by Wells, David
1
[etc.]
There is a well-known method of finding the value of any number in any row of this triangle: the number in the nth place (from left or right) of the mth row is m−1Cn−1 or (m−1)!/(n−1)!(m−n)!. The top-right cell of the original square is actually the middle cell of the 15th row of the triangle, and so it will have the value 14C7 = 14·13·12·11·10·9·8/(7·6·5·4·3·2·1) = 3432.
Now that we've ‘seen the light’ this calculation is far quicker than filling in the values of every cell, simple though that was. However, suppose that we now throw a spanner in the works, literally. We delete one of the cells, let's say, (3,4). The goal of the puzzle is the same, to say how many ways there are from (1,1) to (8,8), but without going through (3,4).
On the face of it, this still a perfectly good mathematical puzzle – although the choice of this one square to be avoided does seem highly arbitrary and therefore inelegant – and the solution can still be easily calculated. We just take away from 3432 the number of ways of going from one corner to the other which do go through (3,4). That in turn is the number of routes from (0,0) to (3,4) multiplied by the number of routes from (3,4) to (8,8), which equals the number of routes from (1,1) to (6,5). So, after a little thought, the answer to this altered puzzle (Figure 5.5) is 3432−10×126 = 2172.
Figure 5.5 One cell blacked out
So far, so good. This is still a mathematical calculation though not as simple or elegant as the solution to the original puzzle. But now suppose that we go further and delete two more squares (Figures 5.6).
Figure 5.6 Three cells blacked out
To answer the same puzzle we now have to subtract from the original total all the routes that go through only one, or only two, or all three of the deleted cells, which means calculating and then subtracting seven different numbers. This is no longer either simple or elegant, and if we go further and deleted, say, half a dozen cells as in Figure 5.7, then not only is the calculation ridiculously complicated but it would actually be much quicker to check the number of routes by starting at (1,1) and working our way up to (8,8), as we have done in the figure.
Figure 5.7 Six cells blacked out
What has happened? Somewhere on the way from the original puzzle to this ‘distorted’ puzzle, the mathematical calculation has become more complicated while the checking solution has become simpler. The mathematical solution has lost its simplicity and elegance and the crude game-like solution has become relatively more efficient.
We can see the same phenomenon in other mathematical recreations. Here is the same chessboard but this time the puzzle is to discover in how many ways it can be filled with dominoes, that is, pairs of squares glued together complete edge to complete edge (Figure 5.8).
Figure 5.8 Chessboard and dominoes
We can think of this puzzle as a simplified version of Dudeney's original pentomino puzzle. Instead of the 12 pentominoes with their varied and apparently arbitrary shapes we have simple little dominoes. Surely the question must be easy to answer? Actually, no, it is rather complicated and was first answered by three physicists who were interested in a problem in statistical mechanics and who referred to the dominoes by the physical term dimers. Temperley and Fischer together, and also Kasteleyn, proved in 1961 that the number of ways of filling an m by n rectangle with mn/2 dominoes is:
The apparently simple original puzzle has produced a really complicated formula – including cosines and π – which applies, however, only to rectangles with at least one side even. Suppose that we try to count the ways of filling an odd-odd rectangle with dimers, inevitably leaving one cell empty (Figure 5.9)?
Figure 5.9 Odd chessboard with dominoes and 1 empty cell
All at once the puzzle becomes far more complex – not least because the position of the empty cell can vary – and less symmetrical and therefore less elegant. Indeed, there is no known formula for the number of dimer patterns with an arbitrary odd cell deleted and as for deleting a number of cells at random as we did for the first problem – don't ask! Instead, as the number of deleted cells increases, it becomes easier and easier to check the number of solutions even as a proof becomes a more and more distant prospect.
It seems that as we add an element of arbitrariness to mathematical problems – an element that is present from the start in many mathematical recreations and which perhaps contributes to their fascination – the possibilities of proof fall and the inevitability of relying for solutions on checking and brute force, rise. We would like to think that behind every problem there is some – maybe very deep and very subtle – structure which guarantees that a more or less short and efficient proof can be found, but this may be over-optimistic.
Part II Mathematics: game-like, scientific and perceptual
Introduction
These three aspects of mathematics could be presented in any order. Which is the best order? Professional mathematical papers are frequently criticised for only presenting the game-like aspects of mathematics while saying nothing about scientific or perceptual aspects, as well as omitting every sign of how their results were discovered in the first place. Very occasional exceptions, such as Euler, are famous for being exceptional. Henri Poincaré asked:
Will we ever be able to say that we understand a theory if we want to give to it straight away its final form, that impeccable logic imposes on it? Which ideas were tried out and discarded, which ideas were tried out and retained? If we don't know these things we will not really understand it, we will even not be able to remember it; at best we will only remember it by learning it off by heart.
