by Wells, David
Hidden Connections, Double Meanings [Wells 1988: 8–9]
Here are four mathematical puzzles expressed in the form of traditional riddles to illustrate the connection:
‘A cube is to an octahedron as a tetrahedron is to…?
‘I am 20 less than my square. What number am I?’
‘What is the difference between the square of a cube and the cube of a square?’
‘I have 4 sides but only a single line of symmetry. My brother has the same property but he and I are different. What are we?’
Another traditional puzzle appeals to me because it sets the solver a trap, albeit a rather obvious one. Here is one version. A snail – or a serpent or a frog! – lies at the bottom of a well, 30 units deep. It climbs 6 units every day but falls back 3 units every night. How long does it take to escape from the well? The obvious answer is that the snail rises 3 units every day-and-night, on balance, so it takes 10 days-and-nights to escape, but this is wrong because it will actually reach the top of the well half-way through the 10th day and after only 9 nights.
This amuses me and must have amused many many people in the past because the puzzle is first found in the Indian Bakhshali manuscipt from the 7th century CE and appears again during the Renaissance in puzzle collections by Dell’Abaco in about 1370 and then by Chuquet in 1484 and in many collections since.
Everyday puzzles
Many everyday situations easily give rise to puzzles. Take a situation, spot a formal feature, strip away everything else and you may have a puzzle which will provide amusement, recreation and fun. The Romans punished companies of soldiers who were cowardly in battle by decimation. They were lined up and every tenth man was executed. Here is a puzzle on the same principle, called after the Jewish historian Josephus who is supposed to have saved himself by just such a trick as the puzzle describes:
Josephus, during the sack of the city of Jotapata by the Emperor Vespasian, hid himself in a cellar with forty other Jews who were determined to commit suicide rather than fall into the hands of the Romans. Not wishing to abandon life, he proposed that they form a circle and that every third person, counting round the circle, should die, in the order in which they were selected. In other words, the count was: ‘one, two three, out, four five six, out, seven…’ Where did he place himself, and a companion who also wished to live, in order to ensure that they were the last two remaining?
Anyone might notice that standing right under a statue on a plinth does not give the best view, but nor does standing far away. Here is the resulting puzzle: from which point does the statue subtend the greatest angle? This puzzle has been re-invented many times, most recently as a problem about rugby:
According to the rules of rugby football, a conversion of a try must be taken on a line extending backwards from the point of touchdown, at right-angles to the goal line. From which point on this line should the conversion be taken if the aim is to maximise the angle subtended by the goal-posts?
Figure 3 Angle of rugby conversion puzzle
The solution, for the mathematician but not necessarily for the practising rugby player, is to draw a circle through the goal posts which touches the perpendicular line on which the conversion must be taken (Figure 3). The kick should be taken from T.
Russian parents used to challenge their children as young as five or six years with Merchant's Problems, tricky questions to make them think. Here is one of them, paraphrased from the late great Russian mathematician Vladimir Arnol'd, which has subsequently become a well-known puzzle in the West:
From a barrel of beer you transfer one teaspoonful of beer into your teacup. Then you transfer one teaspoonful of the mixture back into the barrel of beer. Is there now more beer in the tea or more tea in the beer?
These cunning questions were not unlike some traditional riddles that survived in our society into the Victorian era, and then died out, apart from some degenerate forms which appear in children's comics. Here are some modern mathematical ‘tricky questions’, suitable for pupils: Why is 3 times 4 equal to 4 times 3?
What is the greatest number less than 10?
Does 10 have a square root?
Can you always tell if one fraction is bigger than another?
Does every shape have an area?
The next puzzle is more artificial. It was created by William Rowan Hamilton, a famous nineteenth-century mathematician, who sold the rights to Jacques,
Figure 4 The Icosian Game
manufacturers of fine chess sets, for the sum of £25. It was first sold in London in 1859 as a solid puzzle and later as a flat version called The Icosian Game. The solid wooden puzzle was in the form of an icosahedron with the names of 20 cities marked at its vertices from B for Bruxelles to Z for Zanzibar with a few letters left out. The flat version had the design in Figure 4.
The object of the game was to start at B and travel along the edges of the icosahedron to Z, visiting every city once and only once. Graph theorists today call the solution to such a puzzle, on any graph, a Hamiltonian circuit.
In all these puzzles, the element of challenge and mystery is the hook, and the child – and many an adult – is happy to be caught.
Construction toys illustrate a different type of formality. LEGO ®, named by its Danish inventor for leg godt, meaning to play well, has been awarded the title of ‘the greatest plastic product of all time’. It is also extremely formal and mathematical, like MECCANO and other construction toys. Because the pieces are manufactured to fine tolerances they can easily be joined in innumerable different ways – there are more than 100 million different ways to combine the six eight-stud bricks of the same colour.
