This is the second of two articles and discusses problems relating
to the curvature of space, shortest distances on surfaces,
triangulations of surfaces and representation by graphs.
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
An introduction to some beautiful results of Number Theory
Published December 2001,February 2001,December 2011,February 2011.
There is a legendary story of the sage who posed the question:
'A normal elephant has four legs; if an elephant's trunk is called
a leg, how many legs does it have?' He asked a mathematician, who
continued to stare at a pile of paper on which he was scribbling as
he muttered: 'four and one make five'. Next to him a philosopher
mused enigmatically and puffed for a few moments on his pipe before
observing: 'The fact that it is called a leg, doesn't
change the fact that it is not a leg, so the answer is
four '. 'Excuse me,' said a passing zoologist, 'if a trunk
is classified as a leg, clearly this will also apply to the tail,
so it has six legs, and it's an insect'. A logician joined
the conversation: 'A normal elephant has four legs, but
you did not actually say that this elephant is normal, so
there is insufficient evidence...'
Continuing to seek enlightenment, the sage in his wisdom passed
the query on to a statistician who returned the following day
asserting 'the mean is 0.33'. 'Might I ask how you came by this
information?' queried the sage, concealing his innermost thoughts
behind an inscrutible smile. 'The best way to solve such a question
is to obtain empirical information,' replied the statistician, 'so
I went to the local zoo and got the answer from the horse's mouth,
so to speak. Two elephants refused to respond and the third blew
his own trumpet just once.'
Still bemused the sage went along to the local school which was
deeply embroiled in GCSE investigations and once again stated his
problem. 'That's a very interesting question,' said the
The moral of this story is that, as Humpty Dumpty once said,
'when I use a word, it means just what I want it to mean, and
nothing else'. The term 'proof' is just such a word. In different
contexts it means very different things. To a judge and jury it
means something established by evidence 'beyond a reasonable
doubt'. To a statistician it means something occuring with a
probability calculated from assumptions about the likelihood of
certain events happening randomly. To a scientist it means
something that can be tested -- the proof that water boils at
100o is to carry out an experiment. A mathematician
wants more -- simply predicting and testing is not enough -- for
there may be hidden assumptions (that the water boiling is always
carried out at normal atmospheric pressure and not, say, on the top
of Mount Everest).
When a problem is encountered, the question of providing a
convincing argument to explain the solution often arises. The book
'Thinking Mathematically' by John Mason, Leone Burton and Kay
Stacey has a large number of problem-solving situations. One is the
problem 'into how many squares can you cut a square?'
Faced with such a question, you might begin by thinking 'as many
as you like', or 'infinity'. Then you may begin to realize that a
square could be cut into 4, 9 or 16 by dividing it into pieces of
equal size. If you play about with possible ways of cutting a
square into smaller squares, you may suddenly see that any square
can be be cut into four smaller squares, including the smaller
squares themselves. Aha! A square could be cut into four quarters
and one quarter cut into four again, losing the quarter as a
counted square but gaining four smaller ones - so a square can be
cut into seven squares.
Figure 1: cutting a square into seven squares
Cutting any of the squares in this picture into four squares
gives three extra squares. Thus it is possible to cut a square into
7 squares, 10 squares, 13 squares, 16, 19, 22, ... and so on.
Figure 2: cutting a square into ten squares
It is easy to see that IF a square can be cut into n squares
THEN it can be cut into n+3 squares. It is this general result
which is the key to the solution of the problem. For instance,
if I could cut a square into six smaller squares,
then I could do 6+3=9, 9+3=12, and so on, to get the sequence 6, 9,
12, 15, ... If I could cut a square into five smaller
squares, then I could get the sequence 5, 8, 11, 14, ..., and so
on. But can I ?
One attack suggested by students is to look at a picture like
figure 3, and to propose that, by rubbing out a number of lines in
a three by three subdivision it is possible to "glue four squares
into a single square". This gives one big square and five smaller
squares, making six squares in all.
