View source for Latin Squares in reduced form/Randomizing using Jacobson and Matthews' technique

{{draft task}}

Section 3.3 of [[https://pdfs.semanticscholar.org/4a7c/d245f6f6a4ef933c6cf697832607f71a39c1.pdf Generalised 2-designs with Block Size 3(Andy L. Drizen)]] describes a method of generating Latin Squares of order n attributed to Jacobson and Matthews. The purpose of this task is to produce a function which given a valid Latin Square transforms it to another using this method.

;part 1
Use one of the 4 [[Latin Squares in reduced form]] of order 4 as X0 to generate 10000 Latin Squares using X(n-1) to generate X(n). Convert the resulting Latin Squares to their reduced form, display them and the number of times each is produced.

;part 2
As above for order 5, but do not display the squares. Generate the 56 [[Latin Squares in reduced form]] of order 5, confirm that all 56 are produced by the Jacobson and Matthews technique and display the number of each produced.

;part 3
Generate 750 Latin Squares of order 42 and display the 750th.

;part 4
Generate 1000 Latin Squares of order 256. Don't display anything but confirm the approximate time taken and anything else you may find interesting
 
=={{header|F_Sharp|F#}}==
===The Functions===
<syntaxhighlight lang="fsharp">
// Jacobson and Matthews technique for generating Latin Squares. Nigel Galloway: August 5th., 2019
let R=let N=System.Random() in (fun n->N.Next(n))

let jmLS α X0=
  let X0=Array2D.copy X0
  let N=let N=[|[0..α-1];[α-1..(-1)..0]|] in (fun()->N.[R 2])
  let rec randLS i j z n g s=
    X0.[i,g]<-s; X0.[n,j]<-s
    if X0.[n,g]=s then X0.[n,g]<-z; X0
    else randLS n g s (List.find(fun n->X0.[n,g]=s)(N())) (List.find(fun g->X0.[n,g]=s)(N())) (if (R 2)=0 then let t=X0.[n,g] in X0.[n,g]<-z; t else z)
  let i,j=R α,R α
  let z  =let z=1+(R (α-1)) in if z<X0.[i,j] then z else 1+(z+1)%α
  let n,g,s=let N=[0..α-1] in (List.find(fun n->X0.[n,j]=z) N,List.find(fun n->X0.[i,n]=z) N,X0.[i,j])
  X0.[i,j]<-z; randLS i j z n g s

let asNormLS α=
  let n=Array.init (Array2D.length1 α) (fun n->(α.[n,0]-1,n))|>Map.ofArray
  let g=Array.init (Array2D.length1 α) (fun g->(α.[n.[0],g]-1,g))|>Map.ofArray
  Array2D.init (Array2D.length1 α) (Array2D.length1 α) (fun i j->α.[n.[i],g.[j]])

let randLS α=Seq.unfold(fun g->Some(g,jmLS α g))(Array2D.init α α (fun n g->1+(n+g)%α))
</syntaxhighlight>
===The Task===
;part 1
<syntaxhighlight lang="fsharp">
randLS 4 |> Seq.take 10000 |> Seq.map asNormLS |> Seq.countBy id |> Seq.iter(fun n->printf "%A was produced %d times\n\n" (fst n)(snd n))
</syntaxhighlight>
{{out}}
<pre>
[[1; 2; 3; 4]
 [2; 3; 4; 1]
 [3; 4; 1; 2]
 [4; 1; 2; 3]] was produced 2920 times

