A training guide for the Maths Olympics

The highlight of the MUMS calendar is the annual Maths Olympics. One of the best things you can do in the lead-up to this great event is to engage in the training that up until now has only been practised by one or two teams every year. I do not pretend to offer the final word when it comes to training methods for the Maths Olympics, but I will try to give a glimpse into the methods my team has used in the past seven years we have competed. What worked for us may not necessarily work for your team. Try them and see!

Before we begin, you may want to read the following explanation of the difference between the Maths Olympics and the International Mathematical Olympiad, often known as the maths olympiad.

First, a few words about the Maths Olympics itself. It's a maths relay with teams of five people. The team is split up into two groups on opposite sides of Theatre A with a runner relaying a question from the marker to the groups. Only one side may work on any one question and you  can't work on the next question until you've either solved the previous question or given up on it. There's no penalty for wrong answers so you can guess as much as you want. Calculators aren't permitted for this contest - you're going  to have to use old-fashioned pencil and paper!
Why should you participate in the Maths Olympics? Because it's a fun thing to do. In fact one of the rules explicitly says that "You shall have fun!" What other event at uni allows you to combine the joy of solving interesting problems, team work and physical activity all at the same time?  You will have the chance to watch your lecturers embarrass themselves. The lecturer teams have often fared poorly (they're a bit too old!). It's probably the only chance you'll have to get physical with them - one year the then head of maths was taken out with bruised ribs. Listen out too for the witty comments by the compere.

Even if you don't feel like competing, come along and spectate. The start has always been a great sight - all runners simultaneously hustling for that first question.  It can also be really funny seeing friends of yours being really pumped up after solving questions, or really flustered while attempting them! If that's not enough there's a spectator competition as well - questions already attempted by all the teams are put up on an overhead and spectators who solve them win chocolates for their efforts. For those who prefer to just sit back and watch, you'll find that  the compere hurls sweets into the crowd at regular intervals.

Some tips

The following tips offer some ways to improve your performance and  also enjoyment in the Maths Olympics (to a level even higher than what it already would be!).

Enter early. You won't enjoy it as much if you're not competing! Due to the finite size of the lecture theatre, only about 28 teams can enter. In some years, almost 50 teams have submitted entries, only for many to be turned away.

Bring along lots of pens and paper. You'll need it!
Look at the sample questions. Have a look at sample questions to get a feel for the type of questions posed. Questions tend to be similar in style to the ones in the Australian Mathematics Competition (often referred to as the "Westpac"). As a result, it's not necessary to have done any maths at uni, in fact this may even be a handicap!

The runner can and should help. In years gone past, the runner was forbidden from helping with the question solving. However, due to blatant violations of this rule and the difficulty of enforcement, it was repealed. Keep working when the runner goes off, as the answer may have been wrong. One worthwhile tactic is to have the runner running back and forth guessing answers while the rest of the side tries to actually work out the answer.

Don't give up too easily! If you get stuck, read the question carefully, even if you have no idea, spend a minute as it may be easier than it first looks. Questions are usually designed to have a neat solution that doesn't require too much calculation. If you still have no idea, don't be afraid to abandon the question. One easy trap to fall into is to keep on investing more time in a question because it is psychologically difficult to give up on a question once you've spent a significant amount of time on it.

Guess. Always look for the opportunity to guess the answer. For example, consider the following question:

f(1)=1, f(2n) = f(n) and f(2n+1) = 1 - f(n). What is f(98)?

At first you might think,

f(1) = 1, f(2) = f(1) = 1, f(3) = 1 - f(1) = 0,
f(4) = f(2) = 1, f(5) = 1 - f(2) = 0, f(6) = f(3) = 0.

Hmmm, no obvious pattern... Then you might say,

f(98) = f(49) = 1 - f(24) = 1 - f(12)
= 1 - f(6) = 1 - f(3) = 1 - (1 - f(1)) = 1 - (1 - 1) = 1.

That wasn't too bad was it? However, another possibly quicker approach would be to notice that the rules imply that f(n) must be either 1, or 1 - 1 = 0. Thus a quick way to do this question would be to guess 0 and when that fails, try 1. Remember that there is no penalty for guessing, you can do it as many times as you like (provided the runner doesn't mind!). So let your legs do the thinking for you!

