The Bulgarian original of this article is available here.

First published in Dnevnik 1.238 (27 December 2001), p. 8
Translated by Ivan A Derzhanski.

Awarded Third Prize in the article competition Raising Public Awareness of Mathematics,
European Mathematical Society, 2003.

For Those Who Think Mathematics Dreary

I’m no mathematician. I find counting a bore.
Madonna as Breathless in the film Dick Tracy, 1990

Problems and Sums

Such statements are heard often—even from educated people, who take special pride in making them, as if hatred of mathematics constituted evidence of artisticality and imagination. At the same time hardly anyone boasts of being ignorant of music, literature or painting. No one is entirely indifferent to music, but this doesn’t mean that we’re all musically gifted. On the other hand, it is thought that it takes special talent to appreciate the beauty of mathematics. This is why its exclusion from the sphere of intellectual endeavours worthy of attention seems not to disturb society too much. If one nevertheless ventures to admire its beauty and depth, one is put down as a dreary type or at best an eccentric with a strange hobby.
At times one hears that mathematics is what mathematicians do, and a mathematician is someone who doesn’t ask what mathematics is about.… Somewhat of a vicious circle, isn’t it? But if solving problems isn’t what mathematicians do, what is? Thing is, the expression problem solving has rather different meanings to different people. Here is what problem solving looked like in the eyes of the painter Nikolay Bogdanov-Belsky (1868–1945), whose picture Counting in Our Heads (1896) is on display in the Tretyakov Gallery. The teacher in the picture is the well-known educator Sergey Rachinsky (1833–1902), and the blackboard features an arithmetic expression whose evaluation requires nothing more than familiarity with exponentiation. (That would be the fifth grade by today’s curricula.) It is even easier if a calculator is at hand.
Text Box: Figure 1

Skipping the details, we’ll just note that this is what is called a numeric example. A problem is genuine when it takes creative effort to solve it. In fact there are two categories of problems in mathematics: those that are solved by technique and those that require insight. The category into which a problem falls depends on what one knows. In this sense what is a problem for some is for others a routine exposition of a logically connected sequence of reasoning and calculation.

The transformation of a problem that used to take insight into one that only requires technique can be a considerable step forward.

Some nineteenth-century textbooks talk of division as an arithmetic operation that takes a head, a fact that evokes a smile of condescension on the lips of many. If it is no longer an intellectual art, but an activity accessible to children, it is because an algorithm has been found—a kind of recipe, a breakdown of the problem into a sequence of simple actions. Likewise, if certain aspects of musical composition are algorithmised, some musical problems that used to require insight shall become a matter of technique. It’s a different story that people who have invested serious effort into the development of this kind of insight may not be enthusiastic about the possible algorithmisation of some of its elements. They may feel that no algorithm could embrace or replace their insight. According to some scholars, all problems (in mathematics or the other arts) can be solved by appropriate technique and if there are problems for which none is known as yet, that is due to gaps in our system of knowledge—a mere question of time, though it may be a long time. Others maintain that some problems will simply never be solved, however much time and perseverance we put into attacking them.

Problem solving in the technical sense isn’t really the solving of problems. It is what experts do when they don’t know what to do, that is, it is knowing how to handle situations for which there is no recipe. This is where imagination comes in. Not for nothing did the great mathematician David Hilbert say of one of his students: ‘He did not have enough imagination and therefore became a poet.…’ Another famous scholar, Godfried Harold Hardy, describes mathematicians as people who like painters or poets are makers of patterns, except that theirs are more permanent, being made with ideas. Mathematics used to occupy a central place among the arts, and today most people think that it is the art of doing sums. That’s hardly their fault though. With so many reforms in mathematical education worldwide, experts state that if the fine arts were taught in the same way, they would be reduced to studying techniques for clipping stone and mixing paints.

