Introduction
The two seminars, The 400 problem Part Iand The 400 problem Part II, of which this is obviously the second, are based on the Problem Solving lesson ‘The 400 Problem’ Level 5. The aim in these two seminars is to both solve the problem and to show how mathematics can develop. The emphasis here is on the development of mathematics. It is probably best to at least read through Part I before taking this seminar as there are plenty of references to it in the following.
You may not see at first how to turn this material into a staff seminar but we will get to that at the end. You should find that there is more information here than you can absorb in one go. No matter, just read it through once and go back to the bits that interest you. You certainly won’t need it all for the seminar but you will need a rough idea of what it’s all about.
Some Meta-mathematics
All that we want to say in this part is based on the diagram below. The aim is to try to describe what research mathematicians do and to emphasise that it’s not too different from what we did in Part I.
Figure 1
Problem
Nothing happens in mathematics unless there is a problem. In Part I our problem was the 400 Problem. Start with two subtraction sums.
The letters a and b stand for two of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. They stand for the same numbers in each subtraction sum. What are a and b?
This problem came from a book somewhere. Where do mathematicians' problems come from?
Well, there are several sources. First there are a lot of hard problems floating around that people have tried to solve for years. Every now and again one of these very old problems gets solved. The most recent long-standing problem that succumbed was Fermat’s Last Theorem. A recent less difficult problem was reported on in the August 2004 newsletter of this web site.
Some problems come directly out of the real world. How do ice flows grow and break up? What happens to the various layers of water in a dam as the water is released? What is going to be the value of my shares tomorrow? If I act in a certain way how will my opponent react? These usually land in the area we call Applied Mathematics. Many of these problems can’t be answered with the same certainty that we can answer the 400 problem. When we do get answers they often involve approximations that nevertheless give us a pretty good idea of what is going to happen.
But there are a lot of problems that mathematicians, both Pure and Applied, make up as they go along. We’ll see how this happens in the conjecture, generalisation, and extension sections below. Someone just gets an idea. If they can solve it then it’s all over. If they can’t, then they might ask a friend for help or put it on the web or in a journal or a book. The harder it is and the longer it goes unsolved, the more famous it gets and the more people try to solve it.
The only difference between a problem that a mathematician works on and one that appears in school is that no one has yet solved the mathematician’s problem. But that doesn’t mean that the amount of creativity or originality needed from a student or teacher is any less than that of a mathematician. What’s more, when any one of them gets the solution, they share the same elation.
But if you ain’t got a problem you ain’t got any maths. And if you want a problem to try, then go to the Problem Solving section of our website or the monthly newsletters or somewhere like www.nrich.maths.org.uk. (Or make one up!)
Experiment
The first thing that mathematicians do when faced with a problem is to try to understand it. And then they try to get some information about it that will help them solve it. This is usually done by experimentation – just picking up the problem and playing with it.
In the case of the 400 Problem, this involved first thinking about how to solve the case n = 4, because at that stage that was all the problem was. In Part I we put some of our experimentation in a table.
There are, in fact, many ways to solve this problem. Most problems yield to trial and error one way or another and this one is no exception. You can set up a table and insert values for a and b and see what happens. If you do this systematically starting from a = 0, b = 0, and move up to a = 9, b = 9, through a = 0, b = 1, a = 0, b = 2, etc., you just have to get a solution at some point if one exists. What’s more, if there happened to be more than one solution, you’d be sure to get all of the solutions this way if you didn’t stop when you found the first one.
But this is not an elegant method and it will take a long time. Clearly you might shorten this time if you write a program to do it. Mathematicians too, often write programs to investigate what’s going on in a problem.
On the other hand, you might be able to use trial and improve to speed things up. Is there a way to make your guesses work so that you get to an answer quicker than trying all possibilities? Very quickly you ought to see that if a was anything other than 3, then no value of b would work. But if you have a = 3, there are at worst 10 cases for b. Substituting a few values of b should make you see pretty quickly that b can only be 6.
So there are two more methods and there is at least a third. You can try algebra. The problem can be changed to look like
400 – (100a + 10b + 4) = 400 + 10a + b – 400.
That whittles down to 11(10a + b) = 396 or 10a + b = 36.
Can you think of any other method?
Once the idea of changing the 400 Problem into the n00 Problem comes along then the method of the 400 Problem (whatever method it was that you used), can be employed again and again to build up evidence for the general case.
