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### Number and algebra

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### Working mathematically

### Advanced mathematics

### For younger learners

# Last Biscuit

### Why do this problem?

### Possible approach

### Key questions

### Possible extension

### 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.

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### Nim

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Age 11 to 18

Challenge Level

- Game
- Getting Started
- Student Solutions
- Teachers' Resources

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.

*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?

You could use the interactivity to demonstrate the game, or start with 12 biscuits (or buttons, or pebbles, or...) and put four biscuits (or buttons, or pebbles, or...) into one jar/container and eight into the other jar/container.

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 first consider the cases where there is a small number of biscuits (less than 10 in each
jar).

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?

- What is the smallest 'certain lose' position?
- How could we prevent our opponent from putting us into this position?

The problem allows for a rich analysis and for opportunities to engage in justification and proof:

- 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'.

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.

Ask students to start by considering the cases where there is a small number of biscuits (less than 5 in each jar).

Can they devise a strategy to win for these individual cases?

Can they devise a strategy to win for these individual cases?

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.