[[1; 2; 3; 4]
 [2; 4; 1; 3]
 [3; 1; 4; 2]
 [4; 3; 2; 1]] was produced 2262 times

[[1; 2; 3; 4]
 [2; 1; 4; 3]
 [3; 4; 2; 1]
 [4; 3; 1; 2]] was produced 2236 times

[[1; 2; 3; 4]
 [2; 1; 4; 3]
 [3; 4; 1; 2]
 [4; 3; 2; 1]] was produced 2582 times
</pre>
;part 2
<syntaxhighlight lang="fsharp">
randLS 5 |> Seq.take 10000 |> Seq.map asNormLS |> Seq.countBy id |> Seq.iteri(fun n g->printf "%d(%d) " (n+1) (snd g)); printfn ""
</syntaxhighlight>
{{out}}
<pre>
1(176) 2(171) 3(174) 4(165) 5(168) 6(182) 7(138) 8(205) 9(165) 10(174) 11(157) 12(187) 13(181) 14(211) 15(184) 16(190) 17(190) 18(192) 19(146) 20(200) 21(162) 22(153) 23(193) 24(156) 25(148) 26(188) 27(186) 28(198) 29(178) 30(217) 31(185) 32(172) 33(223) 34(147) 35(203) 36(167) 37(188) 38(152) 39(165) 40(187) 41(160) 42(199) 43(140) 44(202) 45(186) 46(182) 47(175) 48(161) 49(179) 50(175) 51(201) 52(195) 53(205) 54(183) 55(155) 56(178)
</pre>
;part 3
<syntaxhighlight lang="fsharp">
let q=Seq.item 749 (randLS 42)
for n in [0..41] do (for g in [0..41] do printf "%3d" q.[n,g]); printfn ""
</syntaxhighlight>
{{out}}
<pre>
 16  7 41 15 17 40 12  9 10  5 19 29 21 18  8 22  3 36 23 31 11 38 13 30  2 33  6 42 39 14 32 20 28 35 26  1 34 37 27 24  4 25
 38 25 36 32 40 29 35 27  8 26 31 15  9  7 16 11  4  3 12 20 23 33  5 24 41 14 30 34 42 17 39 18 37 22 21 13  1 10  6 19  2 28
  8 34 27 25 21 31  1 23 37 36 26 13 22 24 35 17 10 40 41 30 42  7 15  2 18  3 29 11 32  4 38 39  9  5 16 14 28 12 20 33 19  6
 33 35 13 34 15 24  4 29 41 27  3 17 10 26 39 23 30 32  1 38 16 25 37 14  6 28 19  9 40  5 18  7 42 11 31 20 12 22  2 21  8 36
  2 42 20  1  7 26 11 10 39 41 34 22 40 23 24 29 14 17  5 33 38 30  6 13  3 16 18 19 31 15 28 21 36 37 32 27  8  4 25  9 35 12
 25 33 14 40 28 30 31 24 29  4  8 20 26 38 12 35  2 39 16  6 13 21 18 17  5 41 23  3 36  7 34 22 27  1 10 42 11 19 15 32 37  9
 17 22 35 28 30 18 21  2 15 39  5 40 27 13  1 34 38 37 26 23 41 36  4  3 11  6 20  8  9 10 12 24 31 25  7 29 16 32 42 14 33 19
 14  9 19  7 26 15 10  4 36 25 22 23 39 16  2 40 18  1 38 13 21 37 34 31 35 24 12 27 11  3  5  6 17 20 41 33 32 29  8 30 28 42
  5 27 24 13  2 36 25 30 23  9  6 14 35 15 42 39 16 26 21 34 33 31  3  1 29 12 38 17 37 19 40  4  7  8 22 41 20 28 32 10 18 11
 19 41 28 26  8 10 30 35 18 33 15 27 25 21 29 42 23 12 17  2  5  1 38  6 20  7 34  4 13 36 24 31 14  3 11 32 39 40  9 22 16 37
 41 10  3 19 22  9 27 40  1 29 16 42 33 39 34  7 37 20 11 12  4 18 35  8 28 26 36  5 17 30 25 32  6 15 24 21 13 23 14  2 38 31
 42  3 16 36 33 21 20 14 31 22  9 38 29 19 37 13 28 10 35 18 39 26 25 27  4 30 15 23 41 24 11  1 40  7  5 17  6  2 12  8 34 32
 23 31 34 41 38 33  3 28  4  1 30 25  6  2 20 14 13 24  8 42  7 12 39 32 22 29  5 37 15  9 27 10 35 36 19 40 17 18 16 11 26 21
 37 16 30 11  4 32 42 33 13  6 14  2 15 27 18 31 20 41 39 40  9 24 36  5 10  8  1 26  3 34 22 28 38 19 29 23 21 25 35 12 17  7
  1 19 26 22 16 25 36 39  3 23 41 37 34  6 17 32 40 21 10 27 12  9 31  7 13  4 24 29  8 11  2  5 15 18 35 28 30 20 33 38 42 14
 11 13 23 30 25 41  6 31 14 32 27 36 19 17 10 33 21 15  7  5  8 28 16 35 34 42 40  2 38 39  9 26 20 24 37  4 18  3 22  1 12 29
 24 17 29 38 23 39 32  5 11 15 35 12  8 10 40  1 22 25  2 36 28  4 42 21  9 20  3 31 16 41 13 30 19 34 33 18 27  6  7 37 14 26
 36  4  6 24 12 20  2 34 40 11 32  9 28  8 38 21  5 31 42 17 14 29 19 22 25 15  7 18 30 26  1 13 16 41 23 39 37 33  3 35 10 27
