The Group Action Game or How to win pints from gullible students

by Shirleen Stibbe

M203 Pure Mathematics Summerschool

Note: to play this game, you'll need two overhead projectors; one for showing the game board, and the other for the rules / scoring / group action table etc.

If you're planning to try it, you can dowload pictures of the board and other stuff from this page of my website:

http://www.shirleenstibbe.co.uk/summer-school

This was a 1-hour revision session on Group Actions. The students should have covered the material before they came to summerschool.

I started by showing them four slides designed to remind them of the basic concepts they should have been familiar with (most weren't). That took about 5 minutes.

"And that's all there is to know about group actions. Thank you very much for attending. Apologies that the session was so short – but there just isn't anything more to say about group actions.

Actually, since you have some unexpected free time, maybe you could help me out with something.

I found a sort of game thing in my room when I arrived – must have been left by a tutor in a previous week – and I'm not sure what it is. But since it involves winning pints in the bar, I rather fancy trying it out. Are you in? "

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The Home Position: starting point for each round

Each shape has a direction, and appears twice in the circles.

The inner circles of shapes and numbers are on a separate slide, so that it can be rotated or reflected relative to the outer (fixed) circles.

The reflection guidelines appear on both slides, so they can be matched folllowing an action.

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Movements

a rotate through 90o b rotate through 180o c rotate through !90o d reflect in vertical e reflect in diagonal f reflect in horizontal g reflect in diagonal

Scoring – number of pints Same Shape

Same Direction

Me

You

! ! 0 2 ! " 1 1 " " 2 0

Actions and Scoring

I invited volunteers to generate a letter / number combination by using the first of the action letters that appears in their name (or the dog's name if necessary) and the day (mod 16) of their birthday. I was 12 pints ahead of them before they demanded to choose their own combinations.

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They discovered, fairly quickly, that choosing a reflection and a number on the reflection axis would fix a shape, but reverse its direction – 1 pint each. (Image shows the result of vertical reflection, which fixes the shapes labelled 1 and 9).

They started winning back pints when I allowed them to use two different actions – easy to spot rotation through +90 and -90. It took them longer to find the two winning strategies using three different actions.

By this stage, about half the audience had sussed that it was a group action, and I asked them to identify the group (D4, the symmetries of the square, of order 8) which they got easily.

It was a little more tricky to decide what the objects in the set were. I gave them a hint by confessing that I'd given each object a direction so that I could prevent them winning outright on the reflections. So we ignored the arrows, and eventually decide that the 16 objects were number/shape combinations.

We had some fun working out the orbits. We ignored the direction arrows, and this is what we eventually came up with:

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Orb 1 13 9 5 1

Orb 2 15 11 7 3

Orb 3 16

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12

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14

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10

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I got them to add the stabilisers of each object to an action table, ignoring the direction arrows (life was too short to complete all 128 cells), We didn't need the shapes for the table. It was a good opportunity to remind them of the one-to-one correspondence between the set of stabilisers of X and the set of fixes of G.

The final activity, before we repaired to the bar, was to use the Counting Theorem to confirm the number of orbits we found. A 'phew!' moment when it checked out. Which we celebrated enthusiastically.

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