Non-transitive Dice
Why do this problem?
This problem offers a good opportunity to introduce or practise
using sample space diagrams or tree diagrams. Transitivity is
such a common phenomenon that most students take it for granted so
they may be surprised and intrigued by the existence of
non-transitive dice.
Possible approach
Before the lesson, create a set of the three dice from the
problem.
"We're going to play a game. I need a volunteer to choose one
of these three dice. Then I'll choose one too and we'll roll them
together - the winner is the person whose die shows the bigger
number." Share with the class the numbers that are on each
die.
When the volunteer has chosen one of the dice, choose the
appropriate die from the two remaining. (Red beats blue, blue beats
green and green beats red.)
Roll sufficiently many times for students to doubt whether this is
a fair game!
"I seem to be lucky today, or perhaps my die is stronger than
yours!"
"We're going to play the game
again at the end of the lesson. I want you to explore whether you
think the game is fair or not, and to work out a strategy for
choosing dice that will give you the best chance of
winning."
If it's not possible to make a
set of dice before the lesson, you could introduce the task by
asking students to work out how the game could be used to raise
money at a school fundraiser, and what strategy the stallholders
should use to fleece the public!
As students work on the task in small groups, circulate and observe
the methods they are using. For those who have difficulty getting
started you could use prompts such as:
"What are the possible outcomes when red plays green? What about
when red plays blue? What about when green plays blue?"
"How could you organise the information systematically?"
"Are there any diagrams you could draw that might
help?"
"How could you work out the probability of each of these
outcomes?"
After a while, pause the class to share methods of approaching the
task, and if appropriate introduce sample space diagrams or tree
diagrams as a good organising structure for grouping the different
possibilities. Then give them time to use the different methods to
complete the task.
For those who finish early, challenge them to find other sets of
non-transitive dice. This spreadsheet could
be used to explore or to check.
Finally, bring the class together and invite them to challenge
you to another game, using their strategy, and ask them to explain
how they arrived at their conclusions.
Possible extension
Dicey
Dice invites students to create different sets
of non-transitive dice, using the numbers from 1-6. Again, the
spreadsheet
could be used to consider different possibilities.
A Dicey
Paradox offers a set of four non-transitive dice for students
to compare. Alternatively, students could be invited to devise
their own set of four non-transitive dice.
Possible support
Tricky Track is
a simpler context that can be used to introduce sample space
diagrams or tree diagrams.