10t = 0
That is, 10t = 2nπ for integer n, or t = nπ/5. But at these times the dot is at positions (2nπ, 0), which are the successive points at which the dot hits the ground.
The same kind of calculation shows that any point not on the rim always has non-zero velocity. I’ll omit the details.
Point Placement Problem
It can be proved - not easily - that the process cannot continue past the 17th point.
The first proof was found by Mieczyslaw Warmus, but this wasn’t published; the first published proof was given by Elwyn Berlekamp and Ron Graham in 1970. Warmus then published a simpler proof in 1976. He also proved that there are precisely 1,536 distinct patterns for placing 17 points, which form 768 mirror-image pairs.
Chess in Flatland
White can force a win by moving the knight.
This is the only opening that can force a win, but I’ll omit that part of the analysis.
To see why moving the knight leads to a win, number the cells of the board 1-8 from the left. Use the symbols R = rook, N = knight, K = king, × = takes, - = moves, * = check, † = checkmate. The table shows only some of the possible sequences of moves, namely those in which White makes one move (which eventually leads to a win whatever Black does) at each step. All Black’s possible replies are considered. This technique is called ‘pruning the game tree’, and it works provided White wins for every line of play that is included. What it omits are alternative ways for White to win, if they exist, and any White moves that could lead to a forced loss for White.
The Infinite Lottery
You can’t win. The Infinite Lottery always beats you by forcing you to remove all the balls.
This may seem rather counter-intuitive, given the way the total number of balls can increase by gigantic amounts at each step. But these amounts are finite; infinity isn’t. Raymond Smullyan proved in 1979 that you always lose. His idea is to look at the biggest number in the box, and keep track of the balls that bear that number.
First, suppose that the biggest number in the box is 1. Then all balls bear the number 1. So you have to remove all the balls, one at a time - which means you lose.
Now suppose that the biggest number in the box is 2. You can’t keep discarding 1’s indefinitely, because they will eventually run out. So at some stage you have to discard one of the 2’s and replace it with lots of 1’s. Now the number of 2’s has decreased. The number of 1’s has gone up, but it’s still finite. Again, you can’t keep discarding 1’s indefinitely, so eventually you have to discard another of the 2’s and replace it with lots of 1’s. Now the number of 2’s has decreased again. Every so often you have to discard a 2, so eventually you run out of 2’s altogether. But now all the balls in the box are 1’s - and we’ve already seen that in that case you lose, however many 1’s there may be.
Ah, but maybe the biggest number in the box is 3. Well ... you can’t keep choosing (and discarding) 2’s and 1’s for ever, for the reasons we’ve just discussed. So eventually you have to discard a 3. Now the number of 3’s drops by one, and the same argument shows that you have to discard another 3 at some point, and another, until you run out of 3’s. Now the box contains only 1’s and 2’s, and we’ve just seen that in this case you lose.
Continuing in this way, it’s clear that you lose if the biggest number in the box is 4, 5, 6, . . . , and so on. That is, you lose no matter what the biggest number in the box is. But the number of balls in the box is finite, so there must be some biggest number.
Whatever it is, you lose!
Formally, this is a proof by the Principle of Mathematical Induction. This principle states that if some property of whole numbers n holds for n = 1, and its truth for any n implies its truth for n + 1, then it holds for all whole numbers. Here the property concerned is ‘If the biggest number in the box is n, then you lose.’
Let’s check that. If n = 1, then the biggest number in the box is 1, and you lose.
Now, suppose that we have proved that if the biggest number in the box is n, then you lose. Suppose that the biggest number in the box is n + 1. You can’t keep discarding numbers n or less, because we know that if you do, you lose - that is, you run out of balls numbered n or less. So at some point you must discard one of the balls bearing the number n + 1, and the number of such balls drops by one. For the same reason, that number must drop again, and again ... and eventually you discard all the balls marked n + 1. But now the remaining balls bear numbers n or smaller, so you lose. In short, if the biggest number in the box is n + 1, then you lose. And that’s the other step you need to complete the induction proof.
You can make the game go on for as long as you like, but it must stop after finitely many moves. However, that finite number can be as big as you wish.
Ships That Pass ...
13 ships.
Suppose (the date doesn’t matter, but this choice makes the sums simpler) that the New York ship sets sail on 10 January. It arrives on 17 January, just as the 17 January ship from London departs.
Similarly, the ship that left London on 3 January arrives in New York on 10 January, just as the ship we’re talking about leaves.
So on the high seas, our ship encounters the ones that left London starting on 4 January and ending on 16 January. That’s 13 ships in all.
