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Age 14 to 16

Challenge Level

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This problem offers opportunties to consider different methods of listing systematically. It can be used to introduce or revisit sample space diagrams, and with some students, tree diagrams.

Play the game a few times for real.

"Is this a fair game? How can we be sure?"

Class work in pairs trying to decide and to develop an argument to justify their conjectures.

After about ten minutes, stop to discuss the merits of different arguments and representations. This may be an appropriate point to highlight the benefits of different systematic methods for listing all possibilities, using sample space diagrams and, if pupils have met them before, tree diagrams. This is an opportunity for everyone's method to be valued, and for the whole class to work
collaboratively. They should agree on some representations or methods for identifying fairness.

Finding a fair game can become a class activity:

Students help to create a class list of all distinct starting points for the game (for example, four ribbons can be either $1R$ and $3B$ or $2R$ and $2B$). These are written on the board for $3, 4, 5, \ldots$ribbons.

Distribute the task of checking which combinations are fair and record them on the board as pairs of pupils decide.

There are not many solutions that work and if pupils are to notice a pattern amongst the combinations that are fair they may need to consider up to a total of $16$ ribbons.

Spend some time conjecturing about more than $16$ ribbons and test.

How can you decide if the game is fair?

How many goes do you think we need to be confident of the likelihood of winning?

Are there efficient systems for recording the different possible combinations?

Can you justify your conclusions?

When the work is being shared out amongst pupils in the class the smaller number of ribbons can be given to pupils who struggle most. They can also use physical objects, such as coloured counters, to check their findings.

Justifying the general rule