 20 39  2 12 32  7 22  3 17 10 37  6 18 40 27  5 42 35 28  4 24 14 33 29 30 31 26 13 19 23 36 41  1 21  9 11 15  8 34 16 25 38
 35 18 37  6  5 13 29  8 24 19 38 34 12 31 21 10 33  7  3 41 15 42 20 11 27 40 16 14 23  1  4  2 22 32 28  9 25 30 26 39 36 17
 10 32  9 33 39 19 41 38 35 18 28 26 14 30  7  4  1 22 37 21 31 40 27 15 42 34  2 25  5 12 23 36  8  6 17  3 29 24 11 13 20 16
 13 28 39  2 31  8  9 37 21 16 40 19 42 36 41  3 12 14 20 10 17 34  1 33 32 35 25 30 18 38 15 11 24 23  6 26  4  5 29  7 27 22
  7 40 12 39 18  3 16 21 42 17  1 32  5 33 13  6 41  8 29 14 34 35 24 36 38 25 31 28 26 27 20 37 23  2 30 10 22  9 19  4 11 15
  4 21  7 17 35 34 19 25 12 42 11  1 30 28 36 26 32 23 14 29  2 20  8 41 24 27 22 15 10 18 37  9 39 38 13  6  3 16 31 40  5 33
 34 23 42 14 41 27 37  6  9 31  4  5  7  1 25 16 35 30 33 11 19  3 26 12 17 38  8 20 24 13 29 15 32 28 40 22  2 39 18 36 21 10
 30  6 21  9 20 17  5 32 38 13 12 28 16 35 22 36 34 29 40 39 25 15 14 37 33 11  4 41  1  2 19  3 26 27 42  8 10  7 23 31 24 18
  6 38  8 10 42 35 13  1 16 37 21  3 11 34 32 20 29 18 25 22 36  5 30 26 39 23 28 12  2 31  7 19 33 40 14 24  9 41 17 27 15  4
 29 15  1 21 14 11 26 17 30 38 10 33 36 20  4 18 39 16 31  3 35  2 32 28 19 13 42  7 12  8  6 40  5  9 25 37 24 27 41 23 22 34
 21 36 32  8  6 23 15 19  2 14 18  4  3 11  5 28 26 13 34 25 30 17  7 42 16 22 39 40 29 37 33 12 41 10 27 31 35 38 24 20  9  1
 39 20 31 29 19  4 38 16 27 30 24 11  2  3 33 15  8 28 18 37 10 13  9 23 36  1 17 22 25 32 26 35 12 42 34  7 40 14 21  5  6 41
 12 11 17 42  9  2 14  7 22 24 25 31 38 41 15 19 36 33 32 28  1 10 29 40 23 18 37 39  6 21 35 27  3 16  8 30  5 26  4 34 13 20
 18 29 33 16 27 42 40 26  7  8 39 24 41  5 30 38  6  9 13  1 32 22  2 34 12 37 11 10 35 20 14 17 21  4 15 19 23 36 28 25 31  3
 28  2  4 18 11  5 23 20 25 35 42 30 31 14  3  9 24 27 19  7 22  6 12 10  1 32 41 36 21 33 16 34 29 13 39 15 38 17 37 26 40  8
  3 26 11 35 24 37 17 36  6  7 13 41  4 32  9  2 31 34 22 15 29  8 40 18 21  5 27  1 14 16 10 38 25 33 20 12 19 42 39 28 30 23
 31  5 22 27 10  6  8 13 34  2 33  7 32 42 26 12 19  4 15  9 40 16 28 38 37 39 35 24 20 29 17 23 11 14  3 25 41 21 36 18  1 30
 15 24  5 37  3 28  7 22 19 34 20 18 17 12 23  8 25 11 36 16 27 41 10  4 31  2  9 32 33 42 21 14 13 29 38 35 26  1 30  6 39 40
 27 37 25  5 13 16 24 41 28  3  2 10 23  4 14 30 11 38  6 19 26 32 21 20 40  9 33 35 34 22 42  8 18 17 12 36 31 15  1 29  7 39
 26 30 10  3 36 22 33 11  5 20 29 21 13 25 31 37 17  2  9 35 18 27 23 39 14 19 32 16 28  6  8 42  4 12  1 38  7 34 40 15 41 24
 32  8 18 31  1 14 34 12 33 28 17 39 37  9 19 27  7  5 30 24 20 23 11 25 15 36 21  6 22 40 41 16 10 26  4  2 42 35 38  3 29 13
  9 14 40 23 37 38 18 15 20 12 36  8  1 22 28 24 27 42  4 32  6 11 41 19 26 10 13 21  7 25 30 29 34 39  2 16 33 31  5 17  3 35
 22 12 15  4 34  1 39 42 32 40  7 35 20 29 11 25  9  6 24 26 37 19 17 16  8 21 14 38 27 28  3 33 30 31 18  5 36 13 10 41 23  2
 40  1 38 20 29 12 28 18 26 21 23 16 24 37  6 41 15 19 27  8  3 39 22  9  7 17 10 33  4 35 31 25  2 30 36 34 14 11 13 42 32  5
</pre>
;part 4
Generating 1000 Latin Squares of order 256 takes about 1.5secs
<syntaxhighlight lang="fsharp">
printfn "%d" (Array2D.length1 (Seq.item 999 (randLS 256)))
</syntaxhighlight>
{{out}}
<pre>
256
Real: 00:00:01.512, CPU: 00:00:01.970, GC gen0: 10, gen1: 10
</pre>

