Ian Stewart Page 10

  1 + 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 + 15 + 20 + 24 + 30

  + 40 + 60 + 120

  = 360 = 3×120

  Here are a few other multiply perfect numbers. Many more are known. (The dots between the numbers mean ‘multiply’.)

  Number Discoverer Date

  Triperfect

  23 .3.7 Robert Recorde 155.7

  25 .3.7 Pierre de Fermat 1636

  29 .3.11.31 André Jumeau de Sainte-Croix 1638

  28 .5.7.19.37.73 Marin Mersenne 1638

  Quadruperfect

  25.33 5.7 René Descartes 1638

  23.32 .5.7.13 René Descartes 1638

  29.33 .5.11.31 René Descartes 1638

  28 .3.5.7.19.37.73 Édouard Lucas 1891

  Quintuperfect

  27.34.5.7.112 .17.19 René Descartes 1638

  210.35.5.72.13.19.23.89 Bernard Frénicle de Bessy 1638

  Sextuperfect

  223.37.53.74.113.133.172 .31.41.61.241.307.467.2801. Pierre de Fermat 1643

  227.35.53.7.11.132 .19.29.31.41.43.61.113.127 Pierre de Fermat 1643

  Septuperfect

  246(247-1).192.127.315.53.75 .11.13.17.23.31.37.41.43.61.89.97.193.442151 Allan Cunningham 1902

  Target Practice

  Robin Hood and Friar Tuck were engaging in some target practice. The target was a series of concentric rings, lying between successive circles with radii 1, 2, 3, 4, 5. (The innermost circle counts as a ring.)

  The target.

  Friar Tuck and Robin both fired a number of arrows.

  ‘Yours are all closer to the centre than mine,’ said Tuck ruefully.

  ‘That’s why I’m the leader of this outlaw band,’ Robin pointed out.

  ‘But let’s look on the bright side,’ Tuck replied. ‘The total area of the rings that I hit is the same as the total area of the rings you hit. So that makes us equally accurate, right?’

  Naturally, Robin pointed out the fallacy ... but:

  Which rings did the two archers hit?

  (A ring may be hit more than once, but it only counts once towards the area.)

  For a bonus point: what is the smallest number of rings for which this question has two or more different answers?

  For a further bonus point: if each archer’s rings are adjacent—no gaps where a ring that has not been hit lies between two that have - what is the smallest number of rings for which this question has two or more different answers?

  Answers on page 302

  Just a Phase I’m Going Through

  Over the course of one lunar month, the phases of the Moon run from new moon to full moon and back again, passing through various intermediate shapes known as ‘crescent’, ‘first quarter’, ‘waning gibbous’, and the like.

  The two ‘quarter’ moons are so named because they occur one-quarter and three-quarters of the way through the lunar month, starting from a new moon. At these times the area of the visible part is half the Moon’s face, not one-quarter. But there are two times during the cycle where a crescent moon occurs, whose visible area is exactly one-quarter of the area of the lunar disc.

  When is the area of the crescent one-quarter of the area of the disc?

  • When this happens, what fraction of the radius AB is the width CB of the crescent?

  • At which fractions of a full cycle, starting from the new moon, do these special crescents occur?

  To simplify the geometry, assume that the Moon is a sphere, and the orbits of both the Moon (round the Earth) and the Earth (round the Sun) are circles lying in the same plane, with both bodies moving at a constant speed. Then the length of a lunar month will also be constant. Assume, too, that the Sun is so far away that its rays are all parallel, and the Moon is sufficiently distant for its image as seen from Earth to be obtained by parallel projection - as if every point on the Moon were transferred to a screen along a line meeting the screen at right angles. (However, you have to replace the real Moon by a much smaller one, otherwise its image ‘in’ the eye would be 3,474 kilometres, or 2,159 miles, across.)

  Parallel projection of the Moon’s features on to a screen.

  None of these assumptions is true, but they’re good approximations, and the geometry gets a lot harder without them.

  Answers on page 303

  Proof Techniques

  • Proof by Contradiction: ‘This theorem contradicts a well-known result due to Isaac Newton.’

