17 FAQs

Overview

Some of my colleagues have given me questions for me to answer, and this is the chapter where I intend to do this. Please feel free to give me questions to answer or email directly comments to me at pjt013@gmail.com. The following are some questions which have been specifically given me and which I am in the process of answering or have answered. Basically the questions invite me to comment on the topics in the headers below.

The methodology - or art - of setting problems

This is a very interesting question. Being a problem poser is distinctly different than being a good solver of problems, just as there is a special skill in designing chess problems, as distinct from being a good chess player.

When we committed ourselves to running a competition in Canberra in 1976 we had nothing to draw on. It was also at short notice, and the University of Waterloo gave us approval to use some of theirs. But we had to start and learned somehow. Different people do so in different ways. My main method was to keep my eyes open during the year and get an idea from a real life experience, and jot it down at the time. This also meant that my AMC questions related to real life, which is what we are trying to do.

But I think most problem creators sit down and experiment with things and discover results maybe after recognising an unexpected pattern. One of the best environments I found for creating questions was in the Challenge problems committee. As an example one year, about 1998, Mike Newman and I were working on tessellating P-tiles (pentominoes in the shape of a P) and by experimentation we discovered something which appeared to be a nice result. But we could not prove it. Eventually Andy Liu was shown the result, and Andy, one of the world's greatest problem solvers, showed us a nice proof. This problem was to make the 1999 IMO short list.

I would add that when we embarked upon this project we were worried that we might quickly run out of ideas. It is true that some of our original ideas could be used once only, in fact we found that each year we seemed to have a richer set than we had had the year before.Mathematics is a big subject.

The relevance of problem solving

In the real world, students will need to become problem solvers. In whatever undertaking they pursue they will encounter "problems". Generally, mathematical reasoning provides a model for problem solving on a wider scale. In the real world one might identify a problem, decide what are the variables which affect the outcome, identify interconnecting relations, and if solved in a mathematical sense, relate the solution back to the real world.

Whereas curricula in Australia and elsewhere will to various degrees encourage practice in problem solving, it is not usual for enough time to be spent on this in the classroom. This can be for a number of reasons, such as time (more time needed for essential skills) and the fact that this is something for which teachers are not always adequately trained. The AMC is specifically designed to provide classroom practice in problem solving by presenting problems in contexts to which students can related from their own day-to-day experience.

The relevance of competitive problem solving

I am not sure if I can give a profound answer for this. However there would be a different relevance for different stake-holders. The student is the most important stakeholder, and by going into competitive problem solving they are taking up a challenge and testing themselves against the problem, or testing themselves against others. I would be satisfied with the former. And of course there is relevance for the teacher or trainer as it gives them feedback about their work. Of course there are other stakeholders like parents, population and governments even. Governments for example should have an interest in Olympiad results because they are a measure of how far the students are achieving beyond the classroom.

Quality control, in problem setting in competitions

This is a particularly important matter for the AMC. In the 37 years in which I was involved before retirement I do not recall any typographic or mathematical error which caused us to take special assessment steps. Whereas it is certain it must happen sometime, it will not happen very often, as from the start we have adopted in my view the tightest moderation standards.

In the very first year, 1976, of the local Canberra competition which formed a predecessor to the AMC we adopted three layers of moderation. This means that in each layer there are people who are given the problems, without solutions and they are asked to provide their solutions and give any relevant comments. In the AMC now the first stage moderator from each state or territory,, or New Zealand, who discusses the problems' suitability for their state. Now that there is a national syllabus I assume there would continue to be at least one moderator from each state. In the AMC the second stage moderators are University moderators, and the third level moderators are normally people easily accessible by the Chairman.

On one occasion while I was Chairman I discovered an error after printing the paper. A third stage moderator convinced me of an error which in fact was not one. At some cost I was able to fix by reprinting before the papers went out. But the situation I found a little unsettling so I introduced a fourth stage, in which some members of the Problems Committee itself do further checking.

I guess related to this question there could have been a question on how we check the integrity of the results. We have detected cheating, sometimes by clever methods which evolved, but I am happy to say the incidence is not large, and certainly less than I might have expected. We also check every medal score with the school to determine whether the school regards that student as good and a medal would not be embarrassing. I did actually have a school which declined the student on these grounds once.

What is distinctive about the AMC, as against other competitions?

The answer to the question above should be regarded as part of this answer also.

