Non-Transitive Dice

7541 /www/nrich/html/content/id/7541/ 1 2011-06-10T16:37:49 <mdoxml version="1.0"> Here are three dice that are used to play a game for two players: <div style="float: left;"><mdo:image width="200" height="220" src="dice.jpg" alt="three dice"></mdo:image></div> The red die has the numbers {1, 1, 6, 6, 8, 8} The green die has the numbers {2, 2, 4, 4, 9, 9} The blue die has the numbers {3, 3, 5, 5, 7, 7} Each player chooses a different die. They roll their dice. The winner is the person whose die shows the bigger number. Alison and Charlie are playing the game. Charlie wants to go first so Alison lets him. Was that such a good idea? Can you advise Alison on which die to choose once she knows which die Charlie has selected? <div class="framework">Notes and background Dice of this sort are known as non-transitive dice. You can read more about transitivity in <a href="http://nrich.maths.org/1345&part=">this article</a> or have a go at creating your own in <a href="http://nrich.maths.org/6953&part=">Dicey Dice</a>. This <a href="/content/id/7541/Non%20Transitive.xls">spreadsheet</a> might be useful if you want to create your own. After you've had a go at the problem, you may be interested to read more about dice in the Plus articles <a href="http://plus.maths.org/content/non-transitiv-dice">Curious dice</a> and <a href="http://plus.maths.org/content/let-em-roll">Let 'em roll</a>. You may be familiar with another non-transitive game known as 'rock, paper, scissors'.</div> </mdoxml> <mdoxml version="1.0"> Victor, from St Paul's school in Brazil advised Alison to choose dice in the following way: If Charlie has selected the red die, Alison is better off with the green die because it is better than the red in two numbers, if Charlie has selected the Green Alison is better choosing the Blue because it is better than the green in two ways. If Charlie chooses the blue Alison chooses the Red because it is better than the blue in two ways. Emmanuel from Loreto College and Chris, Natalie and Gino from St Andrews School in Thailand came to the same conclusion. George from Canberra Grammar School explained his answer using some probabilities: It was a good idea to allow Charlie to go first. This allows Alison to make a choice for her die that increases her chances of winning as follows: 1. Charlie will probably rush to pick the green die to have a chance at rolling 9. He only has a $\frac{1}{3}$ chance of rolling 9. Alison should pick the blue die as in the case Charlie does not roll a 9, Alison has at least a $\frac{2}{3}$ chance of rolling a higher number. 2. If Charlie picks the red die, Alison should pick the green die. Alison has a $\frac{1}{3}$ chance of winning the game if she rolls a 9. If Charlie rolls a 1, Alison is certain to win. 3. Lastly, if Charlie picks the blue die, Alison should pick the red die. If Charlie rolls a 7, Alison has a $\frac{1}{3}$ chance of winning. If Charlie rolls any other number, Alison has at least a $\frac{2}{3}$ chance of winning. Josh from Wilson's School listed all the possibilities: Red vs Green: 1-2, 1-2, 1-4, 1-4, 1-9, 1-9 (x2), 6-2, 6-2, 6-4, 6-4, 6-9, 6-9 (x2), 8-2, 8-2, 8-4, 8-4, 8-9, 8-9 (x2) Green wins 20 times, Red wins 16 times. Red vs Blue: 1-3, 1-3, 1-5, 1-5, 1-7, 1-7 (x2), 6-3, 6-3, 6-5, 6-5, 6-7, 6-7 (x2), 8-3, 8-3, 8-5, 8-5, 8-7, 8-7 (x2) Red wins 20 times, Blue wins 16 times. Blue vs Green: 3-2, 3-2, 3-4, 3-4, 3-9, 3-9 (x2) 5-2, 5-2, 5-4, 5-4, 5-9, 5-9 (x2), 7-2, 7-2, 7-4, 7-4, 7-9, 7-9 (x2) Blue wins 20 times, Green wins 16 times. Komal from Alexandra School in India explained: It is not a good idea to go first because whichever die Charlie selects there is another die which has a higher probability of winning against him. Alison can select her die using the following logic - 1. If Charlie selects the red die ${1,1,6,6,8,8}$ then Alison should select the green die ${2,2,4,4,9,9}$: The probability of Charlie rolling a 1 is 2 from 6 and the probability of Alison rolling a greater number is 6 from 6. The probability of Charlie rolling a 6 is 2 from 6 and the probability of Alison rolling a greater number is 2 from 6 The probability of Charlie rolling a 8 is 2 from 6 and the probability of Alison rolling a greater number is 2 from 6 Therefore the total number of ways Alison can win is (6 + 2 + 2) = 10 out of 18. The blue die is ${3,3,5,5,7,7}$. The probability of Charlie rolling a 1 is 2 from 6 and the probability of Alison rolling a greater number is 6 from 6. The probability of Charlie rolling a 6 is 2 from 6 and the probability of Alison rolling a greater number is 2 from 6. The probability of Charlie rolling a 8 is 2 from 6 and the probability of Alison rolling a greater number is 0 from 6 Therefore the total probability of Alison winning is (6 + 2 + 0) = 8 out of 18. There are similar arguments to show that Alison should choose the blue die if Charlie chooses the green die. </mdoxml> <mdoxml version="1.0"> <h3>Why do this problem?</h3> 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. <h3>Possible approach</h3> <div>Before the lesson, create a set of the three dice from the problem.</div> <div> </div> <div>"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.</div> 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 <a href="/content/id/7541/Non%20Transitive.xls">spreadsheet</a> 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. <h3>Possible extension</h3> <div><a href="http://nrich.maths.org/6953&part=">Dicey Dice</a> invites students to create different sets of non-transitive dice, using the numbers from 1-6. Again, the <a href="/content/id/7541/Non%20Transitive.xls">spreadsheet</a> could be used to consider different possibilities.</div> <div> </div> <div><a href="http://nrich.maths.org/623&part=">A Dicey Paradox</a> 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.</div> <h3>Possible support</h3> <a href="http://nrich.maths.org/2150&part=">Tricky Track</a> is a simpler context that can be used to introduce sample space diagrams or tree diagrams. </mdoxml> <mdoxml version="1.0">If Charlie picks the green die, what should Alison do? If Charlie picks the red die, what should Alison do? If Charlie picks the blue die, what should Alison do? </mdoxml> 2 4 0 0 1 1 0 Non-Transitive Dice Alison and Charlie are playing a game. Charlie wants to go first so Alison lets him. Was that such a good idea? Probability Probability Probability Tree diagrams Probability Conditional probability Mathematics Tools Dice