Let our dice select numbers on their faces with equal probability, i.e. fair dice.
Dice may have more or less than six faces. (The possibility of there being a 
3D physical shape that has that many "faces" that allow them to be fair dice,
is ignored for this task - a die with 3 or 33 defined sides is defined by the 
number of faces and the numbers on each face).

Throwing dice will randomly select a face on each die with equal probability.
To show which die of dice thrown multiple times is more likely to win over the 
others:
# calculate all possible combinations of different faces from each die
# Count how many times each die wins a combination
# Each ''combination'' is equally likely so the die with more winning face combinations is statistically more likely to win against the other dice.
<br>

'''If two dice X and Y are thrown against each other then X likely to: win, lose, or break-even against Y can be shown as: <code>X > Y, X < Y, or X = Y</code> respectively.
'''
;Example 1:
If X is the three sided die with 1, 3, 6 on its faces and Y has 2, 3, 4 on its
faces then the equal possibility outcomes from throwing both, and the winners 
is:
    
    X   Y   Winner
    =   =   ======
    1   2   Y
    1   3   Y
    1   4   Y
    3   2   X
    3   3   -
    3   4   Y
    6   2   X
    6   3   X
    6   4   X
    
    TOTAL WINS: X=4, Y=4

Both die will have the same statistical probability of winning, i.e.their comparison can be written as <code>X = Y</code>

;Transitivity:
In mathematics transitivity are rules like: 
    if a op b and b op c then a op c
If, for example, the op, (for operator), is the familiar less than, <, and it's applied to integers
we get the familiar <code>if a < b and b < c then a < c</code>

;Non-transitive dice
These are an ordered list of dice where the '>' operation between successive 
dice pairs applies but a comparison between the first and last of the list
yields the opposite result, '<'.<br>
''(Similarly '<' successive list comparisons with a final '>' between first and last is also non-transitive).''
<br><br>

Three dice S, T, U with appropriate face values could satisfy

    S < T, T < U and yet S > U

To be non-transitive.

;Notes:

* The order of numbers on the faces of a die is not relevant. For example, three faced die described with face numbers of 1, 2, 3 or 2, 1, 3 or any other permutation are equivalent. For the purposes of the task '''show only the permutation in lowest-first sorted order i.e. 1, 2, 3''' (and remove any of its perms).
* A die can have more than one instance of the same number on its faces, e.g. <code>2, 3, 3, 4</code>
* '''Rotations''': Any rotation of non-transitive dice from an answer is also an answer. You may ''optionally'' compute and show only one of each such rotation sets, ideally the first when sorted in a natural way. If this option is used then prominently state in the output that rotations of results are also solutions.
<br><br>

;Task:
;====
Find all the ordered lists of ''three'' non-transitive dice S, T, U of the form 
S < T, T < U and yet S > U; where the dice are selected from all ''four-faced die''
, (unique w.r.t the notes), possible by having selections from the integers 
''one to four'' on any dies face.

Solution can be found by generating all possble individual die then testing all
possible permutations, (permutations are ordered), of three dice for 
non-transitivity.

;Optional stretch goal:
Find lists of '''four''' non-transitive dice selected from the same possible dice from the non-stretch goal.
<br><br>

Show the results here, on this page.
<br>
;References:
* [https://youtu.be/zzKGnuvX6IQ The Most Powerful Dice] - Numberphile Video.
* [https://en.wikipedia.org/wiki/Nontransitive_dice Nontransitive dice] - Wikipedia.
<br> 