Figure 3: cutting a square into six squares (one big, five
This can then be built on by subdividing any of these squares
into four smaller, to cut a square into 6+3=9, 9+3=12, ... to get
the sequence of possibilities: 6, 9, 12, 15, ...
If you could cut a square into five squares, it would
be possible to get the sequence of possibilities 5, 8, 11, 14, ...
Then it would be possible to do all possible numbers 4 and above,
using the combination of the three sequences
4, 7, 10, 13, ...
5, 8, 11, 14, ...
6, 9, 12, 15, ...
But can you cut a square into five smaller squares? One
student, Paul, suggested to me that if you can do n squares you can
do n-3 by joining a block of four squares together as in Figure 3.
Is Paul's suggestion correct? It is certainly true for n=9, as
Figure 3 shows, but is it true for all whole numbers
There is a well-known story of the experimental physicist who
claimed to prove that 60 is divisible by every other number. He
came to this conclusion by considering a sequence of cases to
establish the pattern: 1,2,3,4,5,6 and then moved on to a few
others at random to test out the theory : 10,12, 20, 30, and
concluded that his result was experimentally verified. He was
surpassed in this endeavour by an engineer who noticed that all odd
numbers seemed to be prime... One - well that's an oddity, but
we'll include it in - three, five, seven, good, we're getting
somewhere - nine? Oh, nine... Let's leave that a moment - eleven,
thirteen - fine. The exceptional case of nine must have been an
This story, which I claim bears no relationship to any known
physicist or engineer, living or dead, does illustrate the
important difference between proof by looking at a number of cases
and proper mathematical proof. It is not enough to consider just a
number of cases, for all of them may have some hidden common
assumption. For instance, we might conclude from a number of
experiments that water always boils at 100o C because we
never have the experience of trying to boil water on the top of
Mount Everest. Scientific proof depends on the
predictability of experiments: that we conjecture that
when we carry out an experiment it will have a predicted outcome.
Such proof is not appropriate in mathematics where we must provide
a logical argument that the conclusion follows from explicitly
To help the student focus on the various stages of putting up a
convincing argument, 'Thinking Mathematically' suggests three
The idea is first to get a good idea how and why the result
works, sufficient to believe its truth. Convincing oneself is,
regrettably, all too easy. So pleased is the average mortal when
the 'Aha!' strikes that, even if shouting 'Eureka' and running down
the street in a bath towel is de rigeur, it is very difficult to
believe that the blinding stroke of insight might be wrong. So the
next stage is to convince a friend - another student, perhaps -
which has the advantage that, to explain something to someone else
at least makes one sort out the ideas into some kind of coherent
argument. The final stage in preparing a convincing argument,
according to 'Thinking Mathematically' is to convince an enemy - a
mythical arbiter of good logic who subjects every stage of an
argument with a fine toothcomb to seek out weak links.
A student might very well convince himself of the truth of the
argument "IF I can cut a square into n smaller squares, THEN I can
cut the square into n- 3 squares". He might even convince a friend
by showing pictures such as figure 3. But an enemy might put up
figure 4, where a square is cut into eight smaller squares (seven
the same size, plus one bigger one). Here there is no set of four
smaller subsquares in a group which can be amalgamated into one
larger square to reduce the eight sub-squares to five
Figure 4: cutting a square into eight squares, but not into
Does this blow, which demolishes Paul's theory, show that you
cannot cut a square into five smaller squares? No it does not. It
suggests that Paul's method does not work, but perhaps some
other method will ...
The problem which I leave you to formulate precisely and prove
has two parts:
(a) Find all numbers n such that a square can be cut
into n smaller sub-squares and prove that this is actually
possible for every such number n.
(b) For all the numbers n not included in part (a), prove that
it is not possible to cut a square into such a number of
(b) For all the numbers n not included in part (a), prove that
it is not possible to cut a square into such a number of
You should certainly have a go at this before moving to the next
Proof requires a careful statement of assumptions and a precise
argument showing how a clearly stated result is deduced. It is
surprising how often we miss the fact that a statement has
implicit, unspoken assumptions. Look at the square problem. Into
how many squares can I cut a square? For what numbers is this
If a square is cut into more than one square, there will be a
corner of a smaller square in each corner of the original square.
