Vacation research scholarship I

Over the summer break, Gordon and I each supervised a 2nd year undergraduate student in a research project. At UWA, the first course in group theory is in 3rd year, and we do not teach any combinatorics, so we needed to give each student a crash course before they could sink their teeth into a research problem. One of their outcomes was a blog post, and the first of these is by my student, who was also supervised by Sylvia Morris. His project was on “Sets of type (m,n) in projective spaces”.

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The MDS Conjecture

I will begin with a couple of games you can play with the ternary [4,2]-Hamming code, in much the same way Peter Cameron describes here and in a previous post of mine for a different Hamming code.

Choose a number between 0 and 8 (inclusive). I will ask you four questions, you can answer ‘a’, ‘b’ or ‘c’ and you are allowed to lie at most once. Here are my questions:

  1. Is your number (a) less than 3 (b) between 3 and 5 (c) more than 5?
  2. Is your number (a) 0 mod 3 (b) 1 mod 3 (c) 2 mod 3?
  3. Is your number in (a) {0,5,7} (b) {1,3,8} or (c) {2,4,6}?
  4. Is your numberin (a) {0,4,8} (b) {1,5,6} or (c) {2,3,7}?

From these questions, I can easily determine your number.

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Maths activities for school students

One of the things John and I do around here is run activities for groups of school students  who are visiting the university. Today we had such a visit from a group from Esperance.  The visits often include activities in other disciplines across campus, for example physics demonstrations or building bridges/gyroscopes/electric motors in engineering. This presents an interesting dilemma as to what the maths activity should be.  It needs to be very much hands on so that the students get to do something. John and I are also keen for it to be very different to the sort of maths done at school.

Currently, our main activity revolves around the game of Nim. We introduce the rules of the game, give a short demonstration and then call for challengers. We offer a chocolate bar to anyone who can beat us.  After a couple of games, we then give all the students some counters and get them to play the game amongst themselves so that they can try and work out a winning strategy and continue to try to beat us.  Towards the end of the activity we outline binary numbers, Nim addition and how to use it to win.  The activity goes down really well.  It draws  on their competitive instincts and some students get right into trying to beat us. It has worked with the whole range of high school ages and we once even ran the activity successfully  for a group of primary school students (minus the maths behind it though).

We are interested in learning about  other suitable activities.  It would be good to have a couple up our sleeves as there are sometimes students who attend more than one of these visits. John is looking at developing something involving Conway’s Rational Tangles. He has demonstrated it to a couple of groups of students now and has been very succesful but it is more of a demonstration than an activity. Ideally we should pair it with a couple of other things as we usually have 45-60 minutes. Please let us know of other activities that work well which could be done with or separately to these.

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