Non-transitive dice: Difference between revisions

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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.

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: X > Y, X < Y, or X = Y 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 X = Y

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 if a < b and b < c then a < c

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, '<'.
(Similarly '<' successive list comparisons with a final '>' between first and last is also non-transitive).

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. 2, 3, 3, 4
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.

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.

Show the results here, on this page.

References

The Most Powerful Dice - Numberphile Video.
Nontransitive dice - Wikipedia.

F#

The task (4 sided die)

<lang fsharp> // Non-transitive dice. Nigel Galloway: August 9th., 2020 let die=[for n0 in [1..4] do for n1 in [n0..4] do for n2 in [n1..4] do for n3 in [n2..4]->[n0;n1;n2;n3]] let N=seq{for n in die->(n,[for g in die do if (seq{for n in n do for g in g->compare n g}|>Seq.sum<0) then yield g])}|>Map.ofSeq let n3=seq{for d1 in die do for d2 in N.[d1] do for d3 in N.[d2] do if List.contains d1 N.[d3] then yield (d1,d2,d3)} let n4=seq{for d1 in die do for d2 in N.[d1] do for d3 in N.[d2] do for d4 in N.[d3] do if List.contains d1 N.[d4] then yield (d1,d2,d3,d4)} n3|>Seq.iter(fun(d1,d2,d3)->printfn "%A<%A; %A<%A; %A>%A\n" d1 d2 d2 d3 d1 d3) n4|>Seq.iter(fun(d1,d2,d3,d4)->printfn "%A<%A; %A<%A; %A<%A; %A>%A" d1 d2 d2 d3 d3 d4 d1 d4)</lang>

Output:

[1; 1; 4; 4]<[2; 2; 2; 4]; [2; 2; 2; 4]<[1; 3; 3; 3]; [1; 1; 4; 4]>[1; 3; 3; 3]
[1; 3; 3; 3]<[1; 1; 4; 4]; [1; 1; 4; 4]<[2; 2; 2; 4]; [1; 3; 3; 3]>[2; 2; 2; 4]
[2; 2; 2; 4]<[1; 3; 3; 3]; [1; 3; 3; 3]<[1; 1; 4; 4]; [2; 2; 2; 4]>[1; 1; 4; 4]
Real: 00:00:00.039

[1; 1; 4; 4]<[2; 2; 2; 4]; [2; 2; 2; 4]<[2; 2; 3; 3]; [2; 2; 3; 3]<[1; 3; 3; 3]; [1; 1; 4; 4]>[1; 3; 3; 3]
[1; 3; 3; 3]<[1; 1; 4; 4]; [1; 1; 4; 4]<[2; 2; 2; 4]; [2; 2; 2; 4]<[2; 2; 3; 3]; [1; 3; 3; 3]>[2; 2; 3; 3]
[2; 2; 2; 4]<[2; 2; 3; 3]; [2; 2; 3; 3]<[1; 3; 3; 3]; [1; 3; 3; 3]<[1; 1; 4; 4]; [2; 2; 2; 4]>[1; 1; 4; 4]
[2; 2; 3; 3]<[1; 3; 3; 3]; [1; 3; 3; 3]<[1; 1; 4; 4]; [1; 1; 4; 4]<[2; 2; 2; 4]; [2; 2; 3; 3]>[2; 2; 2; 4]
Real: 00:00:00.152

Extra credit (6 sided die)

Sides numbered 1..6

<lang fsharp> // Non-transitive diceNigel Galloway: August 9th., 2020 let die=[for n0 in [1..6] do for n1 in [n0..6] do for n2 in [n1..6] do for n3 in [n2..6] do for n4 in [n3..6] do for n5 in [n4..6]->[n0;n1;n2;n3;n4;n5]] let N=seq{for n in die->(n,[for g in die do if (seq{for n in n do for g in g->compare n g}|>Seq.sum<0) then yield g])}|>Map.ofSeq let n3=[for d1 in die do for d2 in N.[d1] do for d3 in N.[d2] do if List.contains d1 N.[d3] then yield (d1,d2,d3)] let d1,d2,d3=List.last n3 in printfn "Solutions found = %d\nLast solution found is %A<%A; %A<%A; %A>%A" (n3.Length) d1 d2 d2 d3 d1 d3 </lang>

Output:

Solutions found = 121998
Last solution found is [5; 5; 5; 5; 5; 6]<[3; 3; 5; 6; 6; 6]; [3; 3; 5; 6; 6; 6]<[4; 4; 4; 6; 6; 6]; [5; 5; 5; 5; 5; 6]>[4; 4; 4; 6; 6; 6]
Real: 00:05:32.984

Sides numbered 1..7

<lang fsharp> // Non-transitive diceNigel Galloway: August 9th., 2020 let die=[for n0 in [1..7] do for n1 in [n0..7] do for n2 in [n1..7] do for n3 in [n2..7] do for n4 in [n3..7] do for n5 in [n4..7]->[n0;n1;n2;n3;n4;n5]] let N=seq{for n in die->(n,[for g in die do if (seq{for n in n do for g in g->compare n g}|>Seq.sum<0) then yield g])}|>Map.ofSeq let n3=[for d1 in die do for d2 in N.[d1] do for d3 in N.[d2] do if List.contains d1 N.[d3] then yield (d1,d2,d3)] let d1,d2,d3=List.last n3 in printfn "Solutions found = %d\nLast solution found is %A<%A; %A<%A; %A>%A" (n3.Length) d1 d2 d2 d3 d1 d3 </lang>

Output:

Solutions found = 1269189
Last solution found is [6; 6; 6; 6; 6; 7]<[4; 4; 6; 7; 7; 7]; [4; 4; 6; 7; 7; 7]<[5; 5; 5; 7; 7; 7]; [6; 6; 6; 6; 6; 7]>[5; 5; 5; 7; 7; 7]
Real: 01:21:36.912

Factor

<lang factor>USING: grouping io kernel math math.combinatorics math.ranges prettyprint sequences ;

possible-dice ( n -- seq )

   [ [1,b] ] [ selections ] bi [ [ <= ] monotonic? ] filter ;

cmp ( seq seq -- n ) [ - sgn ] cartesian-map concat sum ;

non-transitive? ( seq -- ? )

  [ 2 clump [ first2 cmp neg? ] all? ]
  [ [ last ] [ first ] bi cmp neg? and ] bi ;

find-non-transitive ( #sides #dice -- seq )

   [ possible-dice ] [ <k-permutations> ] bi*
   [ non-transitive? ] filter ;

! Task "Number of eligible 4-sided dice: " write 4 possible-dice length . nl

"All ordered lists of 3 non-transitive dice with 4 sides:" print 4 3 find-non-transitive . nl

"All ordered lists of 4 non-transitive dice with 4 sides:" print 4 4 find-non-transitive .</lang>

Output:

Number of eligible 4-sided dice: 35

All ordered lists of 3 non-transitive dice with 4 sides:
V{
    { { 1 1 4 4 } { 2 2 2 4 } { 1 3 3 3 } }
    { { 1 3 3 3 } { 1 1 4 4 } { 2 2 2 4 } }
    { { 2 2 2 4 } { 1 3 3 3 } { 1 1 4 4 } }
}

All ordered lists of 4 non-transitive dice with 4 sides:
V{
    { { 1 1 4 4 } { 2 2 2 4 } { 2 2 3 3 } { 1 3 3 3 } }
    { { 1 3 3 3 } { 1 1 4 4 } { 2 2 2 4 } { 2 2 3 3 } }
    { { 2 2 2 4 } { 2 2 3 3 } { 1 3 3 3 } { 1 1 4 4 } }
    { { 2 2 3 3 } { 1 3 3 3 } { 1 1 4 4 } { 2 2 2 4 } }
}

Go

Translation of: Wren

<lang go>package main

import (

   "fmt"
   "sort"

)

func fourFaceCombs() (res [][4]int) {

   found := make([]bool, 256)
   for i := 1; i <= 4; i++ {
       for j := 1; j <= 4; j++ {
           for k := 1; k <= 4; k++ {
               for l := 1; l <= 4; l++ {
                   c := [4]int{i, j, k, l}
                   sort.Ints(c[:])
                   key := 64*(c[0]-1) + 16*(c[1]-1) + 4*(c[2]-1) + (c[3] - 1)
                   if !found[key] {
                       found[key] = true
                       res = append(res, c)
                   }
               }
           }
       }
   }
   return

}

func cmp(x, y [4]int) int {

   xw := 0
   yw := 0
   for i := 0; i < 4; i++ {
       for j := 0; j < 4; j++ {
           if x[i] > y[j] {
               xw++
           } else if y[j] > x[i] {
               yw++
           }
       }
   }
   if xw < yw {
       return -1
   } else if xw > yw {
       return 1
   }
   return 0

}

func findIntransitive3(cs [][4]int) (res [][3][4]int) {

   var c = len(cs)
   for i := 0; i < c; i++ {
       for j := 0; j < c; j++ {
           for k := 0; k < c; k++ {
               first := cmp(cs[i], cs[j])
               if first == -1 {
                   second := cmp(cs[j], cs[k])
                   if second == -1 {
                       third := cmp(cs[i], cs[k])
                       if third == 1 {
                           res = append(res, [3][4]int{cs[i], cs[j], cs[k]})
                       }
                   }
               }
           }
       }
   }
   return

}

func findIntransitive4(cs [][4]int) (res [][4][4]int) {

   c := len(cs)
   for i := 0; i < c; i++ {
       for j := 0; j < c; j++ {
           for k := 0; k < c; k++ {
               for l := 0; l < c; l++ {
                   first := cmp(cs[i], cs[j])
                   if first == -1 {
                       second := cmp(cs[j], cs[k])
                       if second == -1 {
                           third := cmp(cs[k], cs[l])
                           if third == -1 {
                               fourth := cmp(cs[i], cs[l])
                               if fourth == 1 {
                                   res = append(res, [4][4]int{cs[i], cs[j], cs[k], cs[l]})
                               }
                           }
                       }
                   }
               }
           }
       }
   }
   return

}

func main() {

   combs := fourFaceCombs()
   fmt.Println("Number of eligible 4-faced dice", len(combs))
   it3 := findIntransitive3(combs)
   fmt.Printf("\n%d ordered lists of 3 non-transitive dice found, namely:\n", len(it3))
   for _, a := range it3 {
       fmt.Println(a)
   }
   it4 := findIntransitive4(combs)
   fmt.Printf("\n%d ordered lists of 4 non-transitive dice found, namely:\n", len(it4))
   for _, a := range it4 {
       fmt.Println(a)
   }

