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The Essence of Mathematics Through Elementary Problems
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About this text

And as this is done, so all similar problems are done.

Paolo dell’Abbaco (1282–1374)

Trattato d’aritmetica

It is better to solve one problem in five different ways than to solve five problems in one way.

George Pólya (1887–1985)

If you go on hammering away at a problem, it seems to get tired, lies down, and lets you catch it.

Sir William Lawrence Bragg (1890–1971) Nobel Prize for Physics 1915

Young man, in mathematics you don’t understand things. You just get used to them.

John von Neumann (1903–1957)

This is not a random collection of nice problems. Each item or problem, and each group of problems, is included for two reasons:

  • they constitute good mathematics – mathematics which repays the effort of engaging with it for the first time, or revisiting it (should it already be familiar);

and

  • they embody in a distilled form the quintessential spirit of elementary (initially pre-university) mathematics in a style which can be actively enjoyed by committed students and teachers in schools and colleges, and by the interested general reader.

Some items exemplify core general methods, which can be used over and over again (as hinted by the dell’Abbaco quotation). Some items require us to take different views of ostensibly the same material (as illustrated by the contrasting Pólya quote). Many items will at first seem elusive; but persistence may sometimes lead to an unexpected reward (in the spirit of the Bragg quote). In other instances, a correct answer may be obtained – yet leave the solver less than fully satisfied (at least in the short term, as illustrated by the von Neumann quote). And some items are of little importance in themselves – except that they force the solver to engage in a kind of thinking which is mathematically important.

Almost all of the included items are likely to involve – in some degree – that frustration which characterises all fruitful problem solving (as represented by the Bragg quote, and the William Golding quotation below), where, if we are lucky, a bewildering initial fog of incomprehension is sometimes magically by the process of struggling intelligently to make sense of things. And since one cannot always expect to succeed, there are bound to be occasions when the fog fails to lift. One may then have no choice but to consult the solutions (either because some essential idea or technique is not yet part of one’s stock-in-trade, or because one has overlooked some simple connection). The only advice we can give here is: the longer you can delay looking at the solutions the better. But these solutions have been included both to help you improve your own efforts, and to show the way when you get truly stuck.

The “essence of mathematics”, which we have tried to capture in these problems is mostly implicit, and so is often left for the reader to extract. Occasionally it has seemed appropriate to underline some aspect of a particular problem or its solution. Some comments of this kind have been included in the text that is interspersed between the problems. But in many instances, the comment or observation that needs to be made can only be appreciated after readers have struggled to solve a problem for themselves. In such cases, positioning the observation in the main text might risk spilling the beans prematurely. Hence, many important observations are buried away in the solutions, or in the Notes which follow many of the solutions. More often still, we have chosen to make no explicit remark, but have simply tried to shape and to group the problems in such a way that the intended message is conveyed silently by the problems themselves.

Roughly speaking, one can distinguish three types of problems: these may be labelled as Core, as Gems, or as focusing on more general Cognition.

1. Core problems or ideas encapsulate important mathematical concepts and mathematical knowledge in a relatively mundane way, yet in a manner that is in some way canonical. These have sometimes been included here to emphasise some important aspect, which contemporary treatments may have forgotten.

2. Gems constitute some kind of paradigm that all aspiring students of mathematics should encounter at some stage. These are likely to be encountered as fully fresh, or surprising, only once in a lifetime. But they then continue to serve as beacons, or trig points, that help to delineate the mathematical landscape.

3. The third type of problem plays an auxiliary role – namely problems which emphasise the importance of basic cognitive skills for doing mathematics (for example: instant mental calculation, visualisation of abstract concepts, short-term memory, attention span, etc.)

The items are grouped into chapters – each with a recognisable theme. Later chapters tend to have a higher level of technical demand than earlier chapters; and the sequence is broadly consistent with a rising level of sophistication. However, this is not a didactically organised text. Each problem is listed where it fits most naturally, even if it involves an idea which is not formally introduced until somewhat later. Detailed solutions, together with any comments which would be out of place in the main text, are grouped together at the end of each chapter.

The first few chapters tend to focus on more elementary material – partly to emphasise the hierarchical structure of mathematics, partly as a reminder that the essence of mathematics can be experienced at all levels, and partly to offer a gentle introduction to readers who may appreciate something slightly more structured before they tackle selected parts of later chapters. Hence these early chapters include more discursive commentary than later chapters. Readers who choose to skip these nursery slopes on a first reading may wish to return to them later, and to consider what this relatively elementary material tells us about the essence of mathematics.

The collection is offered as a supplement to the standard school curriculum. Some items could (and perhaps should) be incorporated into any official curriculum. But the collection as a whole is mainly designed for those who have good reason, and the time and inclination, to go beyond the usual institutional constraints, and to begin to explore the broader landscape of elementary mathematics in order to experience real, “free range” mathematics – as opposed to artificially reconstituted, or processed products.

It has come to me in a ash! One’s intelligence may march about and about a problem, but the solution does not come gradually into view. One moment it is not. The next and it is there.

William Golding (1911–1993), Rites of Passage