In the child's imagination the pieces always fit perfectly, of course: it is only in real life that this perfect game-like-ness sometimes breaks down. If we think of LEGO-in-the-mind as a construction game in which every move can be made perfectly, then real LEGO ® with real coloured plastic pieces is a very precise but not-quite-perfect model of the mental game.
Continuing our gentle meander from puzzles to abstract games and on to mathematics, we come to those popular mathematical recreations that elegantly and intriguingly link all three themes together in a tight knot of fascination. Several of these recreations – unicursal puzzles, knight tours and the Tower of Hanoi – are presented in Chapter 1.
What do these links tell us about mathematics? That's what this book is about. It's a serious misapprehension that mathematics is all about calculation. It isn't, it's about imagination, insight and intuition. Real mathematics can be illuminated from three different perspectives. It is much like a collection of abstract games, it is much like a science, and it is very much a matter of perception, of seeing.
As a collection of related games, often still changing and evolving, it has both sharp tactics and subtle strategies, standard sequences, powerful methods, familiar manoeuvres, winning ‘moves’ and brilliant combinations, all within the miniature world of the game as created by the rules. It is because mathematics is game-like that mathematicians can prove their conclusions. However, they also ask questions about the game. Often this leads to new ideas, to new possibilities, so the original game is transformed into something else.
As scientists, mathematicians explore their miniature worlds, observe, generalise, create hypotheses and conjectures, perform experiments to test their speculations and draw conclusions – though as long as they are wearing their scientific hats they can never actually prove anything.
As observers, they spot patterns and connections, observe analogies, notice unusual sequences of moves, see one situation in different ways or different situations in the same way, and generalise their results, leading to ideas of structure.
The three aspects cannot be separated. The mathematician as game-player observes and makes conjectures; the mathematician as scientist makes moves and spots possibilities; the mathematician as observer studies objects which are much like the pieces in an abstract game of chess. This book introduces all three aspec
ts.
It is divided into two parts. Part I is about puzzles and games and mathematical recreations. Part II presents mathematics in its three aspects, naturally matching the arguments of Part I, but now illustrated by purely mathematical examples.
The final chapter takes a step back to look at society and culture as a whole and to see what game-like features we can find in it – both illuminating the role of mathematics and helping to explain why maths exists at all.
1 Recreations from Euler to Lucas
Euler and the Bridges of Königsberg
The great Swiss mathematician Leonard Euler was born in 1707 in Basel and published his first paper at the age of 19. From then until his death in St Petersburg in 1783, he wrote a stream of papers on subjects from algebra and calculus to geometry and number theory, to acoustics and music, the theory of lenses, the motion of ships, the atmospheres of the moon and Venus, and many other topics, making him by far the most productive mathematician of all time.
His wonderful memory helped him. He knew Virgil's Aeniad by heart and could start ‘reading’ it from memory, given any page number. He could also do complex calculations in his head: when two pupils summed the first 17 terms of a series but couldn't agree on the fiftieth decimal place, Euler corrected them by doing the calculation mentally. Nothing distracted him, not even his 13 children or his blindness in 1771 after a cataract operation failed: he simply dictated his results to an assistant and wrote his papers faster than ever, nearly 900 in all. Thirty years after his death, the Academy in St Petersburg was still publishing his original works.
He gave his name to several Euler theorems, to the Euler equations and the Euler–Lagrange equation, two Euler formulae, the Euler and Eulerian numbers, and the Euler characteristic. His deepest ideas and discoveries had a lasting importance, yet he was also interested in very simple puzzles and pastimes. Figure 1.1 shows one of them.
The Prussian town of Königsberg lay on the river Pregel which was crossed by seven bridges. The citizens of Königsberg were reputed to enjoy the recreation of starting on one of the banks and then strolling across all seven bridges, once and only once each, to their destination.
Their strolling suggests not just one but several puzzles: is it indeed possible to cross all the bridges exactly once each? If so, can you start wherever you choose? If bridges were added or deleted, would the puzzle still be solvable? To answer these questions a natural first step is to explore the problem like a scientist by trying different starting points and routes. Which are successful? Which seem impossible? Intriguing!
Euler was intrigued and in 1736 he published a paper answering all these questions. Curiously, he didn't simplify the figure. We will, by focusing on the essential features of the puzzle and ignoring everything else (the process called abstraction). We get Figure 1.2.
Figure 1.2 Simplified River Pregel and bridges
As a result of this transformation, the western bank of the Pregel is represented by the single point A, and the south bank by D. The island and the land to the north of it are represented by the points C and B. The seven bridges are the seven lines joining these points. We can now draw the puzzle on a scrap of paper or a restaurant napkin and check possible solutions.