}</lang>

Output:

Number of eligible 4-faced dice 35

3 ordered lists of 3 non-transitive dice found, namely:
[[1 1 4 4] [2 2 2 4] [1 3 3 3]]
[[1 3 3 3] [1 1 4 4] [2 2 2 4]]
[[2 2 2 4] [1 3 3 3] [1 1 4 4]]

4 ordered lists of 4 non-transitive dice found, namely:
[[1 1 4 4] [2 2 2 4] [2 2 3 3] [1 3 3 3]]
[[1 3 3 3] [1 1 4 4] [2 2 2 4] [2 2 3 3]]
[[2 2 2 4] [2 2 3 3] [1 3 3 3] [1 1 4 4]]
[[2 2 3 3] [1 3 3 3] [1 1 4 4] [2 2 2 4]]

Haskell

<lang haskell>{-# language LambdaCase #-} import Data.List import Control.Monad

newtype Dice = Dice [Int]

instance Show Dice where

 show (Dice s) = "(" ++ unwords (show <$> s) ++ ")"

instance Eq Dice where

 d1 == d2 = d1 `compare` d2 == EQ

instance Ord Dice where

 Dice d1 `compare` Dice d2 = (add $ compare <$> d1 <*> d2) `compare` 0
   where
     add = sum . map (\case {LT -> -1; EQ -> 0; GT -> 1})

dices n = Dice <$> (nub $ sort <$> replicateM n [1..n])

nonTrans dice = filter (\x -> last x < head x) . go

 where
   go 0 = []
   go 1 = sequence [dice]
   go n = do
     (a:as) <- go (n-1)
     b <- filter (< a) dice
     return (b:a:as)</lang>

*Main> mapM_ print $ nonTrans (dices 4) 3
[(2 2 2 4),(1 3 3 3),(1 1 4 4)]
[(1 1 4 4),(2 2 2 4),(1 3 3 3)]
[(1 3 3 3),(1 1 4 4),(2 2 2 4)]

*Main> mapM_ print $ nonTrans (dices 4) 4
[(2 2 2 4),(2 2 3 3),(1 3 3 3),(1 1 4 4)]
[(1 1 4 4),(2 2 2 4),(2 2 3 3),(1 3 3 3)]
[(2 2 3 3),(1 3 3 3),(1 1 4 4),(2 2 2 4)]
[(1 3 3 3),(1 1 4 4),(2 2 2 4),(2 2 3 3)]

*Main> length $ nonTrans (dices 4) 5
0

*Main> length $ nonTrans (dices 5) 3
2022

J

Implementation (rotations of results here are also solutions): <lang J>NB. unique list of all y faced dice udice=: Template:1+~./:~"1 (

NB. which dice are less than which other dice? NB. y here is a result of udice lthan=: {{ 0>*@(+/)@,@:*@(-/)"1/~ y}}

NB. "less than loops" length x, for y sided non-transitive dice cycles=: {{

 ud=. udice y
 lt=. lthan ud
 extend=. [:; lt&{{< y,"1 0 y-.~I.x{~{:y }}"1
 r=. ; extend^:(x-1)@,:&.> i.#ud
 ud{~ ~.((i.<./)|.])"1 r #~ lt{~({:,&.>{.)|:r

}}</lang>

Here, extend takes a list of die sequences and appends each possibility of a new die where the new die is not less than the last die in the sequence. So, r is all such lists of length x. We discard the cases where the last element of the list is not less than the first, rotate each dice index list into canonical order, discard duplicates, and then use these indices to select the associated dice.

Task examples:

<lang J> 3 cycles 4 1 1 4 4 2 2 2 4 1 3 3 3

4 cycles 41 1 4 4 2 2 2 4 2 2 3 3 1 3 3 3</lang>JavaTranslation of: Kotlin<lang java>import java.util.ArrayList; import java.util.HashSet; import java.util.List; import java.util.Set; import java.util.stream.Collectors; import java.util.stream.IntStream; public class Main {private static List<List<Integer>> fourFaceCombos() { List<List<Integer>> res = new ArrayList<>(); Set<Integer> found = new HashSet<>();for (int i = 1; i <= 4; i++) { for (int j = 1; j <= 4; j++) { for (int k = 1; k <= 4; k++) { for (int l = 1; l <= 4; l++) { List<Integer> c = IntStream.of(i, j, k, l).sorted().boxed().collect(Collectors.toList());int key = 64 * (c.get(0) - 1) + 16 * (c.get(1) - 1) + 4 * (c.get(2) - 1) + (c.get(3) - 1); if (found.add(key)) { res.add(c); } } } } }return res; }private static int cmp(List<Integer> x, List<Integer> y) { int xw = 0; int yw = 0; for (int i = 0; i < 4; i++) { for (int j = 0; j < 4; j++) { if (x.get(i) > y.get(j)) { xw++; } else if (x.get(i) < y.get(j)) { yw++; } } } return Integer.compare(xw, yw); }private static List<List<List<Integer>>> findIntransitive3(List<List<Integer>> cs) { int c = cs.size(); List<List<List<Integer>>> res = new ArrayList<>();for (int i = 0; i < c; i++) { for (int j = 0; j < c; j++) { if (cmp(cs.get(i), cs.get(j)) == -1) { for (List<Integer> kl : cs) { if (cmp(cs.get(j), kl) == -1 && cmp(kl, cs.get(i)) == -1) { res.add(List.of(cs.get(i), cs.get(j), kl)); } } } } }return res; }private static List<List<List<Integer>>> findIntransitive4(List<List<Integer>> cs) { int c = cs.size(); List<List<List<Integer>>> res = new ArrayList<>();for (int i = 0; i < c; i++) { for (int j = 0; j < c; j++) { if (cmp(cs.get(i), cs.get(j)) == -1) { for (int k = 0; k < cs.size(); k++) { if (cmp(cs.get(j), cs.get(k)) == -1) { for (List<Integer> ll : cs) { if (cmp(cs.get(k), ll) == -1 && cmp(ll, cs.get(i)) == -1) { res.add(List.of(cs.get(i), cs.get(j), cs.get(k), ll)); } } } } } } }return res; }public static void main(String[] args) { List<List<Integer>> combos = fourFaceCombos(); System.out.printf("Number of eligible 4-faced dice: %d%n", combos.size()); System.out.println();List<List<List<Integer>>> it3 = findIntransitive3(combos); System.out.printf("%d ordered lists of 3 non-transitive dice found, namely:%n", it3.size()); for (List<List<Integer>> a : it3) { System.out.println(a); } System.out.println();List<List<List<Integer>>> it4 = findIntransitive4(combos); System.out.printf("%d ordered lists of 4 non-transitive dice found, namely:%n", it4.size()); for (List<List<Integer>> a : it4) { System.out.println(a); } }}</lang>Output:Number of eligible 4-faced dice: 35 3 ordered lists of 3 non-transitive dice found, namely: [[1, 1, 4, 4], [2, 2, 2, 4], [1, 3, 3, 3]] [[1, 3, 3, 3], [1, 1, 4, 4], [2, 2, 2, 4]] [[2, 2, 2, 4], [1, 3, 3, 3], [1, 1, 4, 4]] 4 ordered lists of 4 non-transitive dice found, namely: [[1, 1, 4, 4], [2, 2, 2, 4], [2, 2, 3, 3], [1, 3, 3, 3]] [[1, 3, 3, 3], [1, 1, 4, 4], [2, 2, 2, 4], [2, 2, 3, 3]] [[2, 2, 2, 4], [2, 2, 3, 3], [1, 3, 3, 3], [1, 1, 4, 4]] [[2, 2, 3, 3], [1, 3, 3, 3], [1, 1, 4, 4], [2, 2, 2, 4]]JuliaTranslation of: Python<lang julia>import Base.>, Base.< using Memoize, Combinatorics struct Diename::String faces::Vector{Int}end """ Compares two die returning 1, -1 or 0 for operators >, < == """ function cmpd(die1, die2)# Numbers of times one die wins against the other for all combinations # cmp(x, y) is `(x > y) - (y > x)` to return 1, 0, or -1 for numbers tot = [0, 0, 0] for d in Iterators.product(die1.faces, die2.faces) tot[2 + (d[1] > d[2]) - (d[2] > d[1])] += 1 end return (tot[3] > tot[1]) - (tot[1] > tot[3])end Base.:>(d1::Die, d2::Die) = cmpd(d1, d2) == 1 Base.:<(d1::Die, d2::Die) = cmpd(d1, d2) == -1 """ True iff ordering of die in dice is non-transitive """ isnontrans(dice) = all(x -> x[1] < x[2], zip(dice[1:end-1], dice[2:end])) && dice[1] > dice[end] findnontrans(alldice, n=3) = [perm for perm in permutations(alldice, n) if isnontrans(perm)] function possible_dice(sides, mx)println("\nAll possible 1..$mx $sides-sided dice") dice = [Die("D $(n+1)", [i for i in f]) for (n, f) in enumerate(Iterators.product(fill(1:mx, sides)...))] println(" Created $(length(dice)) dice") println(" Remove duplicate with same bag of numbers on different faces") uniquedice = collect(values(Dict(sort(d.faces) => d for d in dice))) println(" Return $(length(uniquedice)) filtered dice") return uniquediceend """ Compares two die returning their relationship of their names as a string """ function verbosecmp(die1, die2)# Numbers of times one die wins against the other for all combinations win1 = sum(p[1] > p[2] for p in Iterators.product(die1.faces, die2.faces)) win2 = sum(p[2] > p[1] for p in Iterators.product(die1.faces, die2.faces)) n1, n2 = die1.name, die2.name return win1 > win2 ? "$n1 > $n2" : win1 < win2 ? "$n1 < $n2" : "$n1 = $n2"end """ Dice cmp function with verbose extra checks """ function verbosedicecmp(dice)return join([[verbosecmp(x, y) for (x, y) in zip(dice[1:end-1], dice[2:end])]; [verbosecmp(dice[1], dice[end])]], ", ")end function testnontransitivedice(faces)dice = possible_dice(faces, faces) for N in (faces < 5 ? [3, 4] : [3]) # length of non-transitive group of dice searched for nontrans = unique(map(x -> sort!(x, lt=(x, y)->x.name<y.name), findnontrans(dice, N))) println("\n Unique sorted non_transitive length-$N combinations found: $(length(nontrans))") for list in (faces < 5 ? nontrans : [nontrans[end]]) println(faces < 5 ? "" : " Only printing last example for brevity") for (i, die) in enumerate(list) println(" $die$(i < N - 1 ? "," : "]")") end end if !isempty(nontrans) println("\n More verbose comparison of last non_transitive result:") println(" ", verbosedicecmp(nontrans[end])) end println("\n ====") endend @time testnontransitivedice(3) @time testnontransitivedice(4) @time testnontransitivedice(5) @time testnontransitivedice(6)