=={{header|Go}}==
The J & M implementation is based on the C code [https://brainwagon.org/2016/05/17/code-for-generating-a-random-latin-square/ here] which has been heavily optimized following advice and clarification by Nigel Galloway (see Talk page) on the requirements of this task.

Part 4 is taking about 6.5 seconds on my Celeron @1.6 GHz but will be much faster on a more modern machine. Being able to compute random, uniformly distributed, Latin squares of order 256 reasonably quickly is interesting from a secure communications or cryptographic standpoint as the symbols of such a square can represent the 256 characters of the various extended ASCII encodings. 
<syntaxhighlight lang="go">package main

import (
    "fmt"
    "math/rand"
    "time"
)

type (
    vector []int
    matrix []vector
    cube   []matrix
)

func toReduced(m matrix) matrix {
    n := len(m)
    r := make(matrix, n)
    for i := 0; i < n; i++ {
        r[i] = make(vector, n)
        copy(r[i], m[i])
    }
    for j := 0; j < n-1; j++ {
        if r[0][j] != j {
            for k := j + 1; k < n; k++ {
                if r[0][k] == j {
                    for i := 0; i < n; i++ {
                        r[i][j], r[i][k] = r[i][k], r[i][j]
                    }
                    break
                }
            }
        }
    }
    for i := 1; i < n-1; i++ {
        if r[i][0] != i {
            for k := i + 1; k < n; k++ {
                if r[k][0] == i {
                    for j := 0; j < n; j++ {
                        r[i][j], r[k][j] = r[k][j], r[i][j]
                    }
                    break
                }
            }
        }
    }
    return r
}

