Introduction

The subject of Mathematics is all about solving problems. There is a common misconception that every problem has been solved and that all we need to do is to look up the answers at the back of books. Nothing could be further from the truth. There are many problems to which nobody knows the answers. Such matters form the basis for mathematical research.

The last part of the twentieth century can truly be regarded as a "golden age" for Mathematics. The availability of ever more powerful computers enables calculations to be performed in hours or days as opposed to hundreds of years. A particular instance was the proof of the Four Colour Theorem in 1976. Use of a computer helped to solve a problem that had been on the go for nearly 250 years. Whole new areas of Mathematics are springing up in response to the needs of other disciplines such as biology, economics and meteorology. Chaos, fractals and public-key codes are just three topics which have emerged during the last 30 to 40 years, each one generating lots of fascinating problems.

Computers are all very well but the human mind will always have a role to play. Flashes of inspiration and ingenious ideas will always be needed. A recent example was the proof of Fermat’s Last Theorem by Andrew Wiles in 1994 (roughly 350 years after the problem first surfaced in the margin of a book). Within the last few weeks, an American mathematician has claimed to have solved a problem of Kepler on packing identical spheres into a very large box. It has taken nearly 400 years to prove rigorously that fruiterers have had the right idea all along when packing apples and oranges!

In what follows we shall look at a few of my favourite problems which won’t take us hundreds of years to solve. They will nevertheless illustrate a few of the basic techniques used in solving mathematical problems, namely,

Spotting Patterns

Making Conjectures

Logical Argument

Insight.

Problem 1

1998 pupils take part in a knock-out tournament.

How many matches are required to find the winner?

The number 1998 is rather awkward. Things would be much easier if the number of pupils were a power of 2. Let’s see what happens then. Suppose we have 16 pupils. Then the numbers of matches in each round are 8, 4, 2 (semi-finals) and 1 (final), a total of 15. For 32 pupils we get 16 + 8 + 4 + 2 + 1 = 31 matches. Aha! Each time the number of matches is one less than the number of pupils. Is this always the case?

Conjecture: For n pupils we need n - 1 matches.

It turns out to be easy to prove this conjecture if we look at the problem from the correct point of view. Consider the following.

One pupil wins the competition

The remaining n - 1 pupils lose at some stage.

After losing, a pupil is out of the competition.

Each match has exactly one loser.

To get n - 1 losers, we therefore need n - 1 matches.

End of story! Notice that working out byes with 1998 pupils is not the best move (and a complete red herring).

Problem 1 is very simple but nevertheless it illustrates most of the standard tools: trying simpler cases, spotting a pattern, making a conjecture and using a bit of insight to prove the conjecture.

Here is another problem which illustrates the power of insight.

Problem 2

Prove that, for every positive integer n, the fraction
 

is in its lowest terms.

We have to show that the h.c.f. (or g.c.d.) of the integers 21n + 4 and 14n + 3 is always 1. The proof is a one-liner:-
 

This is certainly true as the left-hand side is 42n + 9 - 42n - 8. Why does this solve the problem? If k is a factor of both 14n + 3 and 21n + 4, then k is also a factor of the linear combination 3(14n + 3) - 2(21n + 4) i.e. k must divide 1. The only positive integer k which divides exactly into 1 is 1 itself. Again, end of story! The crucial one-liner can be derived either by a flash of inspiration or via the Euclidean Algorithm for highest common factors.

Note This was the first problem in the first International Mathematical Olympiad (IMO) held in 1959. The IMO is an annual competition for teams of up to six senior pupils from all over the world. Between 70 and 80 countries (including the United Kingdom and Ireland) take part regularly.

Problem 3

Place n points round the circumference of a circle and draw

the chords which join them in pairs. Suppose that no three of

these chords are concurrent inside the circle.

Into how many regions is the interior of the circle divided?

Taking n = 1, 2, 3, 4 and 5 in turn produces 1, 2, 4, 8 and 16 regions as is easily seen from a (large) diagram. Aha!

Conjecture: For n points we get  regions.

Let’s just check this for n = 6. Hello! There only seem to be 31 regions, not the 32 we were expecting. We’ve got the wrong formula! What we should have is
 

For n 4, this last expression can be written as
 

where  is the usual binomial coefficient. Notice that  is the number of chords is the number of points where two chords intersect. (Why?) With that hint, you might like to prove that we now have the right formula. (It helps if you know about mathematical induction.)

In Problem 3 we made a conjecture on the basis of five "experiments" and found that we were barking up the wrong tree. Things can get much worse!

Problem 4

Prove or disprove the statement that
 

The statement is false. A flash of inspiration leads to a particular value of n for which for which n - 81n + 1681 is obviously not prime. Can you spot this value of n? (Hint:- it is bigger than 1000.) If your insight is at a low ebb, you can start trying n = 1, 2, 3,... and see what happens. You should get prime numbers all the way up to n = 80 but then 81 - 81.81 + 1681 = 11681 = 41.

Problems 3 and 4 illustrate the difference between "proof" in the experimental sciences and proof in mathematics. In the physics lab., if you do an experiment 5 times and get the same answer, you will probably feel as though you have verified Ohm’s Law or whatever. In mathematics, as we have seen, 5 experiments are not enough nor are 80, nor are 10, because there are infinitely many positive integers. We need to prove incontrovertibly that a pattern continues for ever. Spotting a pattern is only the start. Rigorous proof using tools such as induction is essential for the mathematician.

Mathematical proofs demand logical argument. You might like to try your hand at the following teaser.

Problem 5

You are given 12 coins and a balance.

One coin is a forgery and is either lighter or heavier than

the others but you are not told which.

Describe how to identify the forgery in at most 2 weightings.

If you think you know a method that works, write out a full explanation and see if it convinces your friends.

To finish this short article, I present a real favourite of mine.

Problem 6

Determine the maximum value of m+ nwhere
 

This problem was set in the 1981 IMO (hence the appearance of 1981). To get started take the condition (n - nm - m) = 1 and show that
 

Try small values of m and look for patterns. You should find lots of patterns upon which to base a conjecture. That leaves the hard part, namely, rigorous justification of your conjecture.

Conclusion

Life in general, and mathematical life in particular, if full of problems. If we knew all the answers, life would be incredibly boring. Fortunately, there is plenty of unfinished business. Tackling mathematical problems can be both frustrating and fascinating but the warm glow of satisfaction when you solve a problem makes the effort worthwhile.

There are many books of mathematical problems which cater for all tastes. I shall mention just one:-
 

As well as presenting problems from the British Mathematical Olympiad, Tony Gardiner has provided an annotated 10-page list of "Some books for your bookshelf", a veritable cornucopia for teachers and students alike. Have fun!