</lang>

Output:All possible 1..3 3-sided dice Created 27 dice Remove duplicate with same bag of numbers on different faces Return 10 filtered dice Unique sorted non_transitive length-3 combinations found: 0 ==== Unique sorted non_transitive length-4 combinations found: 0 ==== 0.762201 seconds (1.72 M allocations: 89.509 MiB, 1.65% gc time) All possible 1..4 4-sided dice Created 256 dice Remove duplicate with same bag of numbers on different faces Return 35 filtered dice Unique sorted non_transitive length-3 combinations found: 1 Die("D 170", [1, 3, 3, 3]), Die("D 215", [2, 2, 2, 4])] Die("D 242", [1, 1, 4, 4])] More verbose comparison of last non_transitive result: D 170 > D 215, D 215 > D 242, D 170 < D 242 ==== Unique sorted non_transitive length-4 combinations found: 1 Die("D 167", [2, 2, 3, 3]), Die("D 170", [1, 3, 3, 3]), Die("D 215", [2, 2, 2, 4])] Die("D 242", [1, 1, 4, 4])] More verbose comparison of last non_transitive result: D 167 < D 170, D 170 > D 215, D 215 > D 242, D 167 = D 242 ==== 1.075354 seconds (8.15 M allocations: 1.075 GiB, 11.05% gc time) All possible 1..5 5-sided dice Created 3125 dice Remove duplicate with same bag of numbers on different faces Return 126 filtered dice Unique sorted non_transitive length-3 combinations found: 674 Only printing last example for brevity Die("D 2938", [2, 3, 3, 4, 5]), Die("D 3058", [2, 2, 3, 5, 5])] Die("D 3082", [1, 2, 4, 5, 5])] More verbose comparison of last non_transitive result: D 2938 > D 3058, D 3058 > D 3082, D 2938 < D 3082 ==== 2.422382 seconds (11.51 M allocations: 2.834 GiB, 18.87% gc time) All possible 1..6 6-sided dice Created 46656 dice Remove duplicate with same bag of numbers on different faces Return 462 filtered dice Unique sorted non_transitive length-3 combinations found: 40666 Only printing last example for brevity Die("D 35727", [2, 3, 3, 4, 4, 5]), Die("D 41991", [2, 3, 3, 3, 3, 6])] Die("D 46484", [1, 2, 2, 6, 6, 6])] More verbose comparison of last non_transitive result: D 35727 > D 41991, D 41991 > D 46484, D 35727 < D 46484 ==== 123.182432 seconds (553.51 M allocations: 392.013 GiB, 17.02% gc timeKotlinTranslation of: Go<lang scala>fun fourFaceCombos(): List<Array<Int>> {val res = mutableListOf<Array<Int>>() val found = mutableSetOf<Int>() for (i in 1..4) { for (j in 1..4) { for (k in 1..4) { for (l in 1..4) { val c = arrayOf(i, j, k, l) c.sort() val key = 64 * (c[0] - 1) + 16 * (c[1] - 1) + 4 * (c[2] - 1) + (c[3] - 1) if (!found.contains(key)) { found.add(key) res.add(c) } } } } } return res} fun cmp(x: Array<Int>, y: Array<Int>): Int {var xw = 0 var yw = 0 for (i in 0 until 4) { for (j in 0 until 4) { if (x[i] > y[j]) { xw++ } else if (y[j] > x[i]) { yw++ } } } if (xw < yw) { return -1 } if (xw > yw) { return 1 } return 0} fun findIntransitive3(cs: List<Array<Int>>): List<Array<Array<Int>>> {val c = cs.size val res = mutableListOf<Array<Array<Int>>>()for (i in 0 until c) { for (j in 0 until c) { if (cmp(cs[i], cs[j]) == -1) { for (k in 0 until c) { if (cmp(cs[j], cs[k]) == -1 && cmp(cs[k], cs[i]) == -1) { res.add(arrayOf(cs[i], cs[j], cs[k])) } } } } }return res} fun findIntransitive4(cs: List<Array<Int>>): List<Array<Array<Int>>> {val c = cs.size val res = mutableListOf<Array<Array<Int>>>()for (i in 0 until c) { for (j in 0 until c) { if (cmp(cs[i], cs[j]) == -1) { for (k in 0 until c) { if (cmp(cs[j], cs[k]) == -1) { for (l in 0 until c) { if (cmp(cs[k], cs[l]) == -1 && cmp(cs[l], cs[i]) == -1) { res.add(arrayOf(cs[i], cs[j], cs[k], cs[l])) } } } } } } }return res} fun main() {val combos = fourFaceCombos() println("Number of eligible 4-faced dice: ${combos.size}") println()val it3 = findIntransitive3(combos) println("${it3.size} ordered lists of 3 non-transitive dice found, namely:") for (a in it3) { println(a.joinToString(", ", "[", "]") { it.joinToString(", ", "[", "]") }) } println()val it4 = findIntransitive4(combos) println("${it4.size} ordered lists of 4 non-transitive dice found, namely:") for (a in it4) { println(a.joinToString(", ", "[", "]") { it.joinToString(", ", "[", "]") }) }}</lang>Output:Number of eligible 4-faced dice: 35 3 ordered lists of 3 non-transitive dice found, namely: [[1, 1, 4, 4], [2, 2, 2, 4], [1, 3, 3, 3]] [[1, 3, 3, 3], [1, 1, 4, 4], [2, 2, 2, 4]] [[2, 2, 2, 4], [1, 3, 3, 3], [1, 1, 4, 4]] 4 ordered lists of 4 non-transitive dice found, namely: [[1, 1, 4, 4], [2, 2, 2, 4], [2, 2, 3, 3], [1, 3, 3, 3]] [[1, 3, 3, 3], [1, 1, 4, 4], [2, 2, 2, 4], [2, 2, 3, 3]] [[2, 2, 2, 4], [2, 2, 3, 3], [1, 3, 3, 3], [1, 1, 4, 4]] [[2, 2, 3, 3], [1, 3, 3, 3], [1, 1, 4, 4], [2, 2, 2, 4]]MiniZincThe model<lang MiniZinc> %Non transitive dice. Nigel Galloway, September 14th., 2020 int: die; int: faces; set of int: values; predicate pN(array[1..faces] of var values: n, array[1..faces] of var values: g) = sum(i in 1..faces,j in 1..faces)(if n[i]<g[j] then 1 elseif n[i]>g[j] then -1 else 0 endif)>0; predicate pG(array[1..faces] of var values: n) = forall(g in 1..faces-1)(n[g]<=n[g+1]); array[1..die,1..faces] of var values: g; constraint forall(n in 1..die)(pG(g[n,1..faces])); constraint forall(n in 1..die-1)(pN(g[n,1..faces],g[n+1,1..faces])); constraint pN(g[die,1..faces],g[1,1..faces]); output[join(" ",[show(g[n,1..faces])++"<"++show(g[n+1,1..faces]) | n in 1..die-1])," "++show(g[die,1..faces])++">"++show(g[1,1..faces])]; </lang>The task(4 sided die)3 die values=1..4Compiling non_trans.mzn, additional arguments die=3; faces=4; values=1..4; Running non_trans.mzn [1, 1, 4, 4]<[2, 2, 2, 4] [2, 2, 2, 4]<[1, 3, 3, 3] [1, 3, 3, 3]>[1, 1, 4, 4] ---------- [2, 2, 2, 4]<[1, 3, 3, 3] [1, 3, 3, 3]<[1, 1, 4, 4] [1, 1, 4, 4]>[2, 2, 2, 4] ---------- [1, 3, 3, 3]<[1, 1, 4, 4] [1, 1, 4, 4]<[2, 2, 2, 4] [2, 2, 2, 4]>[1, 3, 3, 3] ---------- ========== %%%mzn-stat: nodes=669 %%%mzn-stat: failures=413 %%%mzn-stat: restarts=0 %%%mzn-stat: variables=746 %%%mzn-stat: intVars=60 %%%mzn-stat: boolVariables=684 %%%mzn-stat: propagators=205 %%%mzn-stat: propagations=35274 %%%mzn-stat: peakDepth=49 %%%mzn-stat: nogoods=413 %%%mzn-stat: backjumps=256 %%%mzn-stat: peakMem=0.00 %%%mzn-stat: time=0.011 %%%mzn-stat: initTime=0.004 %%%mzn-stat: solveTime=0.007 %%%mzn-stat: baseMem=0.00 %%%mzn-stat: trailMem=0.00 %%%mzn-stat: randomSeed=1599951884 Finished in 174msec4 die values=1..4Compiling non_trans.mzn, additional arguments die=4; faces=4; values=1..4; Running non_trans.mzn [1, 1, 4, 4]<[2, 2, 2, 4] [2, 2, 2, 4]<[2, 2, 3, 3] [2, 2, 3, 3]<[1, 3, 3, 3] [1, 3, 3, 3]>[1, 1, 4, 4] ---------- [2, 2, 2, 4]<[2, 2, 3, 3] [2, 2, 3, 3]<[1, 3, 3, 3] [1, 3, 3, 3]<[1, 1, 4, 4] [1, 1, 4, 4]>[2, 2, 2, 4] ---------- [1, 3, 3, 3]<[1, 1, 4, 4] [1, 1, 4, 4]<[2, 2, 2, 4] [2, 2, 2, 4]<[2, 2, 3, 3] [2, 2, 3, 3]>[1, 3, 3, 3] ---------- [2, 2, 3, 3]<[1, 3, 3, 3] [1, 3, 3, 3]<[1, 1, 4, 4] [1, 1, 4, 4]<[2, 2, 2, 4] [2, 2, 2, 4]>[2, 2, 3, 3] ---------- ========== %%%mzn-stat: nodes=1573 %%%mzn-stat: failures=856 %%%mzn-stat: restarts=0 %%%mzn-stat: variables=994 %%%mzn-stat: intVars=80 %%%mzn-stat: boolVariables=912 %%%mzn-stat: propagators=273 %%%mzn-stat: propagations=85224 %%%mzn-stat: peakDepth=65 %%%mzn-stat: nogoods=856 %%%mzn-stat: backjumps=717 %%%mzn-stat: peakMem=0.00 %%%mzn-stat: time=0.022 %%%mzn-stat: initTime=0.007 %%%mzn-stat: solveTime=0.015 %%%mzn-stat: baseMem=0.00 %%%mzn-stat: trailMem=0.01 %%%mzn-stat: randomSeed=1599951808 Finished in 190msecExtra Credt(6 sided die)3 die values=1..6Compiling non_trans.mzn, additional arguments die=3; faces=6; values=1..6; Running non_trans.mzn [3, 3, 4, 6, 6, 6]<[5, 5, 5, 5, 5, 6] [5, 5, 5, 5, 5, 6]<[2, 3, 5, 6, 6, 6] [2, 3, 5, 6, 6, 6]>[3, 3, 4, 6, 6, 6] ---------- [3, 3, 4, 6, 6, 6]<[5, 5, 5, 5, 5, 6] [5, 5, 5, 5, 5, 6]<[1, 3, 5, 6, 6, 6] [1, 3, 5, 6, 6, 6]>[3, 3, 4, 6, 6, 6] ---------- [2, 2, 4, 6, 6, 6]<[5, 5, 5, 5, 5, 6] [5, 5, 5, 5, 5, 6]<[1, 2, 5, 6, 6, 6] [1, 2, 5, 6, 6, 6]>[2, 2, 4, 6, 6, 6] ---------- [2, 2, 3, 6, 6, 6]<[5, 5, 5, 5, 5, 6] [5, 5, 5, 5, 5, 6]<[1, 2, 5, 6, 6, 6] [1, 2, 5, 6, 6, 6]>[2, 2, 3, 6, 6, 6] ---------- [3, 4, 4, 6, 6, 6]<[5, 5, 5, 5, 5, 6] [5, 5, 5, 5, 5, 6]<[1, 3, 5, 6, 6, 6] [1, 3, 5, 6, 6, 6]>[3, 4, 4, 6, 6, 6] ---------- [3, 4, 4, 6, 6, 6]<[5, 5, 5, 5, 5, 6] [5, 5, 5, 5, 5, 6]<[2, 3, 5, 6, 6, 6] [2, 3, 5, 6, 6, 6]>[3, 4, 4, 6, 6, 6] ---------- [3, 4, 4, 6, 6, 6]<[5, 5, 5, 5, 5, 6] [5, 5, 5, 5, 5, 6]<[3, 3, 5, 6, 6, 6] [3, 3, 5, 6, 6, 6]>[3, 4, 4, 6, 6, 6] ---------- [2, 4, 4, 6, 6, 6]<[5, 5, 5, 5, 5, 6] [5, 5, 5, 5, 5, 6]<[1, 2, 5, 6, 6, 6] [1, 2, 5, 6, 6, 6]>[2, 4, 4, 6, 6, 6] ---------- [1, 4, 4, 6, 6, 6]<[5, 5, 5, 5, 5, 6] [5, 5, 5, 5, 5, 6]<[1, 1, 5, 6, 6, 6] [1, 1, 5, 6, 6, 6]>[1, 4, 4, 6, 6, 6] ---------- [2, 4, 4, 6, 6, 6]<[5, 5, 5, 5, 5, 6] [5, 5, 5, 5, 5, 6]<[2, 2, 5, 6, 6, 6] [2, 2, 5, 6, 6, 6]>[2, 4, 4, 6, 6, 6] ---------- [2, 3, 4, 6, 6, 6]<[5, 5, 5, 5, 5, 6] [5, 5, 5, 5, 5, 6]<[1, 2, 5, 6, 6, 6] [1, 2, 5, 6, 6, 6]>[2, 3, 4, 6, 6, 6] ---------- [ 9 more solutions ] [4, 4, 4, 6, 6, 6]<[5, 5, 5, 5, 5, 6] [5, 5, 5, 5, 5, 6]<[1, 2, 5, 6, 6, 6] [1, 2, 5, 6, 6, 6]>[4, 4, 4, 6, 6, 6] ---------- [ 19 more solutions ] [2, 4, 4, 6, 6, 6]<[5, 5, 5, 5, 5, 6] [5, 5, 5, 5, 5, 6]<[1, 3, 5, 6, 6, 6] [1, 3, 5, 6, 6, 6]>[2, 4, 4, 6, 6, 6] ---------- [ 39 more solutions ] [1, 5, 5, 5, 5, 6]<[1, 2, 2, 6, 6, 6] [1, 2, 2, 6, 6, 6]<[3, 4, 5, 5, 5, 6] [3, 4, 5, 5, 5, 6]>[1, 5, 5, 5, 5, 6] ---------- [ 79 more solutions ] [1, 2, 5, 6, 6, 6]<[2, 3, 3, 6, 6, 6] [2, 3, 3, 6, 6, 6]<[5, 5, 5, 5, 5, 6] [5, 5, 5, 5, 5, 6]>[1, 2, 5, 6, 6, 6] ---------- [ 159 more solutions ] [1, 1, 5, 5, 5, 6]<[2, 3, 3, 4, 6, 6] [2, 3, 3, 4, 6, 6]<[1, 4, 5, 5, 5, 5] [1, 4, 5, 5, 5, 5]>[1, 1, 5, 5, 5, 6] ---------- [ 319 more solutions ] [2, 2, 4, 4, 6, 6]<[4, 4, 4, 4, 4, 6] [4, 4, 4, 4, 4, 6]<[1, 2, 5, 5, 5, 5] [1, 2, 5, 5, 5, 5]>[2, 2, 4, 4, 6, 6] ---------- [ 639 more solutions ] [1, 2, 2, 2, 6, 6]<[2, 3, 3, 3, 3, 5] [2, 3, 3, 3, 3, 5]<[1, 1, 5, 5, 5, 5] [1, 1, 5, 5, 5, 5]>[1, 2, 2, 2, 6, 6] ---------- [ 1279 more solutions ] [1, 1, 2, 6, 6, 6]<[2, 3, 4, 5, 5, 6] [2, 3, 4, 5, 5, 6]<[1, 4, 5, 5, 5, 5] [1, 4, 5, 5, 5, 5]>[1, 1, 2, 6, 6, 6] ---------- [ 2559 more solutions ] [1, 1, 2, 6, 6, 6]<[3, 4, 4, 5, 5, 6] [3, 4, 4, 5, 5, 6]<[2, 5, 5, 5, 5, 5] [2, 5, 5, 5, 5, 5]>[1, 1, 2, 6, 6, 6] ---------- [ 5119 more solutions ] [3, 4, 4, 4, 4, 6]<[1, 3, 5, 5, 5, 5] [1, 3, 5, 5, 5, 5]<[2, 2, 2, 6, 6, 6] [2, 2, 2, 6, 6, 6]>[3, 4, 4, 4, 4, 6] ---------- [ 10239 more solutions ] [3, 3, 3, 3, 4, 5]<[1, 3, 3, 4, 4, 5] [1, 3, 3, 4, 4, 5]<[2, 2, 2, 5, 5, 6] [2, 2, 2, 5, 5, 6]>[3, 3, 3, 3, 4, 5] ---------- [ 20479 more solutions ] [2, 2, 3, 3, 3, 6]<[1, 1, 4, 4, 4, 5] [1, 1, 4, 4, 4, 5]<[1, 1, 1, 5, 5, 5] [1, 1, 1, 5, 5, 5]>[2, 2, 3, 3, 3, 6] ---------- [ 40959 more solutions ] [1, 1, 3, 3, 4, 4]<[1, 1, 1, 4, 6, 6] [1, 1, 1, 4, 6, 6]<[1, 2, 2, 3, 3, 6] [1, 2, 2, 3, 3, 6]>[1, 1, 3, 3, 4, 4] ---------- [ 40076 more solutions ] [2, 2, 2, 4, 5, 5]<[3, 3, 3, 3, 3, 4] [3, 3, 3, 3, 3, 4]<[2, 3, 3, 3, 4, 4] [2, 3, 3, 3, 4, 4]>[2, 2, 2, 4, 5, 5] ---------- ========== %%%mzn-stat: nodes=203916 %%%mzn-stat: failures=185337 %%%mzn-stat: restarts=0 %%%mzn-stat: variables=1658 %%%mzn-stat: intVars=126 %%%mzn-stat: boolVariables=1530 %%%mzn-stat: propagators=451 %%%mzn-stat: propagations=8595895 %%%mzn-stat: peakDepth=109 %%%mzn-stat: nogoods=185337 %%%mzn-stat: backjumps=18579 %%%mzn-stat: peakMem=0.00 %%%mzn-stat: time=11.444 %%%mzn-stat: initTime=0.010 %%%mzn-stat: solveTime=11.434 %%%mzn-stat: baseMem=0.00 %%%mzn-stat: trailMem=0.01 %%%mzn-stat: randomSeed=1599951938 Finished in 12s 600msec3 die values=1..7Compiling non_trans.mzn, additional arguments die=3; faces=6; values=1..7; Running non_trans.mzn [4, 4, 6, 7, 7, 7]<[4, 5, 5, 7, 7, 7] [4, 5, 5, 7, 7, 7]<[6, 6, 6, 6, 6, 7] [6, 6, 6, 6, 6, 7]>[4, 4, 6, 7, 7, 7] ---------- [3, 4, 6, 7, 7, 7]<[4, 5, 5, 7, 7, 7] [4, 5, 5, 7, 7, 7]<[6, 6, 6, 6, 6, 7] [6, 6, 6, 6, 6, 7]>[3, 4, 6, 7, 7, 7] ---------- [3, 3, 6, 7, 7, 7]<[3, 5, 5, 7, 7, 7] [3, 5, 5, 7, 7, 7]<[6, 6, 6, 6, 6, 7] [6, 6, 6, 6, 6, 7]>[3, 3, 6, 7, 7, 7] ---------- [3, 3, 6, 7, 7, 7]<[3, 4, 5, 7, 7, 7] [3, 4, 5, 7, 7, 7]<[6, 6, 6, 6, 6, 7] [6, 6, 6, 6, 6, 7]>[3, 3, 6, 7, 7, 7] ---------- [3, 3, 6, 7, 7, 7]<[3, 4, 4, 7, 7, 7] [3, 4, 4, 7, 7, 7]<[6, 6, 6, 6, 6, 7] [6, 6, 6, 6, 6, 7]>[3, 3, 6, 7, 7, 7] ---------- [2, 3, 6, 7, 7, 7]<[3, 5, 5, 7, 7, 7] [3, 5, 5, 7, 7, 7]<[6, 6, 6, 6, 6, 7] [6, 6, 6, 6, 6, 7]>[2, 3, 6, 7, 7, 7] ---------- [2, 4, 6, 7, 7, 7]<[4, 5, 5, 7, 7, 7] [4, 5, 5, 7, 7, 7]<[6, 6, 6, 6, 6, 7] [6, 6, 6, 6, 6, 7]>[2, 4, 6, 7, 7, 7] ---------- [2, 2, 6, 7, 7, 7]<[2, 5, 5, 7, 7, 7] [2, 5, 5, 7, 7, 7]<[6, 6, 6, 6, 6, 7] [6, 6, 6, 6, 6, 7]>[2, 2, 6, 7, 7, 7] ---------- [2, 3, 6, 7, 7, 7]<[3, 4, 5, 7, 7, 7] [3, 4, 5, 7, 7, 7]<[6, 6, 6, 6, 6, 7] [6, 6, 6, 6, 6, 7]>[2, 3, 6, 7, 7, 7] ---------- [2, 2, 6, 7, 7, 7]<[2, 4, 5, 7, 7, 7] [2, 4, 5, 7, 7, 7]<[6, 6, 6, 6, 6, 7] [6, 6, 6, 6, 6, 7]>[2, 2, 6, 7, 7, 7] ---------- [2, 3, 6, 7, 7, 7]<[3, 4, 4, 7, 7, 7] [3, 4, 4, 7, 7, 7]<[6, 6, 6, 6, 6, 7] [6, 6, 6, 6, 6, 7]>[2, 3, 6, 7, 7, 7] ---------- [ 9 more solutions ] [1, 2, 6, 7, 7, 7]<[2, 3, 5, 7, 7, 7] [2, 3, 5, 7, 7, 7]<[6, 6, 6, 6, 6, 7] [6, 6, 6, 6, 6, 7]>[1, 2, 6, 7, 7, 7] ---------- [ 19 more solutions ] [1, 1, 6, 7, 7, 7]<[2, 5, 5, 7, 7, 7] [2, 5, 5, 7, 7, 7]<[6, 6, 6, 6, 6, 7] [6, 6, 6, 6, 6, 7]>[1, 1, 6, 7, 7, 7] ---------- [ 39 more solutions ] [2, 2, 6, 7, 7, 7]<[4, 5, 5, 7, 7, 7] [4, 5, 5, 7, 7, 7]<[6, 6, 6, 6, 6, 7] [6, 6, 6, 6, 6, 7]>[2, 2, 6, 7, 7, 7] ---------- [ 79 more solutions ] [5, 5, 5, 6, 7, 7]<[1, 6, 6, 6, 6, 7] [1, 6, 6, 6, 6, 7]<[1, 2, 6, 6, 7, 7] [1, 2, 6, 6, 7, 7]>[5, 5, 5, 6, 7, 7] ---------- [ 159 more solutions ] [1, 1, 6, 6, 6, 7]<[3, 3, 3, 4, 7, 7] [3, 3, 3, 4, 7, 7]<[1, 4, 6, 6, 6, 6] [1, 4, 6, 6, 6, 6]>[1, 1, 6, 6, 6, 7] ---------- [ 319 more solutions ] [3, 3, 4, 6, 6, 7]<[4, 4, 5, 5, 5, 7] [4, 4, 5, 5, 5, 7]<[1, 2, 6, 6, 6, 7] [1, 2, 6, 6, 6, 7]>[3, 3, 4, 6, 6, 7] ---------- [ 639 more solutions ] [2, 2, 2, 6, 6, 7]<[5, 5, 5, 5, 5, 7] [5, 5, 5, 5, 5, 7]<[1, 2, 6, 6, 6, 6] [1, 2, 6, 6, 6, 