// 'm' is assumed to be 0 based
func printMatrix(m matrix) {
    n := len(m)
    for i := 0; i < n; i++ {
        for j := 0; j < n; j++ {
            fmt.Printf("%2d ", m[i][j]+1) // back to 1 based
        }
        fmt.Println()
    }
    fmt.Println()
}

// converts 4 x 4 matrix to 'flat' array
func asArray16(m matrix) [16]int {
    var a [16]int
    k := 0
    for i := 0; i < 4; i++ {
        for j := 0; j < 4; j++ {
            a[k] = m[i][j]
            k++
        }
    }
    return a
}

// converts 5 x 5 matrix to 'flat' array
func asArray25(m matrix) [25]int {
    var a [25]int
    k := 0
    for i := 0; i < 5; i++ {
        for j := 0; j < 5; j++ {
            a[k] = m[i][j]
            k++
        }
    }
    return a
}

// 'a' is assumed to be 0 based
func printArray16(a [16]int) {
    for i := 0; i < 4; i++ {
        for j := 0; j < 4; j++ {
            k := i*4 + j
            fmt.Printf("%2d ", a[k]+1) // back to 1 based
        }
        fmt.Println()
    }
    fmt.Println()
}

func shuffleCube(c cube) {
    n := len(c[0])
    proper := true
    var rx, ry, rz int
    for {
        rx = rand.Intn(n)
        ry = rand.Intn(n)
        rz = rand.Intn(n)
        if c[rx][ry][rz] == 0 {
            break
        }
    }
    for {
        var ox, oy, oz int
        for ; ox < n; ox++ {
            if c[ox][ry][rz] == 1 {
                break
            }
        }
        if !proper && rand.Intn(2) == 0 {
            for ox++; ox < n; ox++ {
                if c[ox][ry][rz] == 1 {
                    break
                }
            }
        }

        for ; oy < n; oy++ {
            if c[rx][oy][rz] == 1 {
                break
            }
        }
        if !proper && rand.Intn(2) == 0 {
            for oy++; oy < n; oy++ {
                if c[rx][oy][rz] == 1 {
                    break
                }
            }
        }

        for ; oz < n; oz++ {
            if c[rx][ry][oz] == 1 {
                break
            }
        }
        if !proper && rand.Intn(2) == 0 {
            for oz++; oz < n; oz++ {
                if c[rx][ry][oz] == 1 {
                    break
                }
            }
        }

        c[rx][ry][rz]++
        c[rx][oy][oz]++
        c[ox][ry][oz]++
        c[ox][oy][rz]++

        c[rx][ry][oz]--
        c[rx][oy][rz]--
        c[ox][ry][rz]--
        c[ox][oy][oz]--

        if c[ox][oy][oz] < 0 {
            rx, ry, rz = ox, oy, oz
            proper = false
        } else {
            proper = true
            break
        }
    }
}

func toMatrix(c cube) matrix {
    n := len(c[0])
    m := make(matrix, n)
    for i := 0; i < n; i++ {
        m[i] = make(vector, n)
    }
    for i := 0; i < n; i++ {
        for j := 0; j < n; j++ {
            for k := 0; k < n; k++ {
                if c[i][j][k] != 0 {
                    m[i][j] = k
                    break
                }
            }
        }
    }
    return m
}

