Grime Dice TN
This activity is based around a set of transitive dice which have a rather surprising set of properties. They are NOT the classic 6-faced dice with numbers one through six on each face - they are far more interesting! Played against each other, rather like paper, rock scissors, there is no dominant dice (red dice beats blue dice, blue beats green yet green beats red). Students find them fascinating to play with (you will not reveal this fact beforehand) and this provides the motivation to analyse the probabilities behind them. You will easily be able to transform a set of normal dice into transitive dice or you may wish to make use of a digital simulation provided. Students are free to solve the problems with whatever probability techniques they have built up, although knowledge of possibility spaces and/or tree diagrams would certainly be helpful. Full solutions for the teacher as well as instructions how best to exploit the dice provided.
The following is some practical advice about how the activity might be run.
A dice simulation is provided for those who may not wish to use the physical dice. Once the game has been played enough times for the students to work out the transitive nature of the dice it is time to analyse the theoretical probabilities. Two worksheets are provided to work through the probabilities of the games: one open-ended and one more structured. Teachers will need to decide which is most appropriate for the students in their class (there is no reason why different students in the same class could be using different worksheets).
I really recommend getting a few sets of the dice which can be purchased from mathsgear . However, it is perfectly simple to transform some ordinary 6-faced dice into these transitive dice by sticking labels with the correct number of dots drawn on each face.
This really depends on the class and how much time the teacher is willing to let the students play with the dice. It is really important that the students realise that there is no one best dice. They should play enough times to realise that RED beats BLUE, BLUE beats GREEN and GREEN beats RED. In my experience this can take 20 minutes. The probability analysis will take 40 minutes or more to complete depending on the ability of the students.
Starting and Finishing
Playing the game is with the students is essential for providing the hook. I get the students to play in small teams for a short while then invite them to challenge the teacher. The teacher lets the student pick a dice then he/she should be able to pick a stronger dice. Obviously, in a small number of trials 'surprise outcomes' are more likely; however, the teacher should win regularly in a best of 10 event.
If the students think they have worked out the best strategy and want the teacher to choose first - no problem! The teacher can choose first but require that the game be played with two rolls of each dice and the sum of the two dice being the score. The order of the dice reverses meaning BLUE beats RED, etc.
To get a really good idea how to exploit these dice watch James Grime the inventor of Grime Dice in action explaining how he would use them to win best!
- P(RED beats BLUE) = 21/36 (i.e. more than 50%)
- P(BLUE beats GREEN) = 21/36
- P(GREEN beats RED) = 25/36
- P(BLUE & BLUE beats RED & RED) = 85/144
- P(GREEN & GREEN beats BLUE & BLUE) = 85/144
- P(RED & RED beats GREEN & GREEN) = 671/1296 (this is a slim victory)
A bright student went away and worked out the missing two dice: yellow and magenta. A difficult challenge but possible!
Yellow: 3 3 3 3 8 8
Magenta: 1 1 6 6 6 6