6]>[2, 2, 2, 6, 6, 7] ---------- [ 1279 more solutions ] [1, 2, 2, 6, 7, 7]<[4, 5, 5, 5, 5, 7] [4, 5, 5, 5, 5, 7]<[1, 1, 6, 6, 6, 6] [1, 1, 6, 6, 6, 6]>[1, 2, 2, 6, 7, 7] ---------- [ 2559 more solutions ] [3, 3, 3, 6, 7, 7]<[3, 4, 5, 5, 7, 7] [3, 4, 5, 5, 7, 7]<[2, 6, 6, 6, 6, 6] [2, 6, 6, 6, 6, 6]>[3, 3, 3, 6, 7, 7] ---------- [ 5119 more solutions ] [4, 4, 4, 5, 7, 7]<[5, 5, 5, 5, 5, 7] [5, 5, 5, 5, 5, 7]<[3, 4, 6, 6, 6, 6] [3, 4, 6, 6, 6, 6]>[4, 4, 4, 5, 7, 7] ---------- [ 10239 more solutions ] [3, 4, 5, 5, 7, 7]<[5, 5, 5, 5, 5, 6] [5, 5, 5, 5, 5, 6]<[2, 2, 6, 6, 6, 6] [2, 2, 6, 6, 6, 6]>[3, 4, 5, 5, 7, 7] ---------- [ 20479 more solutions ] [1, 3, 3, 3, 7, 7]<[2, 4, 4, 5, 5, 6] [2, 4, 4, 5, 5, 6]<[1, 2, 4, 6, 6, 6] [1, 2, 4, 6, 6, 6]>[1, 3, 3, 3, 7, 7] ---------- [ 40959 more solutions ] [2, 3, 4, 4, 7, 7]<[3, 4, 4, 5, 5, 5] [3, 4, 4, 5, 5, 5]<[1, 1, 4, 5, 6, 7] [1, 1, 4, 5, 6, 7]>[2, 3, 4, 4, 7, 7] ---------- [ 81919 more solutions ] [2, 2, 2, 2, 7, 7]<[1, 3, 3, 3, 3, 5] [1, 3, 3, 3, 3, 5]<[1, 2, 2, 4, 6, 6] [1, 2, 2, 4, 6, 6]>[2, 2, 2, 2, 7, 7] ---------- [ 163839 more solutions ] [1, 1, 2, 7, 7, 7]<[2, 3, 4, 5, 5, 7] [2, 3, 4, 5, 5, 7]<[2, 3, 5, 5, 6, 6] [2, 3, 5, 5, 6, 6]>[1, 1, 2, 7, 7, 7] ---------- [ 327679 more solutions ] [1, 1, 4, 6, 7, 7]<[2, 2, 3, 5, 7, 7] [2, 2, 3, 5, 7, 7]<[3, 4, 4, 4, 5, 7] [3, 4, 4, 4, 5, 7]>[1, 1, 4, 6, 7, 7] ---------- [ 613827 more solutions ] [3, 3, 3, 4, 4, 5]<[2, 2, 3, 5, 6, 6] [2, 2, 3, 5, 6, 6]<[2, 2, 2, 5, 7, 7] [2, 2, 2, 5, 7, 7]>[3, 3, 3, 4, 4, 5] ---------- ========== %%%mzn-stat: nodes=1659193 %%%mzn-stat: failures=1573389 %%%mzn-stat: restarts=0 %%%mzn-stat: variables=1694 %%%mzn-stat: intVars=126 %%%mzn-stat: boolVariables=1566 %%%mzn-stat: propagators=451 %%%mzn-stat: propagations=54691502 %%%mzn-stat: peakDepth=109 %%%mzn-stat: nogoods=1573389 %%%mzn-stat: backjumps=85804 %%%mzn-stat: peakMem=0.00 %%%mzn-stat: time=128.215 %%%mzn-stat: initTime=0.010 %%%mzn-stat: solveTime=128.205 %%%mzn-stat: baseMem=0.00 %%%mzn-stat: trailMem=0.01 %%%mzn-stat: randomSeed=1599952012 Finished in 2m 9sNimTranslation of: PythonLibrary: itertools<lang Nim> import algorithm, sequtils, sets, strformat import itertools type Die = objectname: string faces: seq[int]Die functions.func `$`(die: Die): string =## Return the string representation of a Die. &"({die.name}: {($die.faces)[1..^1]})"func cmp(die1, die2: Die): int =## Compare two die returning 1, -1 or 0 for operators >, < ==. var tot: array[3, int] for d in product(die1.faces, die2.faces): inc tot[1 + ord(d[1] < d[0]) - ord(d[0] < d[1])] result = ord(tot[0] < tot[2]) - ord(tot[2] < tot[0])func verboseCmp(die1, die2: Die): string =## Compare tow die returning a string. var win1, win2 = 0 for (d1, d2) in product(die1.faces, die2.faces): inc win1, ord(d1 > d2) inc win2, ord(d2 > d1) result = if win1 > win2: &"{die1.name} > {die2.name}" elif win1 < win2: &"{die1.name} < {die2.name}" else: &"{die1.name} = {die2.name}"func `>`(die1, die2: Die): bool = cmp(die1, die2) > 0 func `<=`(die1, die2: Die): bool = cmp(die1, die2) <= 0Added a permutation iterator as that of "itertools" doesn't allow to specify the length.iterator permutations[T](values: openArray[T]; r: int): seq[T] {.closure} =## Yield permutations of length "r" with elements taken from "values". let n = values.len if r > n: return var indices = toSeq(0..<n) var cycles = toSeq(countdown(n, n - r + 1)) var perm = values[0..<r] yield perm if n == 0: return while true: var exit = true for i in countdown(r - 1, 0): dec cycles[i] if cycles[i] == 0: discard indices.rotateLeft(i..indices.high, 1) cycles[i] = n - i else: let j = cycles[i] swap indices[i], indices[^j] for iperm, ivalues in indices[0..<r]: perm[iperm] = values[ivalues] yield perm exit = false break if exit: returnDice functions.func isNonTrans(dice: openArray[Die]): bool =## Return true if ordering of die in dice is non-transitive. for i in 1..dice.high: if dice[i] <= dice[i-1]: return false result = dice[0] > dice[^1]func findNonTrans(allDice: openArray[Die]; n = 3): seq[seq[Die]] =## Return the list of non-transitive dice. for perm in permutations(allDice, n): if perm.isNontrans: result.add permproc possibleDice(sides, maxval: Positive): seq[Die] =## Return the list of possible dice with given number of sides and maximum value.echo &"All possible 1..{maxval} {sides}-sided dice." var dice: seq[Die] var n = 1 for faces in product(toSeq(1..maxval), repeat = sides): dice.add Die(name: &"D{n}", faces: faces) inc n echo &" Created {dice.len} dice." echo " Remove duplicate with same bag of numbers on different faces." var found: HashSet[seq[int]] for d in dice: let count = sorted(d.faces) if count notin found: found.incl count result.add d echo &" Return {result.len} filtered dice."func verboseDiceCmp(dice: openArray[Die]): string =## Return the verbose comparison of dice. for i in 1..dice.high: result.add verboseCmp(dice[i-1], dice[i]) & ", " result.add verboseCmp(dice[0], dice[^1])———————————————————————————————————————————————————————————————————————————————————————————————————when isMainModule:let dice = possibleDice(sides = 4, maxval = 4) for n in [3, 4]: let nonTrans = dice.findNonTrans(n) echo &"\n Non-transitive length-{n} combinations found: {nonTrans.len}." for list in nonTrans: echo "" for i, die in list: echo " ", if i == 0: '[' else: ' ', die, if i == list.high: "]" else: "," if nonTrans.len != 0: echo &"\n More verbose comparison of last non-transitive result:" echo " ", verboseDiceCmp(nonTrans[^1]) echo "\n ===="</lang>Output:All possible 1..4 4-sided dice. Created 256 dice. Remove duplicate with same bag of numbers on different faces. Return 35 filtered dice. Non-transitive length-3 combinations found: 3. [(D16: [1, 1, 4, 4]), (D88: [2, 2, 2, 4]), (D43: [1, 3, 3, 3])] [(D43: [1, 3, 3, 3]), (D16: [1, 1, 4, 4]), (D88: [2, 2, 2, 4])] [(D88: [2, 2, 2, 4]), (D43: [1, 3, 3, 3]), (D16: [1, 1, 4, 4])] More verbose comparison of last non-transitive result: D88 < D43, D43 < D16, D88 > D16 ==== Non-transitive length-4 combinations found: 4. [(D16: [1, 1, 4, 4]), (D88: [2, 2, 2, 4]), (D91: [2, 2, 3, 3]), (D43: [1, 3, 3, 3])] [(D43: [1, 3, 3, 3]), (D16: [1, 1, 4, 4]), (D88: [2, 2, 2, 4]), (D91: [2, 2, 3, 3])] [(D88: [2, 2, 2, 4]), (D91: [2, 2, 3, 3]), (D43: [1, 3, 3, 3]), (D16: [1, 1, 4, 4])] [(D91: [2, 2, 3, 3]), (D43: [1, 3, 3, 3]), (D16: [1, 1, 4, 4]), (D88: [2, 2, 2, 4])] More verbose comparison of last non-transitive result: D91 < D43, D43 < D16, D16 < D88, D91 > D88 ====PerlTranslation of: Go<lang perl>use strict; use warnings; sub fourFaceCombs {my %found = (); my @res = (); for (my $i = 1; $i <= 4; $i++) { for (my $j = 1; $j <= 4; $j++) { for (my $k = 1; $k <= 4; $k++) { for (my $l = 1; $l <= 4; $l++) { my @c = sort ($i, $j, $k, $l); my $key = 0; for my $p (@c) { $key = 10 * $key + $p; } if (not exists $found{$key}) { $found{$key} = 1; push @res, \@c; } } } } } return @res;} sub compare {my $xref = shift; my $yref = shift;my @x = @$xref; my $xw = 0;my @y = @$yref; my $yw = 0;for my $i (@x) { for my $j (@y) { if ($i < $j) { $yw++; } if ($j < $i) { $xw++; } } }if ($xw < $yw) { return -1; } if ($yw < $xw) { return 1; } return 0;} sub findIntransitive3 {my $dice_ref = shift; my @dice = @$dice_ref; my $len = scalar @dice;my @res = (); for (my $i = 0; $i < $len; $i++) { for (my $j = 0; $j < $len; $j++) { my $first = compare($dice[$i], $dice[$j]); if ($first == 1) { for (my $k = 0; $k < $len; $k++) { my $second = compare($dice[$j], $dice[$k]); if ($second == 1) { my $third = compare($dice[$k], $dice[$i]); if ($third == 1) { my $d1r = $dice[$i]; my $d2r = $dice[$j]; my $d3r = $dice[$k]; my @itd = ($d1r, $d2r, $d3r); push @res, \@itd; } } } } } } return @res;} sub findIntransitive4 {my $dice_ref = shift; my @dice = @$dice_ref; my $len = scalar @dice;my @res = (); for (my $i = 0; $i < $len; $i++) { for (my $j = 0; $j < $len; $j++) { for (my $k = 0; $k < $len; $k++) { for (my $l = 0; $l < $len; $l++) { my $first = compare($dice[$i], $dice[$j]); if ($first == 1) { my $second = compare($dice[$j], $dice[$k]); if ($second == 1) { my $third = compare($dice[$k], $dice[$l]); if ($third == 1) { my $fourth = compare($dice[$l], $dice[$i]); if ($fourth == 1) { my $d1r = $dice[$i]; my $d2r = $dice[$j]; my $d3r = $dice[$k]; my $d4r = $dice[$l]; my @itd = ($d1r, $d2r, $d3r, $d4r); push @res, \@itd; } } } } } } } } return @res;} sub main {my @dice = fourFaceCombs(); my $len = scalar @dice; print "Number of eligible 4-faced dice: $len\n\n";my @it3 = findIntransitive3(\@dice); my $count3 = scalar @it3; print "$count3 ordered lists of 3 non-transitive dice found, namely:\n"; for my $itref (@it3) { print "[ "; for my $r (@$itref) { print "[@$r] "; } print "]\n"; } print "\n";my @it4 = findIntransitive4(\@dice); my $count = scalar @it4; print "$count ordered lists of 4 non-transitive dice found, namely:\n"; for my $itref (@it4) { print "[ "; for my $r (@$itref) { print "[@$r] "; } print "]\n"; }} main();</lang>Output:Number of eligible 4-faced dice: 35 3 ordered lists of 3 non-transitive dice found, namely: [ [1 1 4 4] [1 3 3 3] [2 2 2 4] ] [ [1 3 3 3] [2 2 2 4] [1 1 4 4] ] [ [2 2 2 4] [1 1 4 4] [1 3 3 3] ] 4 ordered lists of 4 non-transitive dice found, namely: [ [1 1 4 4] [1 3 3 3] [2 2 3 3] [2 2 2 4] ] [ [1 3 3 3] [2 2 3 3] [2 2 2 4] [1 1 4 4] ] [ [2 2 2 4] [1 1 4 4] [1 3 3 3] [2 2 3 3] ] [ [2 2 3 3] [2 2 2 4] [1 1 4 4] [1 3 3 3] ]Phixwith javascript_semantics requires("0.8.2") -- (added sq_cmp() builtin that returns nested -1/0/+1 compare() results, just like the existing sq_eq() does for equal().) integer mx = 4, -- max number of a die side (later set to 6) mn = 1 -- min number of a die side (later set to 0) function possible_dice(integer sides) -- -- construct all non-descending permutes of mn..mx, -- ie/eg {1,1,1,1}..{4,4,4,4} with (say), amongst -- others, {1,1,2,4} but not {1,1,4,2}. -- sequence die = repeat(mn,sides), -- (main work area) res = {deep_copy(die)} while true do -- find rightmost incrementable side -- ie/eg {1,2,4,4} -> set rdx to 2 (if 1-based indexing) integer rdx = rfind(true,sq_lt(die,mx)) if rdx=0 then exit end if -- set that and all later to that incremented -- ie/eg {1,2,4,4} -> {1,3,3,3} die[rdx..$] = die[rdx]+1 res = append(res,deep_copy(die)) end while printf(1,"There are %d possible %d..%d %d-sided dice\n",{length(res),mn,mx,sides}) return res end function function Dnn(sequence die) -- reconstruct the python die numbering (string) -- essentially just treat it as a base-N number. integer l = length(die), -- (sides) N = mx-mn+1, -- (base) n = 0 -- (result) for k=1 to l do n += (die[k]-mn)*power(N,l-k) end for return sprintf("D%d",n+1) end function function cmpd(sequence die1, die2) -- compares two die returning -1, 0, or +1 for <, =, > integer res = 0 for i=1 to length(die1) do res += sum(sq_cmp(die1[i],die2)) end for return sign(res) end function function low_rotation(sequence set) integer k = find(min(set),set) return set[k..$]&set[1..k-1] end function integer rotations function find_non_trans(sequence dice, integer n=3) atom t1 = time()+1 integer l = length(dice), sk, sk1, c sequence set = repeat(1,n), -- (indexes to dice) cache = repeat(repeat(-2,l),l), res = {} while true do bool valid = true for k=1 to n-1 do sk = set[k] sk1 = set[k+1] c = cache[sk][sk1] if c=-2 then c = cmpd(dice[sk],dice[sk1]) cache[sk][sk1] = c end if if c!=-1 then valid = false -- force k+1 to be incremented next: set[k+2..$] = l exit end if end for if valid then sk = set[1] sk1 = set[$] c = cache[sk][sk1] if c=-2 then c = cmpd(dice[sk],dice[sk1]) cache[sk][sk1] = c end if if c=+1 then res = append(res,low_rotation(set)) end if end if -- find rightmost incrementable die index -- ie/eg if l is 35 and set is {1,2,35,35} -- -> set rdx to 2 (if 1-based indexing) integer rdx = rfind(true,sq_lt(set,l)) if rdx=0 then exit end if -- increment that and reset all later -- ie/eg {1,2,35,35} -> {1,3,1,1} set[rdx] += 1 set[rdx+1..$] = 1 if time()>t1 and platform()!=JS then progress("working... (%d/%d)\r",{set[1],l}) t1 = time()+1 end if end while if platform()!=JS then progress("") end if rotations = length(res) res = unique(res) rotations -= length(res) return res end function function verbose_cmp(sequence die1, die2) -- compares two die returning their relationship of their names as a string integer c = cmpd(die1,die2) string op = {"<","=",">"}[c+2], n1 = Dnn(die1), n2 = Dnn(die2) return sprintf("%s %s %s",{n1,op,n2}) end function function verbose_dice_cmp(sequence dice, set) sequence c = {}, d1, d2 for j=1 to length(set)-1 do d1 = dice[set[j]] d2 = dice[set[j+1]] c = append(c,verbose_cmp(d1,d2)) end for d1 = dice[set[1]] d2 = dice[set[$]] c = append(c,verbose_cmp(d1,d2)) return join(c,", ") end function procedure show_dice(sequence dice, non_trans, integer N) integer l = length(non_trans), omissions = 0, last = 0 if N then printf(1,"\n Non_transitive length-%d combinations found: %d\n",{N,l+rotations}) end if for i=1 to l do object ni = non_trans[i] if sequence(ni) then if i<5 then printf(1,"\n") for j=1 to length(ni) do sequence d = dice[ni[j]] printf(1," %s:%v\n",{Dnn(d),d}) end for last = i else omissions += 1 end if end if end for if omissions then printf(1," (%d omitted)\n",omissions) end if if rotations then printf(1," (%d rotations omitted)\n",rotations) end if if last then printf(1,"\n") if mx<=6 and mn=1 then printf(1," More verbose comparison of last result:\n") end if printf(1," %s\n",{verbose_dice_cmp(dice,non_trans[last])}) printf(1,"\n ====\n") end if printf(1,"\n") end procedure atom t0 = time() sequence dice = possible_dice(4) for N=3 to 4 do show_dice(dice,find_non_trans(dice,N),N) end for -- From the numberphile video (Efron's dice): mx = 6 mn = 0 dice = possible_dice(6) --show_dice(dice,find_non_trans(dice,4),4) -- ok, dunno about you but I'm not waiting for power(924,4) permutes... -- limit to the ones discussed, plus another 4 random ones -- (hopefully it'll just pick out the right ones...) printf(1,"\nEfron's