The Australian Mathematics Competition (AMC) is not simply a stand-alone exam. It is the first step in a complete program of enrichment designed by the Australian Mathematics Trust, which enables students to explore their talents and develop them to their potential. The programs run by the Trust extend to the Mathematics Challenge for Young Australians, enrichment courses, and up to international competition.

Above all, the AMC truly meets my definition of being a Competition. It is the only broad based mathematics competition in Australia which fulfils this definition, as outlined below.

What is a Competition?

It was back in the 1970s that the first major Australia-wide school competition, the Australian Mathematics Competition (AMC), was introduced. Since then, the Australian education scene has seen a proliferation of many similar events, not only called competitions, but sometimes challenges, assessments, etc.

These events might be run by large organisations, small organisations, some from within professional societies, others clearly of a commercial nature, some for individual students, some for teams, some quite broadly pitched, others narrowly focused, and in all the major school subjects.

These events are often seen as similar in what they offer, but they can be quite different, and it really is worth asking what role they have in the school's academic program.

The word "competition"

First it should be noted that competition might not be the best word. We were hesitant about using this word, as it can be taken in a context different to intention, but in the end we settled on it because there were precedents for its use in Canada and the US.

I should also note what was our intention with the use of the word "competition". We do not mean to emphasise students competing against each other. Instead we see it as the student competing against the problems. In other words we see the AMC as a personal challenge in which the student can try to solve problems with the knowledge that if they fail, it has not counted against their personal assessment, but if they succeed there is personal satisfaction and recognition. And if one can solve one mathematics problem it is not only satisfying, it also provides the motivation for wanting to solve another.

Diagnosis and Feedback

We see a competition as not only an opportunity to assess and diagnose, but this is still a major feature of the AMC. This was the original Australian competition, which, because of the large entry numbers, and optical reading of the answers, was able to provide detailed feedback.

AMC Content

The AMC is carefully moderated by experts from each state to ensure that the mathematical content is within the scope of the regional syllabus. Most of the paper, particularly the first half, is what might be called "Curriculum bound", that is set in familiar classroom context. Towards the end of the paper questions might be set in contexts new to the student, albeit still using mathematics known to the student.

This means the AMC is really testing a little more than normal classroom mathematics, identifying students who can apply their knowledge to new situations. The AMT issues a "Mathematics and Problem Solving" Proficiency certificate to students who would otherwise not receive a credit but who have nevertheless indicated satisfactory ability in problem solving (which the AMC measures) and skill.

Challenge

It has wide appeal also because there are questions of all standards, beginning with quite easy questions, which all students should be able to solve, the questions become progressively harder until the last 5 questions which are for very talented students. Students of all standards have an opportunity to achieve and be challenged during this time.

In fact the concept of challenge is of contemporary interest in the profession of mathematics education. An ICMI Study (number 16) Challenging Mathematics in and beyond the Classroom was conducted between 2002 and 2009 with final outcome reported in a book of the same name published by Springer. This international study was co-chaired by me.

Before and After

A competition is an event which is part of a much wider experience. This experience enables a student to spend time in the weeks before the event practicing problems of the type which will be encountered (and all past problems are classified and available). Afterwards there is significant benefit in discussing the solutions, particularly following up the ones they could not do on the day. Any discussion, whether among students or with the teacher, is beneficial. This is why AMT provides fully worked solutions for the problems.

A Basis for further Development

On top of this the most important thing is that a student who has experienced all of this may want to do more. The AMC is just the first in a range of activities run from within the greater mathematics profession of Australia, leading to Australian students possibly representing Australia at the International Mathematical Olympiad and students in some other countries being identified for their national Olympiad programs.

Certainly the next step after the AMC is to participate in the very popular Mathematics Challenge for Young Australians, which includes course work which substantially develops the problem solving skills of the student.

At the end of this, Australian students who participate well in the AMC and the Challenge can be invited to participate in more advanced work, which might be under the tutelage of local academics or former Olympiad team members.

Why did it happen in Canberra - the CAE, or Peter O’Halloran or something else?

Yes, the obvious answer to this question was yes, Peter O'Halloran was the reason, and he was in Canberra. But there is a subtle extension to this answer which implies it wouln't have happened today in the same way. We were in a College of Advanced Education, a new type of tertiary institute in which the emphasis was on teaching and not researched. As mentioned in my memoirs, in 1972 I submitted a research paper to a typist and five minutes later was called into the Head of School's office and told not ever to do this again or even do research. I was told this was a teaching institute only.