[Poincaré 1899]
‘Ideas were tried out and discarded…ideas were tried out and retained?’ Yes, indeed! The process of trying out ideas, of experimenting with this approach then that approach, of abandoning ideas that don't work, is a process of play, of experiment, of conjecturing, testing, and then, very likely, rejecting: ‘Aha! Of course, it's an analogue of the Laplace transform, let's see [CHECKS BY CALCULATION]. Oh, it isn't. What a pity! Back to the drawing board!’
This is a scientific process, quite different from the textbook approach which tells you to identify the problem by TYPE, select the recommended METHOD, insert the specific DETAILS of your problem, and SWITCH THE METHOD ON! Gottlob Frege, who was a mathematician as well as a logician and philosopher, wrote that,
The aim of proof is…not merely to place the truth of a proposition beyond all doubt, but also to afford us insight into the dependence of truths upon one another.
[Frege 1884/1953: 2e]
The explanations in mathematicians’ published papers may be logically convincing yet unconvincing in every other respect which makes them much harder to read and understand. As Hermann Weyl wrote:
We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and of the road; we want to understand the idea of the proof, the deeper context.
[Weyl 1995: 453]
The italics are ours. The emphasis on perception could not be clearer. ‘Aha! Now I see it!’ is the hope and expectation of all mathematicians (and the despair of many school pupils.) To play around with a problem, to ‘suck it and see’, to turn it over and look at it from another perspective, is the very essence of mathematics. It is no coincidence, of course, that playing is just what game players do.
The three aspects cannot be separated. There are few examples of purely perceptual maths and none at all if you argue that perception in mathematics is always game-like because the ideas are already abstract. Nor can mathematics be purely scientific: it must in the final analysis be game-like. That is one of the differences between maths and physics, or chemistry. Experiment may convince a mathematician psychologically that a result must be correct but he or she still wants a proof, not just because proofs are convincing but because the effort of creating a proof f
or a challenging solution creates new ideas that illuminate the problem left behind and the path ahead as the sun rises on yet another region of the endless mathematical landscape that mathematical gold diggers rush to explore.
We shall start with game-like mathematics, then turn to the scientific aspect, and then finally to the foundation of all art and science – and mathematics – perception.
For the reasons already given, the distribution of topics between the following chapters is rather arbitrary. With a slight change in perspective or a different emphasis, an illustrative example in one chapter could readily be moved to another.
6 Game-like mathematics
Introduction
Game-like mathematics is a world of brilliant moves and cunning tactics, subtle and deep strategy, beautiful combinations and transformations, profound intuitions, and proof.
We first glance at the abacus and Cuisenaire rods, and examples of game-like tactics and strategy in ordinary arithmetic. Even small children can use tricks and short cuts to solve arithmetical problems more easily. These are their introduction to game-like tactics and stratagems and to the importance of not simply relying on routine algorithms – let alone on calculators!
Hidden connections in algebra is a fascinating theme: repeatedly, patterns and analogies turn up which seem to have a deeper significance. Usually they do, but how to find it? Geometrical figures always mean something but algebraic expressions or equations often seem to mean little – but then some neat move or an elegant transformation or a simplification is made, and pattern and structure and meaning appear, almost as if by magic.
Jean le Rond d’Alembert (1717–1783) said that, ‘Algebra is generous. She often gives more than is asked for’ [Boyer 1991: 439] but only to those who learn to play the game of algebra fluently and with insight and imagination. Later we shall see how what appears to be a mere coincidence is connected to a surprising theorem by Liouville. This will be followed by a masterpiece by the incomparable Euler, that has also been analysed by George Polya. Polya claimed that it was all about scientific induction but we think it is at least as much about game-like brilliance.
In Chapter 7 we turn to geometry and Euclid, for two thousand years the paradigm of game-like mathematics – which isn't to say that much of geometry hasn't first been discovered by experiment or that geometers haven't made mistakes. They have, but relatively few and geometry is far more notable for its wealth of surprises, the sheer elegance of its arguments and the depth of its insights.
Goodstein's analogies R. L. Goodstein was a rare mathematician who took the analogies between mathematics and abstract games seriously. In his, Recursive Number Theory, he discussed ‘Arithmetic and the Game of Chess’:
To the numerals correspond the chess pieces, and to the operations of arithmetic, the moves of the game. But the parallel is even closer than this, for to the problem of defining number corresponds the problem of defining the entities of the game…
and a little later he added, as an aside, ‘Here at last we find the answer to the problem of the nature of numbers…’ [Goodstein 1957].
In his Essays on the philosophy of mathematics, he went further: ‘[mathematics] is the continual creation of new games and the comparative anatomy of games’ which shows much more insight. Even when he writes ‘If mathematics had no applications then it would indeed be “merely a game”’, he half-rescues himself by continuing, ‘Even so, not a meaningless game, for operating with symbols in a coherent structure is itself meaningful’ [Goodstein 1965: 216, 111–112].
In Axiomatic Projective Geometry, written with E. J. F. Primrose, Chapter X is titled ‘Geometry as a Board Game’, and justifies its title literally: ‘The game is played by placing rows of letters…in columns on a “board”. Each column serves to express an attribute of the geometry…’ and so on [Goodstein 1953].