dice\n") dice = {{0,0,4,4,4,4}, {1,1,1,1,1,1}, -- (rand) {1,1,1,5,5,5}, {1,2,3,4,5,6}, -- (rand) {2,2,2,2,6,6}, {5,5,5,6,6,6}, -- (rand) {3,3,3,3,3,3}, {6,6,6,6,6,6}} -- (rand) show_dice(dice,find_non_trans(dice,4),0) -- and from wp: mx = 9 mn = 1 dice = possible_dice(6) printf(1,"\nFrom wp\n") dice = {{1,1,6,6,8,8}, {2,2,4,4,9,9}, {3,3,5,5,7,7}} show_dice(dice,find_non_trans(dice,3),0) -- Miwin's dice printf(1,"Miwin's dice\n") dice = {{1,2,5,6,7,9}, {1,3,4,5,8,9}, {2,3,4,6,7,8}} show_dice(dice,find_non_trans(dice,3),0) printf(1,"%s\n\n",elapsed(time()-t0))Output:There are 35 possible 1..4 4-sided dice Non_transitive length-3 combinations found: 3 D16:{1,1,4,4} D88:{2,2,2,4} D43:{1,3,3,3} (2 rotations omitted) More verbose comparison of last result: D16 < D88, D88 < D43, D16 > D43 ==== Non_transitive length-4 combinations found: 4 D16:{1,1,4,4} D88:{2,2,2,4} D91:{2,2,3,3} D43:{1,3,3,3} (3 rotations omitted) More verbose comparison of last result: D16 < D88, D88 < D91, D91 < D43, D16 > D43 ==== There are 924 possible 0..6 6-sided dice Efron's dice D1601:{0,0,4,4,4,4} D19837:{1,1,1,5,5,5} D39249:{2,2,2,2,6,6} D58825:{3,3,3,3,3,3} (3 rotations omitted) D1601 < D19837, D19837 < D39249, D39249 < D58825, D1601 > D58825 ==== There are 3003 possible 1..9 6-sided dice From wp D4121:{1,1,6,6,8,8} D68121:{2,2,4,4,9,9} D134521:{3,3,5,5,7,7} (2 rotations omitted) D4121 < D68121, D68121 < D134521, D4121 > D134521 ==== Miwin's dice D9945:{1,2,5,6,7,9} D74825:{2,3,4,6,7,8} D15705:{1,3,4,5,8,9} (2 rotations omitted) D9945 < D74825, D74825 < D15705, D9945 > D15705 ==== "0.3s"Stretch: length-3 combinations of 6-sided diceWith the recent optimisations I can just about stretch to power(924,3), but I think I was right to skip power(924,4) above, it would probably take over two and a half days. I have also estimated that power(3003,3) (ie 1..9 on the sides) would probably take around 4 hours (assuming such attempts didn't run out of memory).t0 = time() mx = 6 mn = 1 dice = possible_dice(6) show_dice(dice,find_non_trans(dice,3),3) printf(1,"%s\n\n",elapsed(time()-t0)) t0 = time() mx = 6 mn = 0 dice = possible_dice(6) show_dice(dice,find_non_trans(dice,3),3) printf(1,"%s\n\n",elapsed(time()-t0))Output:There are 462 possible 1..6 6-sided dice Non_transitive length-3 combinations found: 121998 D58:{1,1,1,2,4,4} D262:{1,1,2,2,2,4} D88:{1,1,1,3,3,4} D58:{1,1,1,2,4,4} D1556:{1,2,2,2,2,2} D87:{1,1,1,3,3,3} D58:{1,1,1,2,4,4} D1556:{1,2,2,2,2,2} D88:{1,1,1,3,3,4} D58:{1,1,1,2,4,4} D1557:{1,2,2,2,2,3} D88:{1,1,1,3,3,4} (40662 omitted) (81332 rotations omitted) More verbose comparison of last result: D58 < D1557, D1557 < D88, D58 > D88 ==== "30.9s" There are 924 possible 0..6 6-sided dice Non_transitive length-3 combinations found: 1269189 D74:{0,0,0,1,3,3} D403:{0,0,1,1,1,3} D116:{0,0,0,2,2,3} D74:{0,0,0,1,3,3} D2802:{0,1,1,1,1,1} D115:{0,0,0,2,2,2} D74:{0,0,0,1,3,3} D2802:{0,1,1,1,1,1} D116:{0,0,0,2,2,3} D74:{0,0,0,1,3,3} D2803:{0,1,1,1,1,2} D116:{0,0,0,2,2,3} (423059 omitted) (846126 rotations omitted) D74 < D2803, D2803 < D116, D74 > D116 ==== "3 minutes and 60s"Python<lang python>from collections import namedtuple from itertools import permutations, product from functools import lru_cache Die = namedtuple('Die', 'name, faces') @lru_cache(maxsize=None) def cmpd(die1, die2):'compares two die returning 1, -1 or 0 for >, < ==' # Numbers of times one die wins against the other for all combinations # cmp(x, y) is `(x > y) - (y > x)` to return 1, 0, or -1 for numbers tot = [0, 0, 0] for d1, d2 in product(die1.faces, die2.faces): tot[1 + (d1 > d2) - (d2 > d1)] += 1 win2, _, win1 = tot return (win1 > win2) - (win2 > win1)def is_non_trans(dice):"Check if ordering of die in dice is non-transitive returning dice or None" check = (all(cmpd(c1, c2) == -1 for c1, c2 in zip(dice, dice[1:])) # Dn < Dn+1 and cmpd(dice[0], dice[-1]) == 1) # But D[0] > D[-1] return dice if check else Falsedef find_non_trans(alldice, n=3):return [perm for perm in permutations(alldice, n) if is_non_trans(perm)]def possible_dice(sides, mx):print(f"\nAll possible 1..{mx} {sides}-sided dice") dice = [Die(f"D{n+1}", faces) for n, faces in enumerate(product(range(1, mx+1), repeat=sides))] print(f' Created {len(dice)} dice') print(' Remove duplicate with same bag of numbers on different faces') found = set() filtered = [] for d in dice: count = tuple(sorted(d.faces)) if count not in found: found.add(count) filtered.append(d) l = len(filtered) print(f' Return {l} filtered dice') return filtered%% more verbose extra checksdef verbose_cmp(die1, die2):'compares two die returning their relationship of their names as a string' # Numbers of times one die wins against the other for all combinations win1 = sum(d1 > d2 for d1, d2 in product(die1.faces, die2.faces)) win2 = sum(d2 > d1 for d1, d2 in product(die1.faces, die2.faces)) n1, n2 = die1.name, die2.name return f'{n1} > {n2}' if win1 > win2 else (f'{n1} < {n2}' if win1 < win2 else f'{n1} = {n2}')def verbose_dice_cmp(dice):c = [verbose_cmp(x, y) for x, y in zip(dice, dice[1:])] c += [verbose_cmp(dice[0], dice[-1])] return ', '.join(c)%% Useif __name__ == '__main__':dice = possible_dice(sides=4, mx=4) for N in (3, 4): # length of non-transitive group of dice searched for non_trans = find_non_trans(dice, N) print(f'\n Non_transitive length-{N} combinations found: {len(non_trans)}') for lst in non_trans: print() for i, die in enumerate(lst): print(f" {' ' if i else '['}{die}{',' if i < N-1 else ']'}") if non_trans: print('\n More verbose comparison of last non_transitive result:') print(' ', verbose_dice_cmp(non_trans[-1])) print('\n ====')</lang>Output:All possible 1..4 4-sided dice Created 256 dice Remove duplicate with same bag of numbers on different faces Return 35 filtered dice Non_transitive length-3 combinations found: 3 [Die(name='D16', faces=(1, 1, 4, 4)), Die(name='D88', faces=(2, 2, 2, 4)), Die(name='D43', faces=(1, 3, 3, 3))] [Die(name='D43', faces=(1, 3, 3, 3)), Die(name='D16', faces=(1, 1, 4, 4)), Die(name='D88', faces=(2, 2, 2, 4))] [Die(name='D88', faces=(2, 2, 2, 4)), Die(name='D43', faces=(1, 3, 3, 3)), Die(name='D16', faces=(1, 1, 4, 4))] More verbose comparison of last non_transitive result: D88 < D43, D43 < D16, D88 > D16 ==== Non_transitive length-4 combinations found: 4 [Die(name='D16', faces=(1, 1, 4, 4)), Die(name='D88', faces=(2, 2, 2, 4)), Die(name='D91', faces=(2, 2, 3, 3)), Die(name='D43', faces=(1, 3, 3, 3))] [Die(name='D43', faces=(1, 3, 3, 3)), Die(name='D16', faces=(1, 1, 4, 4)), Die(name='D88', faces=(2, 2, 2, 4)), Die(name='D91', faces=(2, 2, 3, 3))] [Die(name='D88', faces=(2, 2, 2, 4)), Die(name='D91', faces=(2, 2, 3, 3)), Die(name='D43', faces=(1, 3, 3, 3)), Die(name='D16', faces=(1, 1, 4, 4))] [Die(name='D91', faces=(2, 2, 3, 3)), Die(name='D43', faces=(1, 3, 3, 3)), Die(name='D16', faces=(1, 1, 4, 4)), Die(name='D88', faces=(2, 2, 2, 4))] More verbose comparison of last non_transitive result: D91 < D43, D43 < D16, D16 < D88, D91 > D88 ====Caching informationIn [45]: cmpd.cache_info() Out[45]: CacheInfo(hits=2148761, misses=1190, maxsize=None, currsize=1190)RIt would not be difficult to adapt this code to meet the stretch goal, but readability would suffer. <lang rsplus>findNonTrans <- function() {diceSet <- unique(t(apply(expand.grid(1:4, 1:4, 1:4, 1:4), 1, sort))) #By construction, each row is a unique dice. winningDice <- function(X, Y) #Recall 'Example 1' in the task. { comparisonTable <- data.frame(X = rep(X, each = length(X)), Y = rep(Y, times = length(Y))) rowWinner <- ifelse(comparisonTable["X"] > comparisonTable["Y"], "X", ifelse(comparisonTable["X"] == comparisonTable["Y"], "-", "Y")) netXWins <- sum(rowWinner == "X") - sum(rowWinner == "Y") if(netXWins > 0) "X" else if(netXWins == 0) "Draw" else "Y" } rows <- seq_len(nrow(diceSet)) #Recall that each row of diceSet is a dice. XvsAllY <- function(X) sapply(rows, function(i) winningDice(X, diceSet[i, ])) winners <- as.data.frame(lapply(rows, function(i) XvsAllY(diceSet[i, ])), row.names = paste("Y=Dice", rows), col.names = paste("X=Dice", rows), check.names = FALSE) solutionCount <- 0 for(S in rows) { beatsS <- which(winners[paste("X=Dice", S)] == "Y") #Finds the indices of all T such that S<T. beatsT <- lapply(beatsS, function(X) which(winners[paste("X=Dice", X)] == "Y")) #To begin finding U such that T<U. beatenByS <- which(winners[paste("X=Dice", S)] == "X") #Finds the indices of all U such that S>U. potentialU <- lapply(beatsT, function(X) intersect(X, beatenByS)) #Combining previous two lines. nonEmptyIndices <- lengths(potentialU) != 0 #Most of potentialU is usually empty lists. if(any(nonEmptyIndices)) #If any lists in potentialU are non-empty, then their entry is the index of a U with S>U & T<U. { solutionCount <- solutionCount + 1 diceUIndex <- potentialU[nonEmptyIndices]1 #Assumes that there is only one valid U for this S. diceTIndex <- beatsS[nonEmptyIndices]1 #Finds the T corresponding to the chosen U. cat("Solution", solutionCount, "is:\n") output <- rbind(S = diceSet[S,], T = diceSet[diceTIndex,], U = diceSet[diceUIndex,]) colnames(output) <- paste("Dice", 1:4) print(output) } }} findNonTrans()</lang>Output:Solution 1 is: Dice 1 Dice 2 Dice 3 Dice 4 S 1 1 4 4 T 2 2 2 4 U 1 3 3 3 Solution 2 is: Dice 1 Dice 2 Dice 3 Dice 4 S 1 3 3 3 T 1 1 4 4 U 2 2 2 4 Solution 3 is: Dice 1 Dice 2 Dice 3 Dice 4 S 2 2 2 4 T 1 3 3 3 U 1 1 4 4RakuThanks to Thundergnat for the nice "less-is-more" tweaks now the 4 dice portion takes around 10 (down from 17 ) minutes to run ..Translation of: Go<lang perl6># 20201225 Raku programming solution my @dicepool = ^4 xx 4 ; sub FaceCombs(\N, @dp) { # for brevity, changed to use 0-based dice on inputmy @res = my @found = []; for [X] @dp { unless @found[ my \key = [+] ( N «**« (N-1…0) ) Z* .sort ] { @found[key] = True; @res.push: $_ »+» 1 } } @res} sub infix:<⚖️>(@x, @y) { +($_{Less} <=> $_{More}) given (@x X<=> @y).Bag } sub findIntransitive(\N, \cs) {my @res = []; race for [X] ^+cs xx N -> @lot { my $skip = False; for @lot.rotor(2 => -1) { { $skip = True and last } unless cs[ @_[0] ] ⚖️ cs[ @_[1] ] == -1 } next if $skip; if cs[ @lot[0] ] ⚖️ cs[ @lot[*-1] ] == 1 { @res.push: [ cs[ @lot ] ] } } @res} say "Number of eligible 4-faced dice : ", +(my \combs = FaceCombs(4,@dicepool)); for 3, 4 {my @output = findIntransitive($_, combs); say +@output, " ordered lists of $_ non-transitive dice found, namely:"; .say for @output;}</lang>Output:Number of eligible 4-faced dice : 35 3 ordered lists of 3 non-transitive dice found, namely: [(1 1 4 4) (2 2 2 4) (1 3 3 3)] [(1 3 3 3) (1 1 4 4) (2 2 2 4)] [(2 2 2 4) (1 3 3 3) (1 1 4 4)] 4 ordered lists of 4 non-transitive dice found, namely: [(1 1 4 4) (2 2 2 4) (2 2 3 3) (1 3 3 3)] [(1 3 3 3) (1 1 4 4) (2 2 2 4) (2 2 3 3)] [(2 2 2 4) (2 2 3 3) (1 3 3 3) (1 1 4 4)] [(2 2 3 3) (1 3 3 3) (1 1 4 4) (2 2 2 4)]WrenLibrary: Wren-sort<lang ecmascript>import "/sort" for Sort var fourFaceCombs = Fn.new {var res = [] var found = List.filled(256, false) for (i in 1..4) { for (j in 1..4) { for (k in 1..4) { for (l in 1..4) { var c = [i, j, k, l] Sort.insertion(c) var key = 64*(c[0]-1) + 16*(c[1]-1) + 4*(c[2]-1) + (c[3]-1) if (!found[key]) { found[key] = true res.add(c) } } } } } return res} var cmp = Fn.new { |x, y|var xw = 0 var yw = 0 for (i in 0..3) { for (j in 0..3) { if (x[i] > y[j]) { xw = xw + 1 } else if (y[j] > x[i]) { yw = yw + 1 } } } return (xw - yw).sign} var findIntransitive3 = Fn.new { |cs|var c = cs.count var res = [] for (i in 0...c) { for (j in 0...c) { for (k in 0...c) { var first = cmp.call(cs[i], cs[j]) if (first == -1) { var second = cmp.call(cs[j], cs[k]) if (second == -1) { var third = cmp.call(cs[i], cs[k]) if (third == 1) res.add([cs[i], cs[j], cs[k]]) } } } } } return res} var findIntransitive4 = Fn.new { |cs|var c = cs.count var res = [] for (i in 0...c) { for (j in 0...c) { for (k in 0...c) { for (l in 0...c) { var first = cmp.call(cs[i], cs[j]) if (first == -1) { var second = cmp.call(cs[j], cs[k]) if (second == -1) { var third = cmp.call(cs[k], cs[l]) if (third == -1) { var fourth = cmp.call(cs[i], cs[l]) if (fourth == 1) res.add([cs[i], cs[j], cs[k], cs[l]]) } } } } } } } return res} var combs = fourFaceCombs.call() System.print("Number of eligible 4-faced dice: %(combs.count)") var it = findIntransitive3.call(combs) System.print("\n%(it.count) ordered lists of 3 non-transitive dice found, namely:") System.print(it.join("\n")) it = findIntransitive4.call(combs) System.print("\n%(it.count) ordered lists of 4 non-transitive dice found, namely:") System.print(it.join("\n"))</lang>Output:Number of eligible 4-faced dice: 35 3 ordered lists of 3 non-transitive dice found, namely: [[1, 1, 4, 4], [2, 2, 2, 4], [1, 3, 3, 3]] [[1, 3, 3, 3], [1, 1, 4, 4], [2, 2, 2, 4]] [[2, 2, 2, 4], [1, 3, 3, 3], [1, 1, 4, 4]] 4 ordered lists of 4 non-transitive dice found, namely: [[1, 1, 4, 4], [2, 2, 2, 4], [2, 2, 3, 3], [1, 3, 3, 3]] [[1, 3, 3, 3], [1, 1, 4, 4], [2, 2, 2, 4], [2, 2, 3, 3]] [[2, 2, 2, 4], [2, 2, 3, 3], [1, 3, 3, 3], [1, 1, 4, 4]] [[2, 2, 3, 3], [1, 3, 3, 3], [1, 1, 4, 4], [2, 2, 2, 4]]

@@ Line 378: / Line 378: @@
 }}</lang>
-Here, <tt>extend<tt> takes a list of die sequences and appends each possibility of a new die where the new die is not less than the last die in the sequence. So, <tt>r</tt> is all such lists of length x. We discard the cases where the last element of the list is not less than the first, permute each into canonical order, discard duplicates, and then use these indices to select the associated dice.
+Here, <tt>extend<tt> takes a list of die sequences and appends each possibility of a new die where the new die is not less than the last die in the sequence. So, <tt>r</tt> is all such lists of length x. We discard the cases where the last element of the list is not less than the first, rotate each dice index list into canonical order, discard duplicates, and then use these indices to select the associated dice.
 Task examples: