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## 'Got a Strategy for Last Biscuit?' printed from http://nrich.maths.org/

### 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.