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 waysit 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 pentominoeswith 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 patternswith 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 aspectsof 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
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 waysit 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 pentominoeswith 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 patternswith 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 aspectsof 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
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