In an earlier paper, Geometry in Modern Dress, Goodstein had taken the axioms of projective geometry and presented them lightheartedly, first in terms of typists and the typewriters they used at the International Typewriting Agency, and then in terms of the Game of Letter-Board [Goodstein 1938: 217].
Tactics and strategy
The abacus is one of the oldest calculating devices, whether using a board with pebbles or beads strung on wires or sliding in slots. A few fortunate Romans even used a hand-held abacus, made of a sheet with beads moving in parallel grooves. There is an example in the British Museum.
Mostly, the Romans used loose stones, or calculi (whence our words to calculate, and calculus) on a ruled board or cloth so it is no surprise that they also referred to ludus calculorum, meaning any game using the same equipment.
Addition on an abacus, or soroban, is like playing a simple board game, because you are physically moving pieces to achieve a desired result according to the rules. We might say that this is an empirical kind of arithmetic. Since an abacus expert hardly needs to think while using the device, experts can be extremely fast. In a contest in Tokyo in 1946, Kiyoshi Matsuzaki using a soroban easily defeated Private Thomas Wood using a hand-held electric calculator on addition, subtraction, multiplication and division problems [Tani 1964: 7].
Small children who use Cuisenaire rods to do sums are also playing a game with simple rules – and whatever method you use, these games have tactics and strategies. What is 8 × 35? Well, 8 is 2 × 2 × 2, so to multiply by 8 we double three times: 35 – 70 – 140 – 280, the correct answer, easily calculated mentally.
What is 12 × 35? Since 12 = 8 + 4, the answer from the previous calculation is 280 + 140 = 420. However, you could also notice that 12 = 3 × 2 × 2 and 3 × 35 = 105, so doubling twice we get, 105 − 210 − 420.
As a last resort, you could use your 12 times table: 12 × 30 = 360 and 12 × 5 = 60, making a total of 420. In this case recalling your tables is just as quick as the other routes, provided that you don't try to visualise a standard sum in your head, complete with ‘write down 0 and carry 6’.
These are examples of very basic tactics and strategy. The strategy is to look at the particular features of the numbers you are multiplying by, the tactics actually carry out the strategy. The same approach is vitally important in programming computers. By exploiting specific features of the problem at hand, you can often program it more simply.
There are many other tactics-and-strategies that well-taught children master. Thus any sharp-eyed observer might notice that 9 × 11 = 99 which is 1 less than 10 × 10: similarly 7 × 9 = 63 and 8 × 8 = 64. So what is 15 × 17? 16 × 16
15 × 17
14 × 18
13 × 19
12 × 20
11 × 21
256
255
252
247
240
231
256–0
256–1
256–4
256–9
256–16
256–25
Here is the extended pattern. It seems that if you know your squares, for example that 15 squared is 225, then you can work out the products of numbers on either side of 15, by simply subtracting a smaller square. 152 = 225, so 14 × 16 = 224 and 13 × 17 = 221 and so on. But why should this work? To just spot the pattern and use it is to do science. How can we prove that the pattern is sound? By this tactical sequence of moves:
This pattern works for any starting square, and any pair on either side, so we could represent it (more briefly) in algebra like this:
For small children, and many adults, a geometric picture is more likely to be convincing (Figure 6.1). The 6 by 6 square and the 5 by 7 rectangle overlap. Comparing the row of 6 small squares along the top with the 5 small squares down the right-hand edge, shows that the large square is 1 small square bigger.
Figure 6.1 Illustration of (A − 1)(A + 1)
By exploiting tactics and strategy we can do many calculations more quickly and more enjoyably. For other calculations, however, the Four Rules are next best (after a calculator) though they are no more than clev
er algorithms. Using them is a pretty routine activity, and frankly – for the mathematician – disappointing and unproductive. Indeed, if the Four Rules were all you could use the counting numbers for, mathematics would never have taken off. Fortunately, just as we can see chess as either an abstract model of a real battle between blood-and-guts armies or an abstract game valued for itself, so we can turn aside from mundane arithmetic and consider the counting numbers purely for their own sake – which is just what the Greeks started to do, 2500 years ago. One tool for doing so is algebra.
Sums of cubes and a hidden connection
On page 132 we shall sum the odd squares scientifically, by spotting a pattern. If we do the same for the sum of the cubes, our pattern spotting should lead to this result:
This is a neat result, and also surprising, because the right-hand side is the square of n(n + 1) which is the sum of the counting numbers, so,
Is this a coincidence or is it deep? Is it a special and isolated result or part of a general pattern? We haven't yet defined what we mean by a ‘coincidence’ in mathematics or what we mean by ‘deep’, so let's just say that if a result is connected to many other results (like Pythagoras’ theorem) then it is deep, but if it is an isolated and unconnected result, then it is coincidental. That definition raises the question, are there any coincidental results in mathematics or must everything be connected to something else? Let's do a test, by looking at a neat property of the divisor function. This is denoted by d(n) and is equal to the number of divisors of the integer n. The divisors of 24 are, 24
