Probabilities in Game design

I love it when these little moments crop up unexpectedly and you see another role that mathematics has. So I am casually playing a game of Rubik's Race with one of my children. Follow the link to find out more about the game, but it is fairly simple. Each of 2 players has a 5 x 5 grid of coloured tiles like the one in the picture on the left. There four tiles in six different colours and one empty space that allows for tiles to be slid in to different combinations. Then a roller like the one in the picture on the right is shaken. The outcome is a 3 x 3 combination of coulours and the aim is to slide your tiles around so that the 3 x 3 grid in the middle of your 5 x 5 grid shows the same combination. Easy really and great fun - highly recommended game. Anyway, my daughter shakes and comes up with the combination shown below right and says "hold on, this isn't possible because it shows 5 yellows!" I stop and think for a while and then realise she is right and that since each cube in the shaker has one of each of six colours on it that it is technically possible to get them all the same colour. Rather than concluding that the game designers have missed an obvious trick I guess that they have concluded that the odds of the shaker throwing up an impossible combination are so small that it isn't worth worrying about. At this point you wonder whether someone has actually worked it out? made some assumptions? tested it out? What ever the outcome you realise that probability has played an important part in the design of this game as it has with so many others. Quickly thinking that I might blog about this, I went to take a picture of the impossible combination, only to notice that my daughter has already shaken again! I said, "do it again until you get another 5 yellows" She dilligently sets out on this mission and quickly realises that it doesn't happen very often. She then asks "can it be 5 of any colour?" and you see a lovely example of how intuitive some probability can be! So now we are left with the question 'What is the probability that the shaker will show an impossible combination?'

Happy summing!

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*** For teaching, I am thinking that whilst the theortical probability might get out of hand here, it could be a good example for some experimental probability. How about keeping records of how many times 2 of the same colour turn up and so on....