The team name. A vital part of the Maths Olympics is the ingenious team names that teams come up with every year. Some team names from the past have included,"No Real Solutions", "The Return of the Lemma", "Pythagoras was Wrong!", "We Don't Count", "Meds on Prozac", and "Gottim - Yes". Start thinking now! For those who really want to get into the spirit of things, team mascots, music, costumes and make-up are encouraged. Bring along a cheer squad too.

Lastly, don't take it too seriously. Have fun! Unless you're one of those people who can't stand coming 2nd (the section below is devoted to them!), just sit back  and enjoy the spectacle when the other half of your team is working on a problem. Yell and scream a bit! At the end, even if you receive no monetary reward, I guarantee you will have had one of your most entertaining hours at uni.

Advanced Training Techniques

Caution: The methods described are highly dangerous and may result in physical and mental injury. Do not try these at home! Do not try without expert supervision. The author will not be responsible for any loss of property, physical injury or impairment of mental faculties sustained. You have been warned!

This section is for those who are ultra-competitive perfectionists who are satisfied with nothing less than coming first, or failing that, to perform at their absolute best.

Treat it seriously! You must learn to ignore all your friends who say that you're mad. Treat it as a badge of honour! At least one team in the past has gone to extraordinary lengths to ensure that nothing will stop them. In fact, the current (at the time of writing) president of MUMS has often been heard to say that 50 hours of training is his target (he's been known to achieve this too).

Pick your team early. Nothing is more important than having the right team, and nothing is more frustrating than finding that your plans for the ideal team have been thwarted due to someone having previous commitments to a rival team. So what makes a good team?

  • As the emphasis is on speed, it's important to find people who can think and calculate fast.
  • Depth is also important. While one or two very strong problem solvers on a team can make a big difference to a team's chances, it's risky to rely on them. Everyone has their bad days when they can't solve much. Having a team with plenty of depth ensures that even if one or two people have a bad day, the others will be able to keep going. The chance that everyone will have a bad day is relatively remote.
  • Possibly the most important factors is synergy between team members. All the team members must respect each other and there must not be conflict between team members. Some ways to achieve this are to select people who are friends and make team members feel secure of their place in the team. Nothing is more demoralizing than to be told that "you might be chosen if we feel like it".
Teamwork. So why do we need synergy amongst team members? It's easy to see that often three people working independently will solve a question faster than one person alone, simply because some people are better at some questions than others. What is less clear is that three people working together can dramatically increase the speed at which problems can be solved and also the range of problems that are solved. There are several reasons for this. Some types of problems may be amenable to a "case bash". You try all the cases to find out which one works. By splitting up the cases amongst the three team members, it may be possible to halve the time taken to cover all these cases. The hardest part is often seeing the approach to take. Announcing what approach one is taking to the others may help everyone get onto the right track earlier. Sometimes (due to the tension) someone may come up with the approach but be unable to perform the calculations. So it may be effective to have one person come up with the idea and another carry out the computation. When long and involved calculations are needed, working in parallel so as to be able to check intermediate steps can be helpful.

Optimise your running order. The ideal first runner is a rugby player. Readers who have witnessed the mad rush to pick up the first question will understand why. Most teams aren't lucky enough to find a mathematically gifted rugby player, but it helps to have someone who can negotiate a dense crowd. The next few runners should be in decreasing order of speed. It's no good having a fast runner if they don't get the chance to run! However there is a trade off here - it may be better to have the fitter runners run later when there's likely to more questions that may require a lot of guessing (because the increased difficulty makes it difficult to solve any other way).

Manage your time well. Why do you need a time management strategy? The most important reason is that it is easy to spend too much time on a question. Psychologically, it is difficult to abandon a question after having spent a lot of time on it. It is all very well to invest a lot of time in a question if you end up solving it, but it leaves you vulnerable to various risks:

  • The official answer to the question may be wrong.
  • The question maybe ambiguous (or use a convention with which you are unfamiliar) which means that you may be working on a different problem to the one intended.
  • The question may be much harder than the ordering of the questions suggests. This was dramatically illustrated in 1996, when the team that came 2nd used the high risk strategy of skipping all the 5 point questions and only attempted the 10 point ones. There just so happened to be a significant number of easy questions in the 2nd 10 questions whereas the 1st 10 contained a number of quite difficult questions. No other team got past question 15 (out of 20).
Note that whether the questions are well ordered is highly team dependent as what may be a hard question for the setters of the paper isn't necessarily hard for you, depending on your background. After much experimentation over the years we've found that the most effective time management strategy for us is to rigidly allocate a limit on the maximum amount of time that we spend on each question with this time being proportional to the number of marks allocated to the question.