// 'from' matrix is assumed to be 1 based
func makeCube(from matrix, n int) cube {
    c := make(cube, n)
    for i := 0; i < n; i++ {
        c[i] = make(matrix, n)
        for j := 0; j < n; j++ {
            c[i][j] = make(vector, n)
            var k int
            if from == nil {
                k = (i + j) % n
            } else {
                k = from[i][j] - 1
            }
            c[i][j][k] = 1
        }
    }
    return c
}

func main() {
    rand.Seed(time.Now().UnixNano())

    // part 1
    fmt.Println("PART 1: 10,000 latin Squares of order 4 in reduced form:\n")
    from := matrix{{1, 2, 3, 4}, {2, 1, 4, 3}, {3, 4, 1, 2}, {4, 3, 2, 1}}
    freqs4 := make(map[[16]int]int, 10000)
    c := makeCube(from, 4)
    for i := 1; i <= 10000; i++ {
        shuffleCube(c)
        m := toMatrix(c)
        rm := toReduced(m)
        a16 := asArray16(rm)
        freqs4[a16]++
    }
    for a, freq := range freqs4 {
        printArray16(a)
        fmt.Printf("Occurs %d times\n\n", freq)
    }

    // part 2
    fmt.Println("\nPART 2: 10,000 latin squares of order 5 in reduced form:")
    from = matrix{{1, 2, 3, 4, 5}, {2, 3, 4, 5, 1}, {3, 4, 5, 1, 2},
        {4, 5, 1, 2, 3}, {5, 1, 2, 3, 4}}
    freqs5 := make(map[[25]int]int, 10000)
    c = makeCube(from, 5)
    for i := 1; i <= 10000; i++ {
        shuffleCube(c)
        m := toMatrix(c)
        rm := toReduced(m)
        a25 := asArray25(rm)
        freqs5[a25]++
    }
    count := 0
    for _, freq := range freqs5 {
        count++
        if count > 1 {
            fmt.Print(", ")
        }
        if (count-1)%8 == 0 {
            fmt.Println()
        }
        fmt.Printf("%2d(%3d)", count, freq)
    }
    fmt.Println("\n")

    // part 3
    fmt.Println("\nPART 3: 750 latin squares of order 42, showing the last one:\n")
    var m42 matrix
    c = makeCube(nil, 42)
    for i := 1; i <= 750; i++ {
        shuffleCube(c)
        if i == 750 {
            m42 = toMatrix(c)
        }
    }
    printMatrix(m42)

    // part 4
    fmt.Println("\nPART 4: 1000 latin squares of order 256:\n")
    start := time.Now()
    c = makeCube(nil, 256)
    for i := 1; i <= 1000; i++ {
        shuffleCube(c)
    }
    elapsed := time.Since(start)
    fmt.Printf("Generated in %s\n", elapsed)
}</syntaxhighlight>

{{out}}
Sample run:
<pre>
PART 1: 10,000 latin Squares of order 4 in reduced form:

 1  2  3  4 
 2  1  4  3 
 3  4  2  1 
 4  3  1  2 

Occurs 2550 times

 1  2  3  4 
 2  4  1  3 
 3  1  4  2 
 4  3  2  1 

Occurs 2430 times

 1  2  3  4 
 2  1  4  3 
 3  4  1  2 
 4  3  2  1 

Occurs 2494 times

 1  2  3  4 
 2  3  4  1 
 3  4  1  2 
 4  1  2  3 

Occurs 2526 times


PART 2: 10,000 latin squares of order 5 in reduced form:

 1(165),  2(173),  3(167),  4(204),  5(173),  6(165),  7(215),  8(218), 
 9(168), 10(157), 11(205), 12(152), 13(187), 14(173), 15(215), 16(185), 
17(179), 18(176), 19(179), 20(160), 21(150), 22(166), 23(191), 24(181), 
25(179), 26(192), 27(187), 28(186), 29(176), 30(196), 31(141), 32(187), 
33(165), 34(189), 35(147), 36(175), 37(172), 38(162), 39(180), 40(172), 
41(189), 42(159), 43(197), 44(158), 45(178), 46(179), 47(193), 48(175), 
49(207), 50(174), 51(181), 52(179), 53(193), 54(171), 55(153), 56(204)