In those days the only promotions possible were from Lecturer to Senior Lecturer and these were normally a formality once a Lecturer had been at the top rung for a year and provided conscientious service. But while the teaching load was heavier than at a University, staff were keen to find other interests. I had found an interest in developing good resources, hence my time at the Open University in Britain, but I and others happened to be keen and available to work with Peter.

I don't think this would be possible today. All of the Colleges were converted to Universities as part of the Dawkins reforms, and today Canberra CAE, as the University of Canberra, is just as competitive as a University as the former ones. It has recently broken into world rankings at about the 650 position, comparing with the best universities in the world, and staff are required to have a regular research output. I don't think Peter would have been successful if he had been at a University in 1976, nor would he have been in 2012 at the University of Canberra.

There is another point when answering the first part of this question, why in Canberra. A prominent Victorian educator told me how difficult it was to introduce the AMC in Victoria at all, and said it would have been impossible if the competition had originated in Sydney. Canberra is often viewed elsewhere in Australia rather negatively, but it is at least neutral, so maybe Canberra was just where it all had to start.

The numbers of schools, and which schools, and why they enter

It is difficult to generalise in this answer, as there are differing entry patterns around Australia, and in fact different cross-sections of school type in different states. For example New South Wales has a large network of selective government schools, whereas Melbourne has a greater number of private schools.

While writing this I no longer have access to the statistics, however towards the end of my term the number of secondary schools participating was about 80% of the total, maybe about 2300, whereas the number of primary schools is much smaller, maybe only 20%. Almost everywhere entry is voluntary in government schools, whereas private schools have variable entry patterns. Some focus particularly on year 7, to get a measure of student ability, some focus on top form only, and there are other patterns. There would be overall a proportionately larger number of students from private schools. Interestingly, the country is well represented, and we get letters from there thanking us for providing them a rare chance to do the same things that city students take for granted.

The Challenge and Enrichment are entered by less schools. Typically there might be 650 schools enrolled in the Challenge and say 320 in the Enrichment. Of course there are more schools which sometimes enter. According to our feedback, including the PwC report mentioned below, most schools know about us, but an average school might only enter when they have a particularly talented student. On the other hand there are high achieving schools which enter regularly and use the Enrichment as their main class activity for top form, when the top form has quickly mastered the formal syllabus.

Informatics is more difficult to answer, but in recent years, the Australian Informatics Competition has grown to over 4000 entries, from I imagine over 100 schools who see this as a viable mathematics class activity.

Something about the significance of the declining status of maths, and the declining qualifications of maths teachers

Everyone knows there is a problem in Australia, but no one has the answer, partly because it is a highly complex problem and certainly can't be fixed overnight. Australia's standards are declining in the main two international tests, TIMSS and Pisa, so there is no doubt there is a problem.

The key to standards is the teacher. Every time I ask a good mathematician if they were inspired by a particular teacher they will say yes, and provide at least one name. A teacher must be in control of the subject matter, understand the subject's culture, not be intimidated by a talented student but on the contrary be modest about the treatment of that student, should have a university major in mathematics, and above else must then have an enthusiasm for the subject which passes on to the student. There are now high schools in Australia where not one teacher has a major. Many Physical education teachers have been converted to maths teachers. Whereas many of them are generally positive towards this they don't tick all the boxes above.

In my day as a student there was no problem. If you wanted to be a teacher you could be paid as a student (very attractive) but be bonded for a few years. This can't be done any more so it is necessary to make the profession more attractive. The salary is not the only issue here. In fact salaries are not necessarily that low. But there are more attractive professions in the IT and finance industries, for example, where one doesn't need to worry about discipline issues. There are other ways of making teaching more attractive, even by giving them better workplace and dignity in their schools.

There are other problems. In my day as a student we spent 320 minutes a week in high school mathematics classes. Now this figure is generally accepted to be about 200 minutes. The syllabus has been trimmed down. What is left is more what I would call calculation with interesting parts of what I would call real mathematics missing. So the teacher does have a harder job to start with.