  • Proof by Metacontradiction: ‘We prove that a proof exists. To do so, assume that there is no proof...’

  • Proof by Deferral: ‘We’ll prove this next week.’

  • Proof by Cyclic Deferral: ‘As we proved last week . . . ’

  • Proof by Indefinite Deferral: ‘As I said last week, we’ll prove this next week.’

  • Proof by Intimidation: ‘As any fool can see, the proof is obviously trivial.’

  • Proof by Deferred Intimidation: ‘As any fool can see, the proof is obviously trivial.’ ‘Sorry, Professor, are you sure?’ Goes away for half an hour. Comes back. ‘Yes.’

  • Proof by Handwaving: ‘Self-explanatory.’ Most effective in seminars and conference talks.

  • Proof by Vigorous Handwaving: More tiring, but more effective.

  • Proof by Over-optimistic Citation: ‘As Pythagoras proved, two cubes never add up to a cube.’

  • Proof by Personal Conviction: ‘It is my profound belief that the quaternionic pseudo-Mandelbrot set is locally disconnected.’

  • Proof by Lack of Imagination: ‘I can’t think of any reason why it’s false, so it must be true.’

  • Proof by Forward Reference: ‘My proof that the quaternionic pseudo-Mandelbrot set is locally disconnected will appear in a forthcoming paper.’ Often not as forthcoming as it seemed when the reference was made.

  • Proof by Example: ‘We prove the case n = 2 and then let 2 = n.’

  • Proof by Omission: ‘The other 142 cases are analogous.’

  • Proof by Outsourcing: ‘Details are left to the reader.’

  • Statement by Outsourcing: ‘Formulation of the correct theorem is left to the reader.’

  • Proof by Unreadable Notation: ‘If you work through the next 500 pages of incredibly dense formulas in six alphabets, you’ll see why it has to be true.’

  • Proof by Authority: ‘I saw Milnor in the cafeteria and he said he thought it’s probably locally disconnected.’

  • Proof by Personal Communication: ‘The quaternionic pseudo-Mandelbrot set is locally disconnected (Milnor, personal communication).’

  • Proof by Vague Authority: ‘The quaternionic pseudo-Mandelbrot set is well known to be locally disconnected.’

  • Proof by Provocative Wager: ‘If the quaternionic pseudo-Mandelbrot set is not locally disconnected, I’ll jump off London Bridge wearing a gorilla suit.’

  • Proof by Erudite Allusion: ‘Local connectivity of the quaternionic pseudo-Mandelbrot set follows by adapting the methods of Cheesburger and Fries to non-compact infinite-dimensional quasi-manifolds over skew fields of characteristic greater than 11.’

  • Proof by Reduction to the Wrong Problem: ‘To see that the quaternionic pseudo-Mandelbrot set is locally disconnected, we reduce it to Pythagoras’s Theorem.’

  • Proof by Inaccessible Reference: ‘A proof that the quaternionic pseudo-Mandelbrot set is locally disconnected can be easily derived from Pzkrzwcziewszczii’s privately printed memoir bound into volume 1 of the printer’s proofs of the 1831 Proceedings of the South Liechtenstein Ladies’ Knitting Circle before the entire print run was pulped.’

  Second Thoughts

  ‘This is a one-line proof - if we start sufficiently far to the left.’

  How Dudeney Cooked Loyd

  In Mathematical Carnival, the celebrated recreational mathematician Martin Gardner tells us: ‘When a puzzle is found to contain a major flaw - when the answer is wrong, when there is no answer, or when, contrary to claims, there is more than one answer or a better answer - the puzzle is said to be �
��cooked”.’ Gardner gives several examples, the simplest being a puzzle he had set in a children’s book. In the array of numbers circle six digits to make the total of circled numbers equal 21. See page 304 for Gardner’s answer, why he had to cook the puzzle, how he did that, and how one of his readers cooked his cook. Both solutions are what Gardner calls a quibble-cook, because they exploit an imprecise specification in the question.

