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A Valuable Math Con for Fun and Profit

August 5th, 2012 by Steven Pomeroy | Filed under Math Tricks.




Rock dulls scissors.

 

Scissors cut paper.

 

Paper covers rock.

 

Rock dulls scissors.

 

And so on, and so on . . .

 

Most everybody (if not all) I know has played Rock-Paper-Scissors.  It is an interesting little competition, but have you ever given it any thought beyond its use as a simple game to be played between two people?  It does not seem to follow transitive logic.  For example:

 

If A is over B, and B is over C, then A is over C

 

is an example of a transitive relation.  In the case of the game Rock-Paper-Scissors, there is no transitive relationship; it is intransitive.  To make the difference between transitive and intransitive clear, I’ll give you another example of each.  Jack is taller than Bill, and Bill is taller than Peter.  Therefore, Jack must be taller than Peter.  This physical relationship between Jack, Bill, and Peter is transitive.  However, the relationship Jack is a friend of Bill, and Bill is a friend of Peter does not necessarily mean that Jack and Peter are friends (although they very well may be); the relationship is intransitive.

 

So I can hear you thinking, “That’s all very interesting, but so what?”  So what?  So what if I tell you that you can use this property to make some cash off of some unsuspecting friends – or a LOT of money off of some not so friendly!  Well, never mind what I just said there – I would not want you to engage in anything illicit like gambling.  I will tell you, however, that the relationship “is more likely than” is intransitive, and you can use this to your advantage.  In essence, it will be like knowing in advance which item your opponent will be using during a game of Rock-Paper-Scissors.

 

A method that you can use to cheat your friends is intransitive dice.  It is a set of three color-coded dice with non-standard numbers on the faces.  Each die has three numbers, repeated twice:

 

 

intransitive dice

 Intransitive dice (opposite sides have the same value as those shown)

 

 

Like Rock-Paper-Scissors, one die has an advantage over another.  In this case, the red die has an advantage over the green die, the green die has an advantage over the purple die, and the purple die has an advantage over the red die.

 

The game you play with these dice is simple.  Each player rolls a die, and the player with the higher number wins the round, and is awarded one point.  Players roll 20 times each, and the player with the most points after 20 rounds wins the game.  To play, have your opponent choose a die to roll.  Now comes the trick – if your opponent chooses green, you choose red.  If he (or she) chooses purple, you choose green.  And if your opponent goes for red, you go for purple.  Each of these scenarios gives you the advantage, namely, a 5/9 (or a 55.55% chance) of winning!

 

To see how the probabilities are calculated, realize that there are nine possible outcomes during each round of play – player one gets 1 of 3 possible numbers and player two counters with 1 of 3 possible numbers, giving  a total of 9 possible outcomes (I’ll rule out the possibility of dice landing on their edges).  Now look at the numbers on the red and green dice.  The possible ways for red to win would be:

 

9 over 8

9 over 6

9 over 1

4 over 1

2 over 1

 

Thus, red has 5 possible ways of winning out of 9 possible outcomes.  Look at the green vs. purple battle.  The possible ways for green to win are:

 

8 over 7

8 over 5

8 over 3

6 over 5

6 over 3

 

Again, 5 out of 9 ways to win.  Lastly, look at the ways purple can win over red:

 

7 over 4

7 over 2

5 over 4

5 over 2

3 over 2

 

giving the purple a greater than 55% chance of winning.

 

Pretty cool stuff, eh?  I don’t know if there are any games in casinos that utilize intransitive dice.  If you want intransitive dice, there are some available through the web.  Also, Amazon.com has blank dice available that you can put your own numbers on.  Whatever you do, have fun with it, but please don’t fleece anybody too badly!

 

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