Although we’ve listed ‘experiment’ right at the start, it is a tool that is used all the way through the research/problem solving process. In fact, whenever mathematicians get stuck, they experiment.
Conjecture
As you found out in Part I, ‘conjecture’ is just a fancy word for ‘guess’. Naming something a conjecture gives it a certain status. Everyone knows that this is something that someone (or some group) has come up with over a period of time on the basis of experiments. It is a best guess at the truth but no one has a way of establishing its truth without a shadow of doubt. In other words we can’t prove it, not at that point anyway.
So if you see Holton’s Conjecture you know that you can’t trust it completely. It’s a flag of warning – don’t use it because it may not be true – and a challenge – here’s something that no one can do.
In Part I we moved to the conjecture of the n00 Problem and then to the conjectures of the n00…00 Problem. We managed to prove one of these but the latter two are still Conjecture 1 and Conjecture 2.
But conjectures, as opposed to Conjectures, have more importance than that. They are in fact the lifeblood of research mathematics. Progress in the subject is only made by a series of guesses. Why doesn’t anyone tell you that in school? Hardly a single, worthwhile result was ever made without a whole lot of guessing. No one ever (or very very rarely), found anything useful without making guesses. The guesses enable mathematicians to hone in on the right conjecture and then to the solution of the problem. And very few conjectures are true. In fact most guesses are false but they do lead you on, hopefully in a direction that will get you a theorem sooner or later.
Proof
Proofs are an essential ingredient of mathematics. It’s the thing that sets off maths from every other subject. So research mathematicians aren’t happy until they can prove
Possibly the most famous theorem, because it is one of the few theorems that are mentioned in school, is Pythagoras’ Theorem. This is the one for right angled triangles that says that the square on the hypotenuse is equal to the sum of the squares on the other two sides. In the triangle below, a2 + b2 = h2.
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Now, of course, there are many ways to prove things. In Part I we saw how to prove something by considering all cases. This is called Proof by Exhaustion. There are other named methods too, like Mathematical Induction, Proof by Contradiction, and so on. But in the end, any proof is just a chain of reasoning that takes you from the hypothesis to the conclusion. This chain may involve algebra, or counting, or simply words sometimes with only a few symbols.
Counter-example
Counter-examples are things that you don’t want. They are examples that are counter to a conjecture. If we had found that for n = 3, a = 5 and b = 4, this would have been a counter-example to our conjecture that 10a + b = 9n. One counter-example will ruin any conjecture. A conjecture, if it is going to hold up to become a theorem, has to be true for all values of the hypothesis. Pythagoras’ Theorem is true for all right angled triangles. There are no counter-examples to this theorem otherwise it wouldn’t be a theorem.
It’s at this point that we need to link experiments, conjecture, theorems, and counter-examples, because they are the engine room of mathematics. And they show that the model of the creative side of mathematics above has a load more arrowed lines joining it all up than we have put in the diagram.
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While experimenting, mathematicians get information that leads to conjectures. They then try to prove that these conjectures are true and head for a proof. But in trying to find a proof they often see how to construct an example that doesn’t satisfy the conjecture – they have a counter-example. This forces them to try some more experiments and these lead to more conjectures or maybe to an idea that will yield a proof. Of course, it is possible that after some time trying to find a counter-example, a proof will pop out. But at this point of the game, there is a lot of to-ing and fro-ing between these four central ideas.
Give up
There are two important reasons for giving up on a maths problems: (i) it’s too hard right now; and (ii) you die. There have been a lot of famous examples of the latter. Fermat had to give up on his Last Theorem. He was probably justified in that it took over 350 years for it to be solved. But even giving up because a problem is too hard is no crime. In fact giving up is a perfectly good problem solving strategy.
How does that work? Suppose that you have been working on a problem for an hour or so and haven’t made much obvious progress. Then it’s a good idea to go off and do something completely different. It’s surprising how often the brain will come up with a new idea that will set you going again. This is the old ‘idea in the shower’ syndrome. When you least expect it, the brain, that must have been simmering over the problem while you were doing other things, suggests a way forward.
There are several documented examples of this – times when mathematicians have suddenly found just what they wanted to solve a difficult problem. This happened one day to the mathematician Poincaré. He had been out with his wife and just as he stepped on to a bus to come home, a brilliant idea came to him. Hamilton was walking with his wife on a canal in Dublin and suddenly his important idea of quaternions came out of a clear blue sky.
