Last Biscuit
Why do this problem?
Strategy games are always good for developing mathematical thinking. This game is interesting because although the 'overall' strategy is difficult, students can usefully analyse particular cases. This will require clear recording of results and careful analysis of the logical possibilities.
Possible approach
This problem featured in an NRICH Secondary webinar in June 2021.
You might want to introduce this problem as a follow-up to Have You Got It?
It might become apparent that certain configurations are known to be winning positions (e.g. 2 and 1).
When one of these configurations is found it could be shared with the group. Using known winning configurations might help students find winning configurations for larger numbers of biscuits.
Throughout, clarity of thinking, analysis of the game position and clear recording of results should be encouraged.
Once several winning positions have been found, can any patterns be identified?
Can they use these to find more winning positions?
Key questions
- What is the smallest 'certain lose' position?
- How could we prevent our opponent from putting us into this position?
Possible extension
- Can students prove that (1, 2) is the only winning configuration with a difference of 1 in the total number of biscuits?
- How many winning configurations differ by 2, 3 or n biscuits?
Students might also like to play Nim, which is a version of this game with many more 'jars'.
Possible support
Start by introducing students to Have You Got It? and don't move on until you are satisfied that students can explain clearly why their strategy works.
Can they devise a strategy to win for these individual cases?
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Nim
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.