  9 9 9

  5 5 5

  3 3 3

  1 1 1

  Gardner, a puzzle expert, also mentions a more serious example of cookery involving the two arch-rivals of late 19th and early 20th century puzzling, the American Sam Loyd and the Englishman Henry Ernest Dudeney. The problem was to cut a mitre (a square with one triangular quarter missing) into as few pieces as possible, so that they could be rearranged to make a perfect square. Loyd’s solution was to cut off two small triangles and then use a ‘staircase’ construction - four pieces in all.

  After Loyd had published his solution in his Cyclopaedia of Puzzles, Dudeney spotted an error, and found a correct solution with five pieces. The easier question here is: what was the mistake? The harder one is to put it right.

  Answers on page 304

  Loyd’s four-piece attempt to dissect a mitre into a square.

  Cooking with Water

  Speaking of quibble-cooks: I’m going to set exactly the same puzzle as one I set in Cabinet (page 199), where the answer was ‘impossible’. But I’m looking for a different answer, because this time I’ll allow any suitably clever quibble-cook.

  Three houses have to be connected to three utility companies - water, gas, electricity. Each house must be connected to all three utilities. Can you do this without the connections crossing? (Work in the plane, no third direction to pass pipes over or under cables. And you are not allowed to pass the cables through a house or a utility company building.)

  Actually, I should have said: ‘You are not allowed to pass the cables or pipes through a house or a utility company building.’ I think that was clear from the context, but just in case you disagree, assume that too.

  Connect houses to utilities obeying all conditions.

  Answer on page 305

  Celestial Resonance

  In the earliest days of the telescope, Galileo Galilei discovered that the planet Jupiter had four moons, now named Io, Europa, Ganymede and Callisto. Astronomers now know of at least 63 moons of Jupiter, but the rest are much smaller than these four ‘Galilean’ satellites, and some are very small indeed. The times the Galilean satellites take to go once round Jupiter, in days, are respectively 1.769, 3.551, 7.155 and 16.689. What is remarkable about these numbers is that each is roughly twice the previous one. In fact,

  3.551/1.769 = 2.007

  7.155/3.551 = 2.015

  16.689/7.155 = 2.332

  The first two ratios are very close to 2; the third one is less impressive.

  The simple numerical relationships between the first three periods are not coincidental: they result from a dynamic resonance, in which configurations of moons or planets tend to repeat themselves at regular periods. Europa and Io are in 2:1 resonance, and so are Ganymede and Europa. The ratio is that of the orbital periods of the two moons concerned; the numbers of orbits they make in the same time are in the opposite ratio, 1 : 2.

  Resonances arise because the corresponding orbits are especially stable, so they are not disrupted by any other bodies in the vicinity, such as the other moons of Jupiter. However, to make things more difficult, some types of resonance are especially unstable, depending on the ratio concerned and the physical system involved. We don’t fully understand the reasons for this. But this type of 2:1 resonance is very stable, and this is why we find it in Jupiter’s larger moons.

  The other main orbital resonances within the Solar System are:

  • 3:2 Pluto-Neptune — 90,465 and 60,190.5 days

  • 2:1 Tethys-Mimas — 1.887 and 0.942 days

  • 2:1 Dione-Enceladus — 2.737 and 1.370 days

  • 4:3 Hyperion-Titan — 21.277 and 15.945 days

  where all bodies listed except Pluto and Neptune are moons of Saturn.24

  When thinking about resonances, it is important to realise that any ratio can be approximated by exact fractions, and there can be ‘accidental resonances’ that are unrelated to dynamic influences between the two orbits concerned. All the above are genuine resonances, showing features such as ‘precession of perihelion’ - movement of the orbital position nearest to the Sun - that lock the orbits stably together. Among the accidental resonances that can be found by searching tables of astronomical data are:

  • 13:8 Earth-Venus

  • 3:1 Mars-Venus

  • 2:1 Mars-Earth

  • 12:1 Jupiter-Earth

  • 5:2 Saturn-Jupiter

  • 7:1 Uranus-Jupiter

  • 2:1 Neptune-Uranus

  Some important genuine resonances occur for asteroids - mainly small bodies, most of which orbit between Mars and Jupiter. Resonances with Jupiter cause asteroids to ‘clump’ at some distances from the Sun, and to avoid other distances.25 More asteroids than average have orbits that are in 2:3, 3:4 and 1:1 resonance with Jupiter (the Hilda family, Thule, and the Trojans) because these resonances stabilise the orbits. In contrast, the resonances 1 : 3, 2:5, 3:7 and 1:2 destabilise the orbits: rings and belts are different from individual bodies. As a result, there are very few asteroids at the corresponding distances from the Sun, called Kirkwood gaps.

