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'Got a Strategy for Last Biscuit?' printed from

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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. They will then be able to put their strategy into action by beating the computer, which never makes a mistake in its moves, although this will still require clear application of the procedure of the strategy.

Possible approach

This game is difficult and most students will need to play it (and lose!) several times to start to get a feel for how to win. Note there there is no 'obvious overall strategy', so students should play the game for a while and then come back together to share ideas. Suggest that students consider the cases for low numbers of biscuits first (less than 10 of each type).

It might become apparent that certain configurations are known to be a 'win' for person to play next. When one of these configurations is found it could be shared with the group.

Using known winning configurations might allow students to develop 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 found? Could students make any conjectures as to the form of winning configurations?

Key questions

  • What is the smallest 'certain lose' position?
  • How could we prevent the computer from putting us into this position?

Possible extension

This structure allows a rich analysis which very able students might enjoy. Various questions of proof are as follows:
  • Can you 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?

You might also like to play Nim , which is a version of this game with multiple counters.

Possible support

Focus to start with on numbers of biscuits less than 5 of each type. Can a strategy be devised to win for these individual cases?

Students could play in pairs if the computer is winning too often. They could also play with counters instead of the interactivity.