Teacher training is another part of the problem. Primary school teachers need a lot more than year 10 mathematics. There should be real mathematics content in the training of these teachers. It is now difficult for a primary school teacher to get a subject major. My son Gregory is the only primary teacher in Canberra with a mathematics major, and is in big demand, but now, the way funding works in Universities, only a sub major is possible, because an education faculty can get more money by teaching more of its own units.

Secondary teacher training is as serious a problem. The best way to qualify would be to get a mathematics degree first (even honours, like my brother John and colleague Anna Nakos) and then do a Dip Ed. But Education Faculties get more money by training the student with a four year BEd in which they teach a lot more sociology and psychology and a lot less mathematics. The University of Canberra has just restructured to include the maths and education staff in the one faculty. This is unusual in Australia but appears to be a very promising step. It does happen in Auckland University I and I understand it works well.

Elitism vs excellence

This is a perplexing subject because the use of these words can be used, sometimes deliberately, to give a misleading impression. There is not confusion about the word Excellence, it speaks for itself. It is the word Elite which has different meanings. A lot of people use the words Elite Athlete in the sense of being a very high level athlete, in fact an excellent athlete but one who has reached a level difficult to attain. This gets me to how I wish to use the word. An elite school to me is one which might be very expensive and generally difficult to get into. In other words inaccessible for the general person. My concern is with this meaning, related to accessibility.

A few years ago I was called to a government meeting for a briefing to be given by the then chief scientist, who had been given the job of breaking the news of budget cuts. People representing what I would call scientific enrichment organisations, for example science communicators, science week people etc. It was announced that all projects would have government funding cut, except two and these would be the Prime Minister Awards and the Olympiads. There were several speeches of dissent which followed and a number of these people described us as elitist.

So this is where I often had to clarify our situation in government negotiations. Yes, we get people up to very high standards, but this is expensive, and not many have the cash to afford it. So the need for Government support has always been to enable universal access, so students of all socio-economic backgrounds, can have access to this quality program.

The PwC report

Probably the most difficult part of my job always was to negotiate for government funding. This seemed to get more difficult towards 2010, when a new round of funding was due (we had for many years negotiated and were treated equally with our sister organisation Australian Science Innovations (ASI), which administered the Olympiads in Biology, Chemistry and Physics). We were given funding, but also were subject to an external review, which would be critical in further considerations. We had until this point of time relied on big-noting ourselves, and it was fair that some sort of external review be made to independently evaluate us. We welcomed the opportunity, as we were confident we had right on our side, but were initially nervous also, as it still depended on the agenda.

As it happened the reviewer appointed was PricewaterhouseCoopers, a high profile multinational, so it was clear that the review would have clout. The review started late in 201 and took place through the first part of 2011. The review started with me and my ASI counterpart being thoroughly interviewed, and over the following weeks various key stakeholders in sessions in various cities around the country. Early in 2011 we were given a preliminary debrief and asked for response. The final report was issued in the middle of the year.

The review was very favourable to us. The Science Olympiads had less infrastructure than us and received some special comments about survival, but together we were lauded for benefiting the country, lifting Australia's standards in mathematics and science, and thereby contributing to the economy.

There were some particularly nice details. Part of this included a professional evaluation of our volunteers, something we never knew how to do ourselves, but we were interested to see the evaluation, which was approximately equal to our annual expenses. We were able to extrapolate this to our AMC approximately also. One interesting outcome was that we were better known to schools around the country than some people thought we were, and we were heavily complimented on the success of our media program (as outlined below).

The relationship to government - Government funding

Government funding is very difficult to get, and the methods kept changing over the years, even being moved between two departments, the one containing education and the one containing science. One of the issues was, which we better belonged to, and it was not uncommon to be sent from one to the other. Both would from time to time a program most suitable for us, but the problem was we were in competition with organisations we didn't regard as similar. In the last government we had extra problems with education, as all discretionary money had been ceded for use by the states.

Since retirement I later got to know one of the public servants I had often had difficult negotiations with. He was by then also retired and we were able to speak a little more freely about the massive amount of time spent going back and forth. Of course these people would have their hands tied because they were under instructions from above. I assumed these people had the same frustrations at all the time which was wasted, and he absolutely agreed. He agreed the problem was we never fitted exactly to the purpose of the general funds to which we applied. Their view, like ours, was that the government should have quarantined from above money for the Olympiads. This would not preclude accountability.

What is the role of the media?