This phenomenon is becoming to be known as the Aha syndrome, or the Eureka effect. This is because it was the kind of thing that is supposed to have happened to Archimedes when he discovered the buoyancy principle. You may remember that he is supposed to have found that while he was taking a bath. He was so excited by the discovery that tradition has it that he ran through the street naked crying ‘Eureka’.
But the Aha syndrome is certainly now well known. And it’s not confined to crazy mathematicians. Many people (even maths students) are on record as having had the experience.
Now from two of the examples above you might think that new ideas come when you are out somewhere with your partner. That may be the clue but it’s more likely that, after an intense period of work by one part of the brain, wheels continue to turn elsewhere, without your help, and new ideas are produced. So do give up but don’t give up too soon. You need to give your brain something to reflect and that requires you to have done a lot of work.
Generalise and Extend
When a problem has been solved, there are two other ways that mathematics develops. One of these is by generalising, and the other is by extending, the problem. Let’s take extending first.
An extension of a problem is another problem that is like the first problem in some way. For instance, when we changed the 4s to 5s in the 400 Problem Part Iwe were extending it. The same thing happened when we added another zero to give the 4000 problem. All that extensions require is a small change in one or other of the conditions of the original problem.
A generalisation of a problem is another problem that contains the original problem as a special case. Going from the 400 Problem to the n00 Problem was a generalisation. If we put n = 4 we get back to the original problem, so the 400 Problem is a special case of the n00 Problem.
The important thing about a generalisation is that if you can solve that, then you will also have the solution of the original problem. Once we found that 10a + b = 9n in the n00 Problem we could put n = 4 to find the answer to the 400 Problem.
It turns out that extensions often lead to generalisations. Look at what happened to the 400 Problem. Going to the extensions of the 500 Problem and the 600 Problem, etc., led us to the generalisation of the n00 Problem. Going to the extensions of the 400 Problem to the 4000 Problem and the 40000 Problem led us to the (super) generalisation of the n00…00 Problem.
What extensions and generalisations do for us is to take us back to the first step of the model – problem. And once there, we go through the process all over again. In doing so, mathematics is grown. In the case of the 400 Problem this growth is not very significant. But with other problems it may well lead to whole new branches of mathematics. Think of what happened when someone said, “OK, we have solutions of equations like x – 8 = 0 but what about x + 8 = 0?” That was the way that negative numbers were born. And then there was someone else who thought “OK, we have solutions of equations like x2 = 1 but what about x2 = -1?” Both of these developments led to an enormous increase in the things that mathematics could do and the problems it could solve.
Seminar
Well there may be a lot of good stuff above but how can you work it into a seminar? Here are is one suggestion.
Show the staff the creative model;
Ask them what they think the individual words mean;
Ask them how they think the diagram fits together;
Discuss all of these things with respect to the 400 problem (and its extensions and generalisations)
What was the 400 Problem?
Did this involve experimenting?
What conjectures did we have?
Were there any counter-examples to any of the conjectures?
What proofs did we find?
What did we have to give up on? Why?
What do you think an Aha moment might be? Can you give any examples?
What extensions of the 400 Problem did we look at?
What extensions of the n00 Problem did we look at?
Can you think of any other extensions?
What generalisations of the 400 Problem did we look at?
What generalisations of the n00 Problem did we look at?
Can you think of any other generalisations?
As far as the diagram goes, what differences might you expect to find between a research mathematician, a teacher, and a student? What similarities might you expect?
On the way through you can bring out any of the points that we have made above, especially if no one comes up with them.
As far as more extensions and generalisations go, you could change the minus signs to plus signs. But more interesting is the following:
Is it possible to find a and b where the two subtraction problems below have the same result?
This leads to asking what happens if you extend the number of zeros. It also leads to asking what values of x and y exist for which the following two equal sums have solutions for a and b?
And you can extend and generalise that quite a distance.
Our conjecture is that for the x = 1 and y = 1 situation, a = 4 and b = 2, and solutions to further extensions exist only when there are an even number of ones. For the second situation we think that x has to be equal to y for there to be any answers for a and b. When you extend this problem to xyz… (in other words, extend the 3-digit numbers to 4-digits and more) we think that you may find something interesting.
But we’ve no idea what happens in the second problem if you change 4 to 5 and so on. We’d like to hear what you find.