Ian Stewart
Page 17
• What is the rule for forming the sequence? The title of this section is a hint.
• Roughly how long is the nth term in this sequence? [For experts only]
Answers on page 323
Non-Mathematicians Musing About Mathematics
The things of this world cannot be made known without a knowledge of mathematics. Roger Bacon
I had a feeling once about Mathematics - that I saw it all ... I saw - as one might see the transit of Venus or even the Lord Mayor’s Show - a quantity passing through infinity and changing its sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable, but it was after dinner and I let it go. Sir Winston Spencer Churchill
Mathematics seems to endow one with something like a new sense. Charles Darwin
For a physicist, mathematics is not just a tool by means of which phenomena can be calculated; it is the main source of concepts and principles by means of which new theories can be created. Freeman Dyson
Do not worry about your difficulties in Mathematics. I can assure you mine are still greater. Albert Einstein
Equations are just the boring part of mathematics. I attempt to see things in terms of geometry. Stephen Hawking
Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house. Robert A. Heinlein
Mathematics may be compared to a mill of exquisite workmanship, which grinds your stuff to any degree of fineness; but, nevertheless, what you get out depends on what you put in; and as the grandest mill in the world will not extract wheat flour from peascods, so pages of formulae will not get a definite result out of loose data. Thomas Henry Huxley
Medicine makes people ill, mathematics make them sad, and theology makes them sinful. Martin Luther
I tell them that, if they will occupy themselves with the study of mathematics, they will find in it the best remedy against the lusts of the flesh. Thomas Mann
The greatest unsolved theorem in mathematics is why some people are better at it than others. Adrian Mathesis37
She knew only that if she did or said thus-and-so, men would unerringly respond with the complimentary thus-and-so. It was like a mathematical formula and no more difficult, for mathematics was the one subject that had come easy to Scarlett in her schooldays. Margaret Mitchell
The advancement and perfection of mathematics are intimately connected with the prosperity of the State. Napoleon I
Mathematical propositions express no thoughts ... we use mathematical propositions only in order to infer from propositions which do not belong to mathematics to others which equally do not belong to mathematics. Ludwig Wittgenstein
[Mathematics] is an independent world. Created out of pure intelligence. William Wordsworth
I’m sorry to say that the subject I most disliked was mathematics. I have thought about it. I think the reason was that mathematics leaves no room for argument. If you made a mistake, that was all there was to it. Malcolm X
Like the crest of a peacock, so is mathematics at the head of all knowledge. An old Indian saying
Euler’s Conjecture
Fermat’s Last Theorem states that two non-zero integer cubes can’t add up to a cube, and ditto for fourth, fifth or higher powers. It was famously proved by Andrew Wiles in 1994-5 (Cabinet, page 50). One of the first people to make inroads into the problem was Euler, who proved the Last Theorem for cubes: two non-zero cubes cannot add up to a cube. But he also noticed that three cubes can add up to a cube. In fact,
33 + 43 + 53 = 63
Euler guessed (the fancy word is ‘conjectured’) that you need to add at least four fourth powers to get a fourth power, at least five fifth powers to get a fifth power, and so on.
Unlike Fermat, he was wrong. In 1966 Leon Lander and Thomas Parkin discovered that
275 + 845 + 1105 + 1335 = 1445
This remained the only known example of the failure of Euler’s conjecture until 1988, when Noam Elkies discovered that
2,682,4404 + 15,365,6394 + 187,9604 = 20,615,6734
In fact, Elkies proved that there are infinitely many cases where three fourth powers add up to a fourth power - but most of them require very big numbers. Roger Frye used a computer to search by trial and error, and found the smallest example:
95,8004 + 217,5194 + 414,5604 = 422,4814
The Millionth Digit
Suppose we write out all whole numbers in turn, strung together like this:
1234567891011121314151617181920212223242526. . .
and so on.
What is the millionth digit?
Answer on page 324
Piratical Pathways
Roger Redbeard, the fiercest pirate in the Kidnibbean Sea, has forgotten a vital piece of information - the address of his bank in the Banana Islands, where he keeps his loot safe from the attentions of the tax authorities. He knows which street it is on, but there are more than thirty banks on Taxhaven Street, all nameless, all looking exactly alike.
All is not lost, however, because he has a map.
Redbeard’s map.
The address of his bank is cunningly concealed in this map: it is the number of distinct ways to trace the word PIRATE, starting at the circle marked P and spelling out the word letter by letter to end at the circle marked E. The address is the number of different ways that this can be achieved, always moving along the lines linking the letters.
What is the address of Redbeard’s Bank?
Answer on page 324
Trains That Pass in the Siding
Two trains, the Atchison Flier (A) and the Topeka Bullet (B), are travelling in opposite directions towards each other along the same single-line track. Each consists of one locomotive, at the front, and nine coaches. Both locomotives and all coaches have the same length. The siding can accommodate no more than four coaches or locomotives in total at any one time, while leaving room for trains to pass along the main track.
Can the trains pass each other? If so, how?
Answer on page 325. [Hint: coaches can be decoupled.]
We’re stuck - aren’t we?
Please Make Yourself Clear
The mathematical logician Abraham Fraenkel, who was of German origin, once boarded a bus in Tel Aviv, Israel. The bus was scheduled to depart at 9.00 precisely, but by 9.05 it was still sitting in the bus station.
Aggrieved, Fraenkel waved a timetable at the driver.
‘What are you - a German or a professor?’ the driver enquired.
‘Do you mean the inclusive or, or the exclusive or?’ Fraenkel replied.38
Abraham Fraenkel.
Squares, Lists and Digital Sums
The list
81, 100, 121, 144, 169, 196, 225
consists of seven consecutive squares. It has a curious feature: the sum of the decimal digits of each of these numbers is itself a square. For example 1 + 6 + 9 = 16 = 42.
Find another sequence of seven consecutive squares with the same property.
Answer on page 326
Hilbert’s Hit-List
In 1900, the German mathematician David Hilbert gave a famous lecture to the International Congress of Mathematicians in Paris, in which he listed 23 of the most important problems in mathematics. He didn’t list Fermat’s Last Theorem, but he mentioned it in the introduction. Here’s a potted description of Hilbert’s problems, and their current status.
1. Continuum Hypothesis
In Cantor’s theory of infinite cardinal numbers (Cabinet, pages 157-61), is there a number strictly between the cardinalities of the integers and the real numbers?
Solved by Paul Cohen in 1963 - the answer can go either way depending on which axioms you use for set theory.
2. Logical Consistency of Arithmetic
Prove that the standard axioms of arithmetic can never lead to a contradiction.
Solved by Kurt Gödel in 1931, who proved that this can’t be done with the usual axioms for set theory (Cabinet, pa
ge 205). On the other hand, Gerhard Gentzen proved in 1936 that it can be done using transfinite induction.
3. Equality of Volumes of Tetrahedra
If two tetrahedra have the same volume, can you always cut one into finitely many polyhedral pieces, and reassemble them to form the other?
Hilbert thought not. Solved in 1901 by Max Dehn - Hilbert was right.
4. Straight Line as Shortest Distance Between Two Points
Formulate axioms for geometry in terms of the above definition of ‘straight line’, and investigate what happens.
The problem is too broad to have a definitive solution, but much work has been done.
5. Lie Groups Without Assuming Differentiability
Technical issue in the theory of groups of transformations.
In one interpretation, solved by Andrew Gleason. However, if it is interpreted as the Hilbert-Smith conjecture,39 it remains unsolved.
6. Axioms for Physics
Develop a rigorous system of axioms for mathematical areas of physics, such as probability and mechanics.
Andrei Kolmogorov axiomatised probability in 1933, but the question is a bit vague and is largely unsolved.
7. Irrational and Transcendental Numbers
Prove that certain numbers are irrational (not exact fractions) or transcendental (not solutions of polynomial equations with rational coefficients). In particular, show that, if a is algebraic and b is irrational, then ab is transcendental - so, for example, 2√2 is transcendental.
Solved, affirmatively and independently, by Aleksandr Gelfond and Theodor Schneider in 1934.
8. Riemann Hypothesis
Prove that all non-trivial zeros of Riemann’s zeta function, in the theory of prime numbers, lie on the line ‘real part = ’.
Unsolved. Possibly the biggest open problem in mathematics (see Cabinet, page 215).
9. Laws of Reciprocity in Number Fields
The classical law of quadratic reciprocity, conjectured by Euler and proved by Gauss in his Disquisitiones Arithmeticae of 1801, states that if p and q are odd primes then (see page 62 for notation) the equation p ≡ x2 (mod q) has a solution if and only if q ≡ y2 (mod p) has a solution, unless p and q are both of the form 4k - 1, in which case one has a solution and the other does not. Generalise this to other powers than the square.
Partially solved.
10. Determine When a Diophantine Equation has Solutions
Find an algorithm which, when presented with a polynomial equation in many variables, determines whether any solutions in whole numbers exist.
In 1970, Yuri Matiyasevich, building on work by Julia Robinson, Martin Davis and Hilary Putnam, proved that there is no such algorithm.
11. Quadratic Forms with Algebraic Numbers as Coefficients
Technical issues, leading in particular to an understanding of the solution of many-variable quadratic Diophantine equations.
Partially solved.
12. Kronecker’s Theorem on Abelian Fields
Technical issues generalising a theorem of Kronecker about complex roots of unity.
Still unsolved.
13. Solving Seventh-Degree Equations using Special Functions Niels Henrik Abel and Évariste Galois proved that the general fifth-degree equation can’t be solved using nth roots, but Charles Hermite showed that it can be solved using elliptic modular functions. Prove that the general seventh-degree equation can’t be solved using functions of two variables.
A variant was disproved by Andrei Kolmogorov and Vladimir Arnold. Another plausible interpretation remains unsolved.