When in the seventeenth century the famous number theorist Pierre de Fermat wrote in the margin of a page in Diophantus’ Arithmetica that he had a proof of the insolubility of the equation xn + yn= zn for n > 2 in integer numbers, but the margin was too small to contain it, few could have realised what impact his remark would have on the development of mathematics for the next 350 years. Today we can’t be sure whether Fermat did in fact have a solution of the problem, which was to become known as Fermat’s Great Theorem. What matters is that Fermat’s Great Theorem stimulated mathematical thinking and research for several centuries. Because of the accessible statement and the huge prize promised to the first solver, many amateurs have also tried their hand at this problem. In 1993 Andrew Wiles, professor of mathematics at Princeton University, announced a proof of Fermat’s challenge during a lecture at Cambridge. Now this is a genuine problem. But it’s not the volume of the solution, taking up about 200 pages as it does, that makes Fermat’s Great Theorem into a problem. It isn’t its difficulty either. Raising the number 2001 to the 2001th power, especially by hand, isn’t easy at all, but that difficulty is of a technical nature. The difficulty of genuine problems lies in finding an approach, organising the premises, using the statements and facts already known and proven, as well as ordering them into a logical sequence.

The example in the picture Counting in Our Heads and Fermat’s Great Theorem are the two extremes of what the wide public is accustomed to calling a problem. This doesn’t mean that mathematics is in the middle, though. Nor does it mean that one has to solve problems or become a mathematician in order to discover the magic of mathematics. We’ll tell you about that magic. Just read on.

Ptolemy and Cleopatra

Here is a problem by the linguist Alfred Zhurinsky (1938–1991).
Several words of Swahili are presented with their translations in a different order: 
  • mtu, mbuzi, jito, mgeni, jitu, kibuzi
  • ‘giant’, ‘little goat’, ‘guest’, ‘goat’, ‘person’, ‘large river’
Pair up the words with their translations.
Swahili is widespread in East Africa, where 5 million people speak it natively and 30 million more use it regularly. This problem, however, is designed for people who neither know it nor have a dictionary at hand. Otherwise it wouldn’t be a problem, but an exercise in applying skills already learnt, recipes already known. As we said, a genuine problem appears when no recipe is known. All we have here is data, within which we need to discover some similarities and correspondences.
But in order to determine the correspondences between the two groups of data (the Swahili words and their glosses), we first need to note the regularities within each group—that some of the words begin or end in the same way (m-buzi, m-geni; ji-tu, m-tu) and that some of the glosses have something in common with others (as ‘little goat’ has with ‘goat’, for one thing, and then there is ‘giant’, which means ‘large person’). These regularities attribute to each group a certain internal structure, which we can represent in tabular form. To obtain the answer, we only have to compare the two structures by superimposing the two tables onto one another.
jito jitu
mbuzi mgeni mtu
‘large river’ ‘giant’
‘goat’ ‘guest’ ‘person’
‘little goat’

It seems that each Swahili word is composed of two parts—one to show the size (e.g., ki- ‘little’) and another to express the basic concept (e.g., -buzi ‘goat’). The same is true of the English expression little goat, but in goat nothing expresses the default size, as the m- in Swahili mbuzi appears to do.

Could we go on filling in the cells in the tables? The guess that mto is ‘river’ is correct. We’d be wrong about kito and kitu though—these words mean respectively ‘gem, jewel’ and ‘thing, something’; ‘little river’ is kijito and ‘little person, dwarf’ is kijitu. This is in accordance with a rule that is not reflected in the problem.

Was there no less laborious way to find the answer? There are dictionaries of Swahili, as well as people who know the language. It isn’t so with all problems, however. When for example Jean-François Champollion applied the same method to decipher the Egyptian hieroglyphic script, he accomplished something that no one had known how to do for two millennia. For a start he compared two names, one () from the Rosetta Stone, the other () from the Philae Obelisk, which appeared to belong respectively to Ptolemy and Cleopatra, judging by the Greek texts that accompanied the two hieroglyphic inscriptions. Champollion conjectured that the recurring characters corresponded to identical sounds, hence also to identical letters in the Greek form of the names—for example the reclining lion, which is in fourth position in Ptolemy and in second position in Cleopatra, was to be read as l, the two eagles in Cleopatra as a and so on.

What kind of problems are these actually, linguistic or mathematical ones? The former, to be sure, because they’re about linguistic objects—the structure, meaning and written representation of words of human languages. But the latter too—what is mathematics all about, if not the discovery of structures, regularities and correspondences?

It may not be a straightforward matter to find a correspondence in a given context. Here are two problems offered by Douglas Hofstadter, a leading researcher in artificial intelligence:
What in 12344321 corresponds to 4 in 1234554321?

