86 lines
3.7 KiB
Text
86 lines
3.7 KiB
Text
Let our dice select numbers on their faces with equal probability, i.e. fair dice.
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Dice may have more or less than six faces. (The possibility of there being a
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3D physical shape that has that many "faces" that allow them to be fair dice,
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is ignored for this task - a die with 3 or 33 defined sides is defined by the
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number of faces and the numbers on each face).
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Throwing dice will randomly select a face on each die with equal probability.
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To show which die of dice thrown multiple times is more likely to win over the
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others:
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# calculate all possible combinations of different faces from each die
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# Count how many times each die wins a combination
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# Each ''combination'' is equally likely so the die with more winning face combinations is statistically more likely to win against the other dice.
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<br>
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'''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.
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'''
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;Example 1:
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If X is the three sided die with 1, 3, 6 on its faces and Y has 2, 3, 4 on its
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faces then the equal possibility outcomes from throwing both, and the winners
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is:
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X Y Winner
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= = ======
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1 2 Y
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1 3 Y
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1 4 Y
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3 2 X
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3 3 -
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3 4 Y
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6 2 X
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6 3 X
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6 4 X
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TOTAL WINS: X=4, Y=4
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Both die will have the same statistical probability of winning, i.e.their comparison can be written as <code>X = Y</code>
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;Transitivity:
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In mathematics transitivity are rules like:
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if a op b and b op c then a op c
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If, for example, the op, (for operator), is the familiar less than, <, and it's applied to integers
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we get the familiar <code>if a < b and b < c then a < c</code>
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;Non-transitive dice
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These are an ordered list of dice where the '>' operation between successive
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dice pairs applies but a comparison between the first and last of the list
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yields the opposite result, '<'.<br>
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''(Similarly '<' successive list comparisons with a final '>' between first and last is also non-transitive).''
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<br><br>
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Three dice S, T, U with appropriate face values could satisfy
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S < T, T < U and yet S > U
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To be non-transitive.
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;Notes:
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* 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).
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* A die can have more than one instance of the same number on its faces, e.g. <code>2, 3, 3, 4</code>
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* '''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.
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<br><br>
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;Task:
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;====
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Find all the ordered lists of ''three'' non-transitive dice S, T, U of the form
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S < T, T < U and yet S > U; where the dice are selected from all ''four-faced die''
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, (unique w.r.t the notes), possible by having selections from the integers
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''one to four'' on any dies face.
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Solution can be found by generating all possble individual die then testing all
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possible permutations, (permutations are ordered), of three dice for
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non-transitivity.
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;Optional stretch goal:
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Find lists of '''four''' non-transitive dice selected from the same possible dice from the non-stretch goal.
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<br><br>
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Show the results here, on this page.
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<br>
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;References:
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* [https://youtu.be/zzKGnuvX6IQ The Most Powerful Dice] - Numberphile Video.
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* [https://en.wikipedia.org/wiki/Nontransitive_dice Nontransitive dice] - Wikipedia.
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<br>
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