The Largest Number is Forty-Two
The complicated calculation is pure misdirection. The fallacy is the assumption that such a number n exists. This illustrates a key aspect of mathematical proofs: if you define something by requiring it to possess some particular property, you can’t assume that ‘it’ has that property unless ‘it’ exists.
In this case, it doesn’t.
1
Contrary to widespread belief, mathematicians do go to parties.
2
‘Cossike practice’ refers to algebra: the Renaissance Italian algebraists referred to the unknown, which we now call x, as cosa, Italian for ‘thing’. As in cosa nostra, ‘this thing of ours’, referring to the Mafia. ‘Surde nombers’ are things like square roots, and the word ‘surd’ still exists in English, though it is seldom used nowadays.
3
From the Latin geminus, meaning ‘twin’.
4
As a practical matter, it is probably a good idea to fit the cat with one of those things that vets use to stop them licking wounds; otherwise the cat will scoff the butter and ruin the experiment.
5
Such as Discworld dwarf bread.
6
Many historians think that Archimedes got there first.
7
There seems to be no agreed name for such puzzles. ‘Changeone-letter-at-a-time-puzzles’ is common, but neither concise nor imaginative.
8
It is the favourite number of the Bursar of Unseen University, who is as mad as a hatter.
9
The first property is ‘every sub algebra is an n-step subideal’, and the second is ‘nilpotent of class n’. For example, if every sub-algebra is a 4-step subideal then the algebra is nilpotent of class 5plexplexplexplex, which is bigger than Skewes’ number because 5plex is a lot bigger than 34.
10
Euler wrote widely about almost everything that had even the slightest connection with mathematics.
11
If that doesn’t sound very mathematical, it can be stated more technically: any smooth vector field on a sphere has a singular point. Hope that helps.
12
Throughout this book, ‘doughnut’ refers to an American one, with a hole. British doughnuts are, or used to be, a single lump, generally filled with jam. Two nations divided by a common culinary heritage. Younger readers may not understand this footnote - all doughnuts have holes, don’t they?
13
Such procedures are often likened to tr
apdoors, where it is easy to go in but hard to get out. I’m inclined to compare them to catflaps. Our cat Harlequin knows how to go out of a catflap, by pushing, but most of the time she imagines that the way to get back in is to reverse the procedure, and sit outside trying to pull the flap open. It wouldn’t surprise me if she took that to the logical extreme and tried to come in tail first. She forgets the secret short cut, and we lie in bed listening to the racket thinking: ‘Harley! Push!’
14
Fermat proved this theorem before Gauss invented modular arithmetic, but not from that point of view.
15
This is the time-honoured formula for: ‘Somebody told me this, but I can’t provide a shred of evidence.’
16
When I was doing a lecture tour of Oregon, I once stayed in the Sylvia Beach Hotel, whose rooms have literary themes: the Oscar Wilde room, the Agatha Christie room. Mine was the Dr Seuss room, with a 15-foot (5-metre) Cat in the Hat painted on one wall.
17
Look up glaber in Latin.
18
This is the only time that the old-fashioned ‘division’ symbol ÷ will appear in this book. Oops.
19
He would also have realised that he need dig only along an arc of a circle, centre Buccaneer Bay and radius 99 nautical perches.
20
Or earlier. The warp drive was invented in 2063 by Zefram Cochrane of Alpha Centauri, but the early version used a fusion plasma as an energy source. By the 22nd century and the first series of Star Trek, the warp drive was powered by a gravimetric field displacement manifold (or warp core) that used antimatter to create energy. In 1994, in our own universe, the physicist Miguel Alcubierre discovered a ‘warp drive’ that does not conflict with relativity, yet allows faster-than-light travel. The trick is the oft-repeated science fiction mantra that ‘while there is a limit to the speed with which matter can travel through space, there is no limit to the speed with which space can travel through space’. Alcubierre found a solution of Einstein’s equations for gravity in which the space ahead of a spacecraft contracts while the space behind expands. The spaceship surfs this wave, carried along by a warp bubble of entirely normal space, relative to which it is stationary. Unfortunately, it takes a lot of negative-energy matter to build an Alcubierre drive, and we don’t have any.
21
This is the kind of energy that things acquire when they move, and in classical mechanics it is half the mass times the square of the speed.
22
However, uncharged particles are not always the same as their antiparticles. The neutron, an uncharged particle, is made from quarks, which individually have non-zero charges. The antineutron is made from the corresponding antiquarks, so the neutron and antineutron are different.
23
The CAT scanner was pioneered by EMI, primarily a recording company. It is suspected that the millions of dollars came from the sale of Beatles records.
24
As of 2006, the International Astronomical Union declared that Pluto is no longer considered a ‘planet’, but a ‘dwarf planet’ or ‘plutoid’. Not all astronomers approve of this.
