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Poison, Antidote, Water

Stage: 4 and 5 Challenge Level: Challenge Level:1

Why do this problem?

This problem provides a precursor to work on abstract group theory and advanced mathematical reasoning. There is a simple concrete context in which the concept of associativity can be explored before any technical definitions are introduced. During these tasks the key concepts of group theory and group tables will arise naturally.

Possible approach

First part of the task (Poison, Antidote, Water)
Describe the concept of the game of Poison, Antidote, Water and use the interactivity to play the game a few times with volunteers to see if anyone is Poisoned. This should be sufficient for students to understand the nature of the task: does the order in which I uncover the cards affect the outcome of the game?
Encourage students to start off this task experimentally: write out a couple of different lists of Ps, As and Ws. For their lists, does the order seem to matter?
Second part of the task (Scissors, Paper, Stone)
Once the first part has been done, it should be easy to set the task of analysis of Scissors, Paper, Stone. Students will quickly realise that they can create examples where the order does make a difference. 
Third part of the task (comparison)
This need not take long, but should focus the minds of the students on the fact that structural comparisons can be made; moreover several discussion points might emerge which you can either note or develop according to your mathematical confidence. The main points to note are these: some operations are associative (order matters), whereas others are not; 'group tables' are a good way of representing such games; some cards take the role of an 'identity'.
Final part 
Set this part without preamble and ask students to work on it in pair or alone, according to their preference. It is hoped that students will soon realise that a 'group table' will be useful and the concepts of 'identity' and 'commutativity' very handy. Depending on the abilities of the group this part can usefully be tackled before or after encountering group tables. You might wish to give a hint that assigning the role of an 'identity' to one of the cards makes the problem simpler.

Key questions

Are any lists of Ps, As, and Ws easier to analyse than others?
What sorts of lists seem to result in being poisoned and which do not?

Possible extension

Consideration of the final part should be sufficient extension with this challenge.

Possible support

Focus on the earlier parts of the task: the main goal of the problem is to raise awareness of the concept of associativity and practise mathematical reasoning. For a weaker group you might try this problem after encountering group theory and group tables as a way of reinforcing the concepts already learned.