Problem Description

We are given a set of 4 special dice. They are six sided dice but have between 1 and 7 dots per side and the number per side may be repeated.  The dice are "fair", i.e .,each side is equally likely to be the top face when the die is thrown.

In this particular set, one of the dice has 1,1,5,5,5,5 dots on its sides. The whole 
set has a peculiar property. If we label them appropriately as A, B, C, and D, then roll 
them in pairs and compare the number of dots on the top face, we get the following 
results:

Can you find the configuration of the other three dice?.

Once a set is found, you can define a game where player 2 (you) has a definite advantage, Just let player 1 select which die to roll and you can always choose one which will, on average,  beat it.

Background & Techniques

Here's a program which solves puzzles illustrating the non-transitive nature of probability.  The best description I've found is at this Math Association page 

Transitivity is an important property of the "size" relationship that allows us to solve 
algebra and logic puzzles. Transitivity for relationship "R" means that if A "R" B is true and B "R" C is true then A "R" C is true. For example: If A=B and B=C then A=C; If A>B and B>C then A>C.

It does not hold true however for the more complex "defeats" relationship. For example, in the "Rock, Scissors, Paper" game: "Rock defeats Scissors" and "Scissors defeats Paper" does not imply that  "Paper defeats Rock".

The default parameters will solve the puzzle as stated above page, but you can experiment with other configurations.

Run times depend on the number of sides per die, the maximum dots per side and the the maximum number of dice allowed in a set that forms the non-transitive loop, where the last die in the set defeats the first die.   The minimum probability box sets  minimum chance of winning if played as a game where you let your opponent choose his die first and then you choose the die preceding his choice in the set. The  begin search from box can be used to specify a starting die configuration in case a prior partial search was interrupted.

Finally, since dice are difficult to physically produce, you can click on any solution line to view and/or print a set of cards representing a set of dice.  This has the advantage that we can consider "dice" with 3, 5, 7, 9, sides which would impossible to construct in the real world as fair dice.  Fronts of the cards represent the dots on a die side and unique card back designs distinguish the dice.

 
Non-programmers are welcome to read on, but may want to skip to the bottom of this page to download executable version of the program.

We use our TGraphSearch control to define the nodes of a graph - nodes are all possible arrangements of dots on  die sides.  For each of these we define edges as  those dice which this node beats with at least the minimum probability specified.  Now it is simply a matter of performing  a depth first search looking for a "path", ordered set of dice, with the property that the last die defeats the first with probability greater than or equal to the specified minimum.  

Searches can be long, so the Solve button label is changed to a "Stop" button while the search is running.   My usual technique to interrupt any program with long running loops is to use the stop button click to set a Tag property to a non-zero value.  The loop initializes Tag to zero and then checks the value within the loop  and stops on a  nonzero value.  It's important  to call Application. ProcessMessages occasionally within the loop to allow a Stop button click to do it's thing.  

A separate Tabsheet to display and/or print a playing card version is made active when the user clicks on any solution.  The search is also stopped if it is still running when the click is made.  The playing cards simulate dice results by using one card for each die face and assigning a unique random card back design for each die.  In the current version, cards are laid out assuming landscape page orientation.    

I've added units UTGraphSearch  and UCardComponentV2 to the DFF Library zip file as part of our ongoing move to keeping commonly used items in a single location.    

Running/Exploring the Program 

Suggestions for Further Explorations