As implied from the PwC report, AMT has a very good history when it comes to using the media. This is very important to us because mathematics and science are much more difficult to get into the Australian media than sport (although it is the lifeblood of sport also). The reasons for our success can be attributed to the energy and organisation of Janine Bavin, who was our formal media and sponsorship contractor during my term as Executive Director. It also involved the need for easy communication between Janine and the office, particularly she should be able to get on the phone directly to the Executive Director and the Manager at any time. I always enjoyed working with Janine, and spent a lot of time focused on this aspect of the Trust's activities.

PwC highlighted the excellent media exposure we had during this time. Perhaps the main achievements were getting on to ABC's 730 report and 7 o'clock television news, frequent updates and interviews from the likes of Adam Spencer on Sydney ABC breakfast and Red Symons on Melbourne ABC breakfast, but it goes much further than that. The job is not done by writing a release and sending it off on a fax machine or placing on a web site. Most of the work is follow up and Janine was always maximising this. In one year during the 24 hour period of an Australian Mathematics Competition I was interviewed no less than 12 occasions, mostly by the ABC, which was always an enjoyable occasion as the reporters asked a lot of things in depth which sometimes I had not thought of myself. Janine would have set most of these up. Another memorable one was not long before retirement by arrangement I appeared on Rod Quinn's all night show, which went all over Australia, where Rod set the ball rolling by asking why mathematics was so important and the whole hour from 4am to 5am was devoted to the subject, most of which was spent fielding calls. I didn't know so many people were awake and active at that hour of the day! I also enjoyed going into the Canberra studios of the ABC where I was sometimes interviewed, particularly by Genevieve Jacobs.

As part of the strategy it is required for Olympiad team members to agree in writing to being available to the media. Not all of them are suitable, but some are outstanding, and we have means of knowing which ones will present the best. A particular favourite of Adam Spencer was Max Menzies, and he always asked for him during that time. But there were many others. Graham White was also excellent. I remember one night while walking through the streets of Ljubljana in 2006 after the Closing Ceremony I received a call on my mobile from the ABC in Sydney (Adam Spencer) and I was able to pass the phone to Graham who was interviewed live in the streets while people were waking up in Sydney to listen to Adam's program.

Students were also often interviewed by overseas media. Robert Newey, from Cobar, one of the great characters of the 2010 and 2011 IOI teams, and still active today, is seen being interviewed by the Canadian media after the Opening Ceremony of the 2010 IOI.

The inter-relationship of the Trusts’ competitions

The best way to describe the inter-relationship of the core events is to provide the Trust's pyramid of activities in mathematics and informatics, ranging from events for students of all standards under normal class conditions with teacher supervision, to representing Australia at international level. The pyramids show the normal way of progressing through the system, and they can be illustrated as such, with colour distinction between pre-Olympian and Olympian, and between open (school-based) and invitational events:

Mathematics Pyramid: Open Events

Australian Mathematics Competition (AMC)

Mathematics Challenge for Young Australians: Challenge Stage

Mathematics Challenge for Young Australians: Enrichment Stage

Australian Intermediate Mathematics Olympiad

Mathematics Pyramid: Invitational Events

AMOC Senior Contest

AMOC School of Excellence

Australian Mathematical Olympiad (AMO)

Asian Pacific Mathematics Olympiad (APMO)

AMOC Selection School

IMO team mentoring

Pre-IMO School

Entry numbers start at several hundred thousand at the base of the pyramid (AMC), through tens of thousands in the Challenge, to a hundred in the Australian Mathematical Olympiad and to just 6 in the IMO team.

Informatics Pyramid: Open Events

Australian Informatics Competition (AIC)

Australian Informatics Olympiad (AIO)

Informatics Pyramid: Invitational Events

AIOC School of Excellence

Australian Informatics Invitational Olympiad (AIIO)

French Australian Regional Informatics Olympiad (FARIO)

Asian Pacific Informatics Olympiad (APIO)

AIOC Selection School

Pre-IOI School

Entry numbers start at several thousand at the base of the pyramid (AIC), through a couple of hundred in the AIO, and to just 4 in the IOI team.

The Blazer Ceremony

I give a history of the blazer ceremony in the early stages of Chapter 12 (AMOC) and 13 (AIOC).

