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
nineteenthcentury 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 x^{n} +
y^{n}=
z^{n}
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 2001^{th}
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.
Several
words of Swahili are presented with their translations in a different order:



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 JeanFrançois 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? 
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? 





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 fifthgeneration
Koch curves:



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


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).




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).




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 threedimensionality.
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
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.