One possible answer (the digit 4) involves the choice of the same object, another one (the digit 3) preserves the object’s rôle in the context (proximity to the centre), which is certainly an intuitive thing to want to do.
In 1980, who was in the UK what the First Lady was in the US
—Margaret Thatcher, Queen Elizabeth II, Prince Philip or Denis Thatcher?
What is a single rôle in one setting can be mapped to four possible rôles in the other, depending on whether we focus on the function of the US President as head of state or head of government and whether we are looking for the spouse of the leader or the female member of the couple.
Finding an analogy between two sets is an interesting problem in artificial intelligence.
The contrasting of two objects is related to a concept central to mathematics, namely function. This word makes most students think of a square or trigonometric function. But the objects in the correspondence need not be numbers at all; they may be given in a less traditional way, for instance:
Figure 1
In words, this correspondence implies that we replace every segment by a ‘bent segment’, where the shape of the bend is a square whose side is three times shorter than the segment. Let us start with a single segment and observe its metamorphoses after we’ve applied the function F several times:
Figure 2

These lines are called Koch curves of the first, second, third and fourth generation, respectively. They belong to a more general class of curves known as fractals.

And here are a square and an octagon whose edges are replaced by fifth-generation Koch curves:
Figure 3

Popular Bulgarian embroidery! This was the last place you ever expected to encounter mathematics, was it not?…

Amorous Turtles

The amorous turtles problem, which is of great antiquity, can be found in books on popular mathematics:
In every corner of a square room there is a turtle. The first is in love with the second, the second with the third, the third with the fourth, and the fourth with the first. (Not an uncommon situation in pop songs.) Each turtle starts walking directly towards the object of its feelings with the same constant speed. Will they ever meet and what will their paths be?

The problem can be solved in a variety of ways (including some that apply knowledge of what is called higher mathematics), but one can get a good idea of the movement by the use of a relatively simple computational model which employs turtle robots. By commanding each turtle to turn towards the object of its passion and move one step forwards and repeating this sufficiently many times, we get the following picture:
Figure 4

The path each turtle walks is a part of a famous curve known as a logarithmic spiral. If we connect the consecutive positions of the turtles by segments, we shall notice that they are always in the corners of a square whose side shrinks with the speed of the movement of the turtles:

Figure 5

The logarithmic spiral occurs in nature—it is the shape of the shell of the oceanic snail Nautilus (Figure 6). We also use it in a more prosaic (but still important) context—when plotting the growth of a deposit with fixed compound interest (Figure 7).
Figure 6
Figure 7

When constructing a computational model of a problem, we usually mean to solve an entire class of related problems, not just the one at hand. It is easy to vary the starting conditions. For example, the turtles may love one another crosswise (the first loving the third, the second the fourth). Or we can get one of them to hate its neighbour and flee from it (Figure 8). As one sees, that makes it a leader—all the others follow it.… We can also vary their number (Figure 9).
Figure 8
Figure 9

What is more, the interesting graphics obtained by varying the starting conditions suggest the idea that this model can serve to generate computational variations to the theme of Los Angeles (Figure 10)—a picture by the painter Victor Vasarely, whose pictures give an exceptional illusion of three-dimensionality. We place six turtles in the corners of a regular hexagon and connect the positions of the turtles after every beat by segments, alternating the colour of the trace (Figure 11). From there on we can create variants by modifying one parameter or the other in the initial setting of the problem (for example the strength of the attraction—Figure 12). Finally, we can locate the turtles in randomly chosen points of the screen and let them go (Figure 13). A sufficient number of experiments can even produce realistic images.
Figure 10
Figure 11

Figure 12

Figure 13

We hope that with the right attitude to the teaching of mathematics, when digital technologies are used to stimulate the spirit of discovery, students will come to see mathematics in a new way—as an area in which interesting experiments can be made and hypotheses formulated. Even if they happen to reinvent the bicycle, the students may feel the joy of the process of invention itself and acquire habits of creative thinking.

They may never become so skilled in mathematics as Monet, Mondrian and Vasarely were in painting, but at least they will have a chance of appreciating mathematics as the art it is. No lesser thing, that.

Three friends dear,
versed in matters drear:
Sava Grozdev, Ivan Derzhanski, Evgenia Sendova

Converted to HTML by Ivan A Derzhanski, 14 November 2003.