Trial and error correctly suggests that the puzzle cannot be solved, and Euler explained why. His insight, which was original then but might today be spotted by a smart child, was that whenever you visit a point and then leave it, two of the lines from the point are ‘used up’. Therefore, if the puzzle is to be solved, any point that you visit which is not your start or end point must have an even number of lines leaving it. Only your start and end points, if necessary, can be visited by an odd number of lines. It follows that Euler's original problem cannot be solved, since A, B, C and D are all joined by either 3 or 5 lines. Behind the surface appearance not all the meeting points are the same – there is a hidden significance in the odd vertices.
This explanation also illustrates the amazing sense of conviction we can get from good visual proofs, sometimes called ‘proofs by showing’ or ‘proofs without words’. No complex calculation is needed, no algebra, just looking ‘in the right way’ until the penny drops.
Today this puzzle is called the Königsberg bridge problem. Euler's paper was titled Solutio problematis ad geometriam situs pertinentis, or in English, The solution of a problem relating to the geometry of position. Euler referred to geometry although it has nothing to do with lengths or angles or circles or triangles or even straight lines but then added of position because only the relative position of the banks of the river and its bridges mattered.
The problem of the Bridge of Königsberg is now considered to be the first ever result in a new branch of mathematics, originally called analysis situs or the analysis of situation but now called topology from the Greek root topos = place.
It is indeed a remarkable problem, an everyday puzzle which could have been posed by the citizens of many towns and cities throughout the centuries. It needs no mathematical skill to understand and only a little ingenuity to solve. Euler's paper was his simplest ever. It is also totally abstract like all the purest mathematics. The town of Königsberg, the river Pregel, its banks and bridges are mere distractions which we can strip away to leave the core puzzle. No wonder that endless analogous puzzles appeared in later puzzle collections such as this ‘figure tracing’ poser (Figure 1.3) from a popular Victorian puzzle book, Everybody's Illustrated Book of Puzzles, by Don Lemon [Lemon 1890: 66].
The reader is instructed: ‘Starting from A, make this figure with one continuous line, without taking the pencil from the paper or going over any line twice, finishing at B.’
Euler knew that the Bridges of Königsberg was a new kind of problem: he opened his paper by referring to Leibniz who had introduced the term analysis situs, without giving any problems involving it:
In addition to that branch of geometry which is concerned with magnitudes and which has always received the greatest attention, there is another branch, previously almost unknown, which Leibniz first mentioned, called the geometry of position…This branch is concerned only with the determination of position and its properties; it does not involve measurement…
[Wilson 1985: 790]
We will add that because no measurement is needed, merely ‘sight’ of the lines and where they meet, you can solve such unicursal puzzles in your head if you have a good visual imagination.
Euler and knight tours
Euler was also interested in another, much older puzzle. Figure 1.4 shows a part of a chess board and the possible moves of a knight: two squares north, south, east or west and then one square at right-angles.
Figure 1.4 Knight and knight's moves
The knight's tour puzzle is to place the knight on a square of your choice and then to make a tour of 63 moves visiting every other square on the chess board. As in the Königsberg puzzle, no square can be visited twice. A harder puzzle is to make 64 moves that will bring you back to your starting square, a re-entrant solution. To get an idea of the possibilities and the difficulty of the puzzle and its pleasures, we can try it on smaller boards (Figures 1.5 and 1.6).
On the 3 by 3 board, we easily visit 8 squares ending where we started by an elegantly symmetrical route, but we cannot include the central square because it is not a knight's move away from any other.
Figure 1.5 Knight tours on 3 by 3 board
Figure 1.6 Knight tours on 4 by 4 board
A little experiment will also suggest that there are no tours on the 4 by 4 board either and we can prove this conclusion also though the reasoning is a little more complicated. In the above figure, b can only be reached from a or c. Let's suppose we go to b from a. Then we must move a-b-c and then d, otherwise d can never be reached: but now we are stuck at d. It follows that since we cannot afford to reach b from any other square, we must either start or end there. So our route must go b-a-d-c or b-c-d-a: but then we have the identical problem with the other pair of corners that cannot no
w both be visited, and a little more experiment shows that a solution is impossible.
Figure 1.7 Knight tour on 5 by 5 board
On the 5 by 5 board, however, we can find many routes like the easy solution in Figure 1.7 which wraps repeatedly round the central square, where the tour ends, or the harder-to-find symmetrical tour in Figure 1.8.
Figure 1.8 Symmetrical knight tour
These solutions, however, do not end where they start. Must this be so? Yes, and we can prove this conclusion also by a little reasoning. First we colour the board black and white like a real chess board. If the four corner squares are coloured black, we will see that in every solution we have to start and end on a black square. Why? Because there are 13 black squares and only 12 white squares and with every knight move you go either from black to white or from white to black. Therefore any sequence which visits them all must go: black – white – black – white – black – white…black, and the last square can never be a knight's move from the first.
The many solutions on the 5 by 5 board suggest that on even larger boards there will be even more solutions. Indeed, there are, including this early solution by the Muslim chess player, Al Mani, which illustrates the bizarre combination of pattern and idiosyncracy in such solutions (Figure 1.9).