Thus, if there is more than one square, there must be at least
four . There cannot be two or three. Perhaps you might like to
try to extend this argument to cover other cases which you suspect
cannot be done (if there are any...).
I have given this problem to hundreds of undergraduates over the
years and we have all eventually agreed on which values of
n cannot be done. It has become quite a party piece which I have
also tried out with many sixthformers.
It was ten years after I first met the problem that a perceptive
fourteen year old girl in a problem-solving session came up with an
original thought. She suggested that the problem had not
explicitly stated that the paper could not be cut and then
glued together again in a different way. Her solution for n=2 is
given in figure 5.
Figure 5: sticking together bits to cut a square into two
This illustrates the fact that we must be extremely careful
about how we phrase our assumptions. Before Figure 5 the (unspoken)
assumption had been that we must make single straight line cuts to
form whole subsquares and we are not allowed to cut into smaller
parts (say triangles) and to stick them together again. The
original problem is better specified by saying:
Square problem version 2 : A square is
cut into n smaller squares by making single straight line cuts,
without joining together cut parts into larger wholes. What are the
possible values of n?
The exceptional cases found earlier would still be exceptions to
this better phrased problem. Figure 5 would now fail to be a
counter example to this because it breaks the rule about not
sticking together cut parts into larger wholes.
However, Figure 5 does suggest a different problem:
Square problem version 3 : Into how
many subsquares n is it possible to cut a square, if it is allowed
to join cut parts into large wholes?
The answer to problem version 3 is likely to be different from
that to version 2. You should see if any or all of the 'impossible'
numbers from version 2 now become 'possible'. For instance, in
Figure 4 we can clearly take any four of the smaller squares and
move them together to glue them into one medium size square. Thus,
if we allow sticking together we can re-form the square in
Figure 4 into five squares of different sizes: one large,
one medium, and three little ones. With a little ingenuity perhaps
you can solve the case n=3 for version 3 of the problem. Perhaps
now you can specify the solutions of both problems. They
will be different. This shows that precision in making mathematical
statements is all important.
Three men were going by train to a conference in a distant
region of the United Kingdom. The engineer looked out of the window
and said, 'Look, all the sheep in Scotland are black'. The
theoretical physicist thought for a moment and said, 'No, there
exists a field in Scotland in which all the sheep are black'. There
was silence from the logical mathematician who mused for some time
in the corner of the compartment before declaring 'No, there exists
a field in Scotland in which all the sheep are at least half
Once we have got precise statements of the assumptions (P)
underlying a theorem and what it is we are trying to prove (Q),
then a mathematical proof of the theorem is in the form "IF P is
true THEN Q is true". In everyday language the conventions are
sometimes different. "If your father comes home before six o'clock
then you can have some chocolate before dinner-time". Here the
assumption P is "father comes home before six o'clock" and the
deduction Q is "you can have chocolate before dinner-time".
Presumably father brings the chocolate and if he arrives
sufficiently early you can have some without spoiling your
appetite. But also contained in the statement that IF father does
not come home before six o'clock, THEN you will NOT have chocolate
before dinner-time. There is often an implication in everyday
language that IF P happens THEN Q will follow, but IF P FAILS THEN
Q FAILS ALSO.
In mathematics such an assumption is not made. Here a
proof in the form IF P THEN Q simply requires that if P is true,
then Q must be true also. If P is false, then no implication as to
the truth or falsehood of Q is necessary.
Consider the example:
If x > 6 then x > 3 .
In mathematics this is considered a true statement. If x is a
number bigger than 6 then it must also be bigger than 3. However,
consider this as separate statements, where P is " x > 6 " and Q is "x >
3 ". What happens for various values of x? If x=7 then P is
true and Q is also true. In fact, when x is a number bigger than 6
then P is true and it will follow also that Q is true.
But if x=5 then P is false but Q is true, and if x=3 then P is
false and Q is false. Thus when P is false, Q can be true or false.
We simply have no interest what happens in this case.
Students often make quite serious errors in proof on examination
papers. As a Senior Examiner in Mechanics for many years I
regularly had to mark questions which said something like this:
A particle mass M rests on a rough plane with
coefficient of friction $\mu$, inclined to the horizontal at an
angle $\alpha$. Show that if the particle slides down the plane
then $\tan\alpha> \mu$.
You might feel that this is a trivial matter. But logically it
is totally erroneous. In mathematics it often happens that IF P
THEN Q is true but IF Q THEN P is false. For instance, it is true
that IF x > 6 THEN x > 3 , but the other way round: IF x > 3 THEN x > 6 is
clearly false. Thus it is important to distinguish between the two.
The statement IF Q THEN P is called the converse of the
statement IF P THEN Q. It is important to distinguish between the
proof of a statement and the proof of its converse. One may be true
and the other may be false. Another example occurred with the case
of "into how many squares can I cut a square" (version 2). It is
true to say that if a square can be cut into n pieces then it can
be cut into n+3 pieces. The converse, that if it can be cut into
n+3 pieces it can be cut into n pieces is false, as can be seen
from the case n=3,5.
Oh, you may say, that's cheating, we don't normally meet
functions like that in the calculus... No we don't. Nor do we
normally have the personal experience of boiling water on the top
of Mount Everest, which would prove that water doesn't always boil
at 100o C.
To be sure that the mathematics will always work it is necessary
to state precisely the assumptions and to take great care over the
deductions. This proves to be rather hard. Indeed it tends to be
the province of university pure mathematics rather than
You may find the accent on precise proof in mathematics rather
esoteric. Other scientists are known to make such jibes at
mathematicians. Indeed they say you can tell whether someone is an
engineer, physicist or mathematician by setting fire to his
wastepaper basket. The engineer will make a cursory calculation and
swamp the basket with enough water to put out the fire and more.
The physicist will sit down, calculate exactly how much water is
needed and pour the exact quantity on the fire. The mathematician?
The mathematician will sit down and calculate exactly how much
water is needed.
Thus the mathematician stands accused of developing a precise
theory that is devoid of application. This could not be further
from the truth. In our universities computer scientists are growing
increasingly worried that students no longer seem to understand the
finer points of proof. This is especially true since the demise of
Euclidean geometry which was largely concerned with the ritual of
deducing one statement about a geometrical figure from given
assumptions. It may have serious consequences. As we use
increasingly more sophisticated software to run our lives we need
computer scientists and programmers who can write provably correct
software that does not contain horrendous bugs - unlike the kind of
software that caused the stockmarket crash because it was designed
to sell under certain conditions which occurred late one Friday and
caused the computers to attempted to outdo each other as the
selling fed back into the system causing even more selling and then
eventual collapse of the market.
It is therefore even more important in today's technological
climate to pay attention to the niceties of well-formulated
statements and logical deduction. "To prove or not to prove" is a
question that can have only one answer, for proof is an essential
component of technological order in the future.
Professor David Tall
Professor in Mathematical Thinking at the University of
Educated at Victoria School (1945-1952), The Grammar School,
Wellingborough (1952-1960) and Wadham College Oxford (1960-1966)
where I obtained first class honours in Mathematics, The Junior
Mathematics Prize, (1963) and DPhil in Mathematics (1967). From
1966 to 1969 I was a lecturer in Mathematics at Sussex University.
Since 1969, I have been on the staff of the University of Warwick,
as a lecturer in mathematics with special interests in education in
the Mathematics Institute (1969-1980), then within what has become
the Institute of Education, being awarded a personal chair in
[This article was originally published in Mathematics Review in
January 1991. NRICH has the permission of the editors and authors
to reprint material from it as the Mathematics Review magazine
closed down after 17 issues. Ed.]