A few more illustrative mathematical puzzles

In Chapter 3 I listed my two most favourite AMC problems, both drawn from early years and composed by Bob Bryce (the sock problem and the Southern Cross problem). I have been asked to give some more and as I listed in Chapter 1.4 of ICMI Study 16, I wish to give some problems which have been listed in AMT-related events classified according to problem solving method. I believe this is a nice collection of methods which are accessible to students in the top 10% of a classroom, but not seen in the classroom, so they are ideal useful extensions.

Diophantine equations

Diophantine equations, which are linear equations with integer solutions, provide an excellent extension path for secondary students. The following problem is taken from an AMC paper and had a good response.

Problem 1 (Ages 13 to 15): Red rose plants are for sale at $3 each and yellow ones for $5 each. A gardener wants to buy a mixture of both types (at least one of each) and decides to buy 13 in total, with more yellow ones than red ones. The number of dollars he spent could be

(A) 51 (B) 67 (C) 65 (D) 58 (E) 57.

Discussion: Because of the finite nature of the problem, the student could canvass all the possibilities, working out the amounts when the number of yellow flowers varies from 7 to 12 inclusive (yielding all odd numbers between 53 and 63 inclusive). However, the problem could be done algebraically. Here, the students are challenged to define suitable variables and construct the necessary functions. Logical thinking, along with strict attention to the conditions of the problem, will lead students of average ability to the solution (which is borne out by the statistics of the competition).

Pigeonhole principle

This elementary idea, thought to have been first articulated as such by Dirichlet and often known as Dirichlet’s principle, is simply a statement that if there are pigeons to be placed into pigeonholes, and there are more pigeons than pigeonholes, then some pigeonhole will contain more than one pigeon. The statement can be extended to cover cases where the number of pigeons is more than double, treble, and so on, the number of pigeonholes, requiring the existence of pigeonholes with at least three, four, and so on, pigeons inside. The following is an example of an accessible problem whose solution is best wrapped up using this idea.

Problem 2 (Ages 13 to 15): Ten friends send greeting cards to each other, each sending 5 cards to different people. Prove that at least two of them sent cards to each other.

Discussion: The words “at least” are the ones which give the experienced student the clue that the pigeonhole principle will be useful here. However the student lacking such experience might ask how many routes from sender to recipient are possible. Since each of the ten friends can send to nine others, there are 90 available routes. However, each pair of friends is involved in 2 routes, so that there are 45 pairs. If more that 45 cards are sent, then by the pigeonhole principle, two of the transmissions must be on the same route in opposite directions. In this case since each student actually sends 5 cards, there are 50 transmissions altogether and thus two friends do send cards to each other.

Such challenges can generate discussion as to other situations where the pigeonhole principle is applicable, such as in combinatorics, number theory and geometry. However, while a useful tool, it does require special circumstances for its application. Two problems can look quite similar. One can be handily dispatched by the principle and the other can be very difficult indeed. It requires judgment and insight to detect when the principle can be used and to identify the pigeons and the pigeonholes.

Discrete optimization and graph theory

Discrete optimization is quite a different skill than that found in calculus. The standard method, which should be applied in an optimization problem with integer variables, involves two steps, one showing existence, and the other showing optimality, that is, giving an argument to show that the proposed solution cannot be exceeded. The following example, from the International Mathematics Tournament of Towns, is one in which there is a nice use of Eulerian graph theory, which is also a useful tool in networking problems.

Problem 3 (Ages 15 to 18): A village is constructed in the form of a square, consisting of 9 blocks, each of side length l, in a 3 by 3 formation. Each block is bounded by a bitumen road. If we commence at a corner of the village, what is the smallest distance we must travel along bitumen roads, if we are to pass along each section of bitumen road at least once and finish at the same corner?

Discussion: This problem is also an excellent interactive classroom problem. Students can try for some time to improve their first results until everyone is convinced they have a result which cannot be beaten. The shortest route does turn out to be of length 28l, with existence shown by the diagram.

The optimality part of the proof is a little more difficult, requiring the graph theory reminiscent of the Königsberg bridges. It is noted that there are three types of node, those with 2, 3 and 4 joining lines. In the case of the even-numbered ones a shortest path can go through them optimally, with an inward route matched by an outward route. However there are eight nodes with three joining lines, which means they have to be visited twice, involving an apparent wasted visit. Assuming they are in four pairs and that an extra route can be shared between a pair, there are at least four extra routes required. Since there were 24 routes in any case, the shown route travelling along 28 links is optimal.