  Kirkwood gaps and Hilda clumps (1 AU is the Earth-Sun distance).

  Similar effects occur in Saturn’s rings. For instance, the Cassini Division - a prominent gap in the rings - is caused by a 2:1 resonance with Mimas, which this time is unstable. The ‘A ring’ does not slowly fuzz out, because a 6:7 resonance with Janus sweeps material away from the outer edge.

  One of the weirdest resonances occurs in the rings of Neptune, a ratio of 43 : 42. Despite the big numbers, this one seems to be a genuine dynamic effect. Neptune’s Adams ring is a complete, though narrow, ring, and it is much denser in some places than others, so the dense regions create a series of short arcs. The problem is to explain how these arcs are spaced along that orbit, and a 43:42 resonance with the moon Galatea, which lies just inside the Adams ring, is thought to be responsible. The arcs should then be placed at some of the 84 equilibrium points associated with this resonance, which form the vertices of a regular 84-sided polygon, and pictures from Voyager 2 support this.

  A section of the Adams ring: grey, resonance islands; black, ring material.

  Resonances are not confined to the orbital periods of moons and planets. Our own Moon always turns the same face towards our planet, so that the ‘far side’ remains hidden. The Moon wobbles a bit, but 82 per cent of the far side is never visible from Earth. This is a 1:1 resonance between the Moon’s period of rotation around its axis and its period of revolution around the Earth. This type of effect is called spin-orbital resonance, and again there are plenty of examples. It used to be thought that the planet Mercury did the same as our Moon, so that one side — facing the Sun - was tremendously hot, and the other one tremendously cold. This turned out to be a mistake, caused by the difficulty of observing the planet when it was close to the Sun and the absence of any surface markings visible in the telescopes then available. In fact, the periods of revolution and rotation of Mercury are 87.97 days and 58.65 days, with a ratio of 1.4999 - a very precise 3:2 resonance.

  Astronomers now know that many stars also have planets; indeed, a total of 344 ‘extrasolar’ planets has been found26 since the first one was detected in 1989. For example, two planets of the star Gliese 876, known as Gliese 876b and Gliese 876c, are in 2:1 resonance. Extrasolar planets are normally detected either by their (tiny) gravitational effects on their parent star, or by changes in the star’s light as and if the planets pass across its face as seen from Earth. But in 2007 the first telescopic image of such a planet was obtained, around a star rejoicing in the name H
R8799.27 The main difficulty here is that the light from the star swamps that of the planet, so various mathematical techniques are used to ‘subtract’ the star’s light. Early in 2009, it was discovered that one of these stars can be extracted, by similar image-processing methods, from a photo of the star taken by the Hubble telescope in 1998, but that’s an aside. The point is that the dynamics of this three-planet system is unstable, so that we would be unlikely to observe it unless the planets are in 4:2:1 resonance. An important consequence of this line of thinking is that such resonances improve the chances of other stable planetary systems existing. Which, perhaps, also improves the prospect of alien life existing somewhere.

  A good site for this topic is:

  en.wikipedia.org/wiki/Orbital_resonance which has a long list of ‘accidental’ resonances, more explanation of the dynamics involved, and an animation of Jupiter’s moons’ 1:2:4 resonance. There is also an animation showing how planets cause a star’s position to ‘wobble’ at: www.gavinrymill.com/dinosaurs/extra-solar-planets.html

  Calculator Curiosity 2

  What’s special about the number 0588235294117647? (That leading zero does matter here.) Try multiplying it by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, and you’ll see. You do need a calculator or software that works with 16-digit numbers. I find that the human brain, a piece of paper and a pen does that job pretty well.