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The Hair Colour Game

Age 7 to 11
Challenge Level

Ed from St Peter's College gave a good solution to the problem:

The first cube (or pair of cubes) means a boy with dark hair, the second cube is a boy with light coloured hair, the third is a girl with dark hair and the fourth is a girl with light coloured hair.
The next day there were eight groups because they added eye colour cubes.
If there was no one with dark coloured hair and blue eyes, you could still place it on the diagram, to show all the possibilities.  

Daniel and Theo sent a clear and colourful tree diagram which you can see here.
Arjun from Vidyashilp Academy thought about the number of groups you would end up with for more cubes.  He says:
For $1$ cube combination there are $2$ possibilities = $2^1$
For a $2$ cube combination there are $4$ possibilities =$2^2$
So, for a $3$ cube combination there are $8$ possibilities =$2^3$
Thus I can say that the total number of combination are $2^n$ where n is number of cubes.

I wonder how we know this for certain?  Could you convince us that this is always the case, Arjun?

Well done to everyone!