Diagrams: Composers of problems such as this often face the issue of whether or not to include a diagram in the problem statement. The level of challenge can be quite different for such problems dependent on whether or not a diagram is provided. In inclusive competitions and in classroom use diagrams are more common. In advanced competitions such as the International Mathematical Olympiad, constructing the diagram is usually part of the challenge and diagrams are rarely if ever provided in the description of the problem.

Cases

Quite often experimentation with a situation leads to a conclusion that a result can only be established after an exhaustive consideration of a mutually exclusive set of cases. This method is usually known as proving by cases. The challenge is two-fold. First, one needs to identify the cases that might apply and to describe them in a way that is clear, efficient and preferably non-overlapping. Secondly, one needs to ensure that the cases are exhaustive, that nothing is left out. This can be illustrated by the following problem, one of the more challenging problems from the AMC.

Problem 4 (Ages 15 to 18): The sum of n positive integers is 19. What is the maximum possible product of these n numbers?

Discussion: This problem is also excellent for classroom interaction. Students can try to obtain maximum products with various selections but soon discover that high numbers adding to 19 don’t seem to help, while at the other extreme the number one is also useless. Students will eventually see that a summand m bigger than 4 can always be replaced to good effect by 2 and m − 2. They will also see that 4 can always be replaced by 2 and 2. They will also be able to formally dispose of the case of an integer being 1. So they are left to consider only sums that use the numbers 2 and 3. The situation is finally resolved by noting that replacing any three 2s in the sum by two 3s will increase the product.

In this problem, students might ask whether the number 19 plays an essential role, or whether the same argument is available when this number is replaced by some other. A common practice in competition problems is to ask students to look at a particular instance of a general result; a student who is aware of this may often establish the general result, often by a more effective argument as the solver is not distracted by irrelevant factors particular to the situation.

A secondary challenge for a more senior student studying calculus is to decide whether there is a continuous version of the challenge and formulate it exactly.

Proof by contradiction

One of the most famous proofs in mathematics, that the square root of 2 is irrational, is made by contradiction and is accessible from school mathematics. It is often seen that a direct proof promises to look very complicated and these are the occasions to try contradiction. The following problem, taken from the International Mathematics Tournament of Towns, is most easily solved by contradiction.

Problem 5 (Ages 15 to 18): There are 2000 apples, contained in several baskets. One can remove baskets and/or remove any number of apples from any number of baskets. Prove that it is possible to have an equal number of apples in each of the remaining baskets, with the total number of apples being at least 100.

Discussion: This is hardly in the form that a student might encounter at school, and the initial challenge is to figure out what is being asked. The indeterminacy of the situation and the variety of possibilities for removal of apples and baskets boggles the mind. An efficient way to control the situation is to suppose that the result is false. As the reader will see in the solution, within this large contradiction are a number of smaller ones to be negotiated.

Suppose that we have a configuration of apples and baskets for which the result fails to hold. It does not change the problem if we assume that all empty baskets have initially been removed. Then the number of remaining baskets does not exceed 99; otherwise, we could leave an apple in each basket and get a contradiction.

In a similar way, we see that number of baskets with at least two apples cannot exceed 49; the number of baskets with at least three apples cannot exceed 33, and so on. We now estimate the number of apples in the baskets. This cannot exceed

99 + 49 + 33 + 24 + …. < 100(1 + 1/2 + 1/3 + 1/4 + … + 1/99) < 100(1 + 1/2 + 1/4 + 1/4 + … + 1/64) < 100(7) = 700.

However, this contradicts the assumption that there are at least 2000 apples.

Enumeration

Combinatorial problems are popular in challenges because they can be less dependent on classroom knowledge and therefore be fair ways of identifying potential problem solvers. Enumeration is a popular source of such problems. Enumeration problems, properly set, can be solved in the time allocated and they have the advantage of challenging the student later to try to generalize, to enable similar problems to be solved from an algorithm.

A good example is the derangement class of problems, of which the following problems from the AMC are good accessible examples.

Problem 6 (Ages 13 to 15): In how many ways can a careless office boy place four letters in four envelopes so that no one gets the right letter?

Discussion: It is possible for a junior high school student to list and count all the cases. Then the student might find the answer if there had been five letters instead of four. If she looks at higher cases too difficult to count, she might find the famous derangement formula.