25
‘Clump’ is a metaphor: they don’t look like the asteroid belts in Star Wars, and what clumps is the distances, not the asteroids themselves. Actually, nowhere in the asteroid belt looks like an asteroid belt in Star Wars. If you stood on a typical asteroid and looked around for the nearest one, it would be about a million miles (1.6 million kilometres) away. No exciting chase scenes, then.
26
As of 1 April 2009. See The Extrasolar Planets Encyclopaedia at www.exoplanet.eu for the latest information.
27
That is, it is item HR8799 in the Yale Bright Star Catalogue. The HR prefix refers the earlier Harvard Revised Photometry Catalogue, and most of the stars listed in the Yale catalogue come from this.
28
Except when checking that it has only one side, by colouring it. Then the ink doesn’t soak through. If it did, an ordinary cylinder would have one side. Because of difficulties like this, mathematicians approach the whole topic differently, talking of ‘orientations’ rather than ‘sides’.
29
More properly the Journal für die reine und angewandte Mathematik (Journal for Pure and Applied Mathematics).
30
Whose total population far exceeds that of the world’s baseball-playing countries.
31
In Terry Pratchett’s Discworld novel Pyramids, there is an Ephebian philosopher named Xeno, who proved that an arrow cannot hit a running man. Other philosophers agreed, with the proviso that ‘it is fired by someone who has been in the pub since lunchtime’. Xeno also claimed that the tortoise is the fastest animal on the Disc, but actually it is the ambiguous puzuma, which travels close to the speed of light. If you see a puzuma, it’s not there.
32
Indeed, a continuum, which according to Cantor is a bigger kind of infinity than that of the whole numbers (see Cabinet, page 160).
33
It is also the name of the principal villain in Stargate SG-1, a Goa’uld System Lord, in case that’s more familiar.
34
It was probably suggested by John of Palermo, but it is the emperor’s question all the same, just as the Great Pyramid was indisputably built by the pharaoh Khufu. Emperors are like that. Hans Christian Anderson’s story of the emperor’s new clothes is completely unconvincing: any little boy who dared to contradict the emperor would have ended up in jail. The cliché ‘the emperor has no clothes’ affirms the imperial status - what people usually mean is that the clothes contain no emperor, which isn’t quite the same thing.
35
You can give a slipstick to a pig, but it’s still a pig.
36
Stress the third syllable, arithmetic, not arithmetic. An older term is ‘arithmetic progression’.
37
Who appears to be pseudonymous.
38
That is, do you allow both attributes, or only one?
39
The group of p-adic integers has no faithful group action on a manifold. Hope that helps.
40
Look up the Latin funiculus.
41
That is, Olaf Tryggvason, the son of Tryggve Olafsson, who was king from 995 to 1000. Before the game of dice, Olaf had proposed marriage to Sigrid the Haughty, the Queen of Sweden, in an attempt to unite Scandinavia. She wasn’t keen.
42
Who seems to have been Olof the Treasurer, from the dates. As it happens, he was the son of Eric the Victorious and Sigrid the Haughty. It’s a small world.
43
According to which a monkey did write the collected works of Shakespeare, though not - to begin with - on a typewriter. It performed the feat indirectly, by producing descendants that evolved into . . . Shakespeare. This is a far more efficient approach.
44
I’m not counting x0 here, although that’s an integer too. However, it’s an arbitrary starting-point, which is a reason - not a terribly good one, but a reason nonetheless - for omitting it from the count. I mention this only because dozens of readers will write to me about it if I don’t. Anyway, if I included x0, then 42 would become 43, and the gratuitous link to Hitch Hiker wouldn’t work.
45
To play the game you will need to make a set of cards - I don’t know anywhere that sells them. It’s worth the effort.
46
They thought it was a bar.
47
Abbott Abbott = A2?
48
He also sees the circles ‘edge on’. Just as we see only a 2D projection - or a stereo pair of projections - of a 3D object.
49
Alas! Poor Pluto.
50
17 May, 2.46 p.m.
51
17 May, 2.47 p.m.
52
Except when you’re trying to stuff it into the cat basket, to take it to the vet.
53
This was unfair to pigs, and ignored a long tradition of political piggery, including the book Lipstick on a Pig: Winning in the No-Spin Era by Someone Who Knows the Game by Victoria Clarke, an Assistant Secretary in George W. Bush’s administration. See: en.wikipedia.org/wiki/Lipstick_on_a_pig.
54
I made this up, but it wouldn’t surprise me to find it’s been used for centuries in parts of Lincolnshire.
55
In A Brief History of Time, where he mentions editorial advice that every formula halves a book’s sales. So he could have sold twice as many. Ye gods.
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