This has the several advantages:

  • It deals with all the issues mentioned above because it ensures that you attempt every question and allocate sufficient time to have a good go at even the last few questions. This protects you against mistakes in questions, solutions or misordering of questions.
  • It is a very simple guideline to implement. This eliminates disagreements over whether a question should be passed or not. By guaranteeing that each question gets a reasonable amount of working time, it reduces the chance of panic at the end when the pressure is greatest.
So how should you implement this strategy? We had a sheet on which we recorded how long we'd spent on each of the preceding questions. That way we always knew exactly how much time we'd spent on the current question and how much time was left.

It's also important to only implement this strategy if you have trained with it. Because people are reluctant to abandon questions after having spent a lot of time on it, every team member must be convinced that on average, this strategy works well. This confidence in the strategy can only come through successful implementation in practice.

Train physically. Mens sana in corpore sano - a healthy mind in a healthy body. For the really dedicated, there is the physical training. Contrary to popular belief, the primary reason for training physically is not to reduce the time taken to run up and the down the stairs. The real reason is so that the runner can contribute towards solving the questions as well. Thinking requires lots of oxygen to reach the brain and those who have competed in the maths olympics before will realise that it is very difficult to think when you are tired from running and the blood is coursing through your brain. You can only avoid this by being fit and training especially to counteract this problem. Some of the members of my team used to practise doing questions and then running up and down
the stairs at home (and in the actual lecture theatre!) a couple of times in between each question.

The second reason to train physically is to enable exhaustive (both in the mathematical and physical sense!) guessing as a technique to solve problems. There have been times when we guessed the answer to a question after about 8 attempts. To be able to do this, the runner needs to be extremely fit.

It's also important to practise going down stairs. Going up stairs is relatively easy, as long as your legs are strong enough, you just go up 2-3 steps at a time, depending on the length of your stride.

Going down is much more involved. First, going down stairs one step at a time is very slow, because you are limited by how fast you can physically move your legs. The solution? Go down the stairs 2 steps at a time! This produces dramatic speedups but requires training months in advance. Because of the speed at which you go down the stairs, much skill is needed to avoid injury. Skill is also needed to maintain a smooth descent for greatest speed. To develop the necessary technique, practise descending slowly and smoothly in a controlled manner and then gradually increase the speed until you can just glide down the stairs at full pelt.

For those who are interested, there are physiological reasons why you need to train specifically to go down stairs. Most of the exercise that we normally do is concentric exercise, where the muscles exert force and contract in the process. Descending stairs requires eccentric exercise, where the muscles exert force and lengthen in the process. Because we don't normally do eccentric exercise (except in some more exotic sports) even very fit people who could run comfortably up many flights of stairs would have very sore muscles if they tried to run back down those stairs. For those who are interested, there has been recent research conducted by Proske and Morgan at Monash University into this. See http://www.monash.edu.au/pubs/montage/Montage_97-01/p8-9.html.

Train with more difficult problems. By choosing difficult problems, you have the chance to practise teamwork, communication and the division of different parts of problems amongst team members. With easy problems, the communication overhead makes it not worthwhile. Also difficult problems give you the opportunity to practise passing questions.

You now have all the tools that you need to compete in the Maths Olympics either to have fun, or to win. If you think of innovations, or disagree with some of my suggestions, feel free to write to me.

Go get 'em guys!!

(written in 1998)

About the author:

Lawrence Ip competed in Maths Olympics from 1991 through 1997, first competing when he was in year 11 at school. He captained the winning team in 1992, 1994, 1995 and 1997. Since retiring from competition he has spent the last 5 years pursuing a PhD in quantum computing at UC Berkeley. He often dreams of making a comeback but in moments of reason realises he is too old for this kind of thing.

Postscript, an overdue acknowledgement:

Although I was the one who wrote this training guide, the development of these training methods was not a solo effort. My main collaborator was Chaitanya Rao. At the time this guide was first written he was too shy to have his contributions recognised.

Last modified August 29, 2010