Problem 7 (Ages 13 to 16): In the school band, five children each own their own trumpet. In how many ways can exactly three of the children take home the wrong trumpet, while the other two take home the right trumpet?

Discussion: This is a variation of the derangement problem in which some matchings are correct and others incorrect. It is possible again for students to count the cases, as the statistics from this inclusive competition showed, and to look for generalizations.

Invariance

Discovering an invariant in a problem can lead to a simple resolution of an otherwise intractable problem. The method of invariance applies in a situation where a system changes from state to state according to various rules, and some property which is important to the statement of the problem remains unchanged in each transition.

This method is very well illustrated by the following famous problem from the International Mathematics Tournament of Towns, not just for its mathematical properties, but for other various associated aesthetic features.

Problem 8 (Ages 15 to 17): On the island of Camelot live 13 grey, 15 brown and 17 crimson chameleons. If two chameleons of different colours meet, they both simultaneously change colour to the third colour (e.g. if a grey and brown chameleon meet they both become crimson). Is it possible they will all eventually be the same colour?

Discussion: If the original number of chameleons had been 15 of each color, it is clear that pair-wise choices of chameleons of the same color pairs would lead to 45 chameleons all of the third color. However with this starting configuration all attempts to obtain the same result fail. The student needs to find a basic property of the starting numbers 13, 15, 17 which remains unchanged during every meeting of two chameleons of different colors. In fact, no two of the three numbers of colored chameleons leave the same remainder upon division by 3.

Inverse thinking

Sometimes there can be useful challenges involved by thinking in the inverse direction. Here is a problem from the Mathematics Challenge for Young Australians.

Problem 9 (Ages 14 to 16): A Fibonacci sequence is one in which each term is the sum of the two preceding terms. The first two terms can be any positive integers. An example of a Fibonacci sequence is 15, 11, 26, 37, 63, 100, 163, ...

• Find a Fibonacci sequence which has 2000 as its fifth term.
• Find a Fibonacci sequence which has 2000 as its eighth term.
• Find the greatest value of n such that 2000 is the nth term of a Fibonacci sequence.

Discussion: Generally one thinks of a Fibonacci sequence in the forward direction. Here, as is common in an inverse thinking scenario, instead of being given the data and then finding the results, here we are given the results and are asked to find the data. It is a challenge for students to think this way.

The student can do this by searching through various second-last terms and working back. In doing so, depending on which term they choose, they can work back uniquely but some choices will not go back far. If the second last term is less than 1000, the third last term is greater than 1000 and that is as far as we can go, as the next term would be negative. We do not do much better if the second last term is too high.

The student can eventually focus in on a small range of values for which the sequence can be traced back a few terms, and then finally the one which goes back optimally. The Golden Ratio can be discovered in extended thinking of this problem, which makes a nice surprise.

Colouring problems

There is a famous problem in which an 8 by 8 checkerboard has it top-left and bottom-right squares removed. One is then asked whether 31 dominoes (1 by 2) can be placed over the remaining 62 squares. At first sight this can seem a tantalising problem with the student trying for some time to show an arrangement. However, as with the chameleon problem, a solution is not reached and the student is left wondering why not. In the end, the reason is obvious. Each domino necessarily has one of each colour in the normal checkerboard coloring scheme. However the two squares removed are of the same colour, leaving an imbalance.

The following problem, taken from the International Mathematics Tournament of Towns, is an extension of this idea.

Problem 10 (Ages 14 to 18): A 7 by 7 square is made up of sixteen 1 by 3 tiles and one 1 by 1 tile. Prove that the 1 by 1 tile lies either at the centre of the square or adjoins one of its boundaries.

Discussion: This problem has a rather surprising result and at first sight, with all the combinations possible, seems almost impossible to prove. But an extension of the domino question above, coloring with 3 colors instead of 2 and looking at the resultant way in which a 3 by 1 domino might cover squares of the board, makes the problem accessible.

Concluding comments

The common feature of all these problems is that no difficult calculations are needed anywhere, which helps to ensure that students in a normal class can reach out to the problem from their normal experience. All of the problems require the discipline of clear thinking, which will enhance the general problem solving capacity of the student.

Major advances in our society often emerge from the disciplined solution of apparently simple problems. For example, Euler’s analysis of the bridge problem led to a higher level of knowledge which is central to modern technology today. For him to have been able to develop an idea beyond what was known is similar to the challenges which people continually face on a day-to-day basis.