### 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.