14. Finiteness of Complete Systems of Functions
Extend a theorem of Hilbert, about algebraic invariants for specific transformation groups, to all transformation groups.
Proved false by Masayoshi Nagata in 1959.
15. Schubert’s Enumerative Calculus
Schubert found a non-rigorous method for counting various geometric configurations by making them as singular as possible (lots of lines overlapping, lots of points coinciding). Make this method rigorous.
Progress in special cases; no complete solution.
16. Topology of Curves and Surfaces
How many connected components can an algebraic curve of given degree, defined in the plane, have? How many distinct periodic cycles can an algebraic differential equation of given degree, defined in the plane, have?
Limited progress in special cases; no complete solution.
17. Expressing Definite Forms by Squares
If a rational function always takes non-negative values, must it be a sum of squares?
Solved by Emil Artin, D. W. Dubois and Albrecht Pfister. It is true over the real numbers, but false in some more general number systems.
18. Tiling Space with Polyhedra
General issues about filling space (Euclidean or not) with congruent polyhedra. Also mentions sphere-packing problems, notably the Kepler conjecture that the most efficient way to pack spheres in space is the face-centred-cubic lattice.
The Kepler problem has been solved, with a computer-aided proof, by Thomas Hales (see Cabinet, page 231). The main question about polyhedra asked by Hilbert has also been solved.
19. Analyticity of Solutions in Calculus of Variations
The calculus of variations emerged from mechanics, and answers questions like: ‘Find the shortest curve with the following properties.’ If a problem in this area is defined by nice (‘analytic’) functions, must the solution be equally nice?
Proved by Ennio de Giorgi in 1957 and, with different methods, by John Nash.
20. Boundary Value Problems
Understand the solutions of the differential equations of physics, inside some region of space, when properties of the solution on the boundary of that region are prescribed. For example, mathematicians can find how a drum of given shape vibrates when its edge is fixed, but what if the edge is constrained in more complicated ways?
Essentially solved, by numerous mathematicians.
21. Existence of Differential Equations with Given Monodromy
A famous type of complex differential equation, called Fuchsian, can be understood in terms of its singular points and its monodromy group (which I won’t even attempt to explain). Prove that any combination of these data can occur.
Answered yes or no, depending on interpretation.
22. Uniformisation using Automorphic Functions
Algebraic equations can be simplified by introducing suitable special functions. For instance, the equation x2 + y2 = 1 can be solved by setting x = cos θ and y = sin θ for a general angle θ. Poincaré proved that any two-variable algebraic equation can be ‘uniformised’ in this manner using functions of one variable. Technical question about extending these ideas to analytic equations.
Solved by Paul Koebe soon after 1900.
23. Development of Calculus of Variations
In Hilbert’s day, the calculus of variations was in danger of becoming neglected, and he appealed for fresh ideas.
Much work has been done, but the question is too vague to be considered solved.
In 2000, the German historian Rüdiger Thiele discovered, in Hilbert’s unpublished manuscripts, that he originally planned to include a 24th problem:
24. Simplicity in Proof Theory
Develop a rigorous theory of simplicity and complexity in mathematical proofs.
This is closely related to the concept of computational complexity, and the notorious (and unsolved) P = NP? problem (see Cabinet, page 199).
Match Trick
Remove exactly two matches to leave two equilateral triangles.
Answer on page 327
Take two matches away, and leave two triangles.
Which Hospital Should Close?
Statisticians know that strange things happen when you combine data. One of them is Simpson’s paradox, which I will illustrate with an example.
The Ministry of Health was collecting data on the success of surgical operations. Two hospitals - Saint
Ambrose’s Infirmary and Bumbledown General - were in the same area, and the ministry was going to close the less successful of the two.
• Saint Ambrose’s Infirmary reported operating on 2,100 patients, of whom 63 (3%) died.
• Bumbledown General reported operating on 800 patients, of whom 16 (2%) died.
To the minister, the situation was perfectly obvious: Bumbledown General had a lower death rate, so he would close Saint Ambrose’s Infirmary.
Naturally, the Chief Executive of Saint Ambrose’s Infirmary protested. But he explained that there was a good reason for reconsidering, and asked the minister to break down the figures into two categories: male and female. The minister was reluctant to do so, on the grounds that it was obvious that Bumbledown General would still do better overall. However, it was easier to look at the new data than to argue, so he obtained the corresponding figures, classified by sex.
• Saint Ambrose’s Infirmary operated on 600 females and 1,500 males. Of these, 6 females died (1%) and 57 males died (3.8%).
• Bumbledown General operated on 600 females and 200 males. Of these, 8 females died (1.33%) and 8 males died (4%).
Note that the numbers add up correctly, to give the original data.
Strangely, Bumbledown General had a worse death rate than Saint Ambrose’s Infirmary in both categories. Yet, when the figures were combined, Saint Ambrose’s Infirmary had a worse death rate than Bumbledown General.