PART 3: 750 latin squares of order 42, showing the last one:

29  2 17 41 34 30  8 33 39  7 20 27 12  6 31 14 40 35 25  9 10 32 19 16 24 42  3 26  5 23  1 28  4 13 38 18 21 37 22 15 36 11 
17 15 11 31  9 38 26 10  1 28 37  8 34 41 21 22 12  5 35 36 13 20 29 42 18  3 19 24 39 32 27 23 16 25 33  4 40  6  2 30  7 14 
36 42 35 39 15 34 37 18 32 25 22 31  4 17  3 19 13 11  8 23 12 24 28 27 16  1  6  9 29 40  7  5  2 14 30 26 41 10 21 33 38 20 
21 13 16 42  3 32  2 26 27 17 15 11 25 37 29  6 19 10 12  7 31 18 36  9 39 41 30 40 35 33 22  1 28 38 24  8 34 23  4 20 14  5 
22 39 13  7 38  9 34 41 37 36 35  6 21 26 17 16  4 30 40 20  8 15 25 19 32  2 11 28 23 24 31 10 42  3 27 12 33 14  1 29  5 18 
33 36 34  3 13  4  7 14  2 29  6 12 31 23 26 17  8 20 32 21 19 41 37  5 38 30 25 11 24 35 42 27 18 16 39 15 10 22 28  1  9 40 
14 31  7 22 39 23 32 34 16 33 24  4 40 42 12 25 35 26 18 28 11  3 15 21 20  9 13 19  1 10  2 41 29  6 17 30  5 38 37  8 27 36 
 9  3  6 30 19 39 14 16  4 15 29 28 23 24 32 10 18 41 37 38 40 34  8 25  2 22 31  5 17 26 36 33 13 21 12 35  7 20 11 27 42  1 
 2 18 28  5  6  7 40 35  3 20  8 34 42 39 37 33 26 23 22 13 14  4 12 15 17 25 36 31 16 29 38 19 32 41  1 27 24 11 30  9 10 21 
27 34 19 15 33 22  5 36  9 30 14  1 24  8 38 42 41 39  7 40  4 37 11 23 29 26 18 12  3 21 35 16 20 10 31 25 17 28  6 32  2 13 
41 16  1 35 22 13 20 29  6 38  5 24 19 10 25 27 17 18 11 32  9  7  2 36  4 34 40 21 33 12  8 30 15 42 37 23 14 26  3 39 31 28 
 7  1 15 16 27 31 18 24 20  8 36 38 10 34  9  4 42 29  2  3 26 39  5 22 41 21 37 30 14 11 33 35 25 23 40 28 13 19 17  6 32 12 
 1 10 20 32 23  5 30 12  8  9 21 36 15 14 18 37 33 31 26 39 41 16  6 24 22 35 29 42 27 28  3 38 11  2  7 34  4 40 19 17 13 25 
 6 32 42 11 20 40 27 25 41 22 17 16 26 29 15  7 23 36 39 34 28 13 18  3 10 37  8 14  2 31  4 24  5 19  9 21 38  1 33 12 30 35 
35 40 30 19 21 12 17  4 22 27  3 20 11  9  8 23 24 42 14 10 39 28 26 29 33 13 41 16 34 25 32 37  7 18  5  6 15  2 36 38  1 31 
15 26 40  1 28 20  9 21  7  5 13 18 30 22 10  8  3 25  6  2 17 36 38 31 14 19 35 23 12 27 11 39 24  4 41 32 29 34 42 16 37 33 
 3  6 26 12 32  1 13  8 42 37 25  7  9 16 35  5 29 21 24 27 34 17 14  2 15 11 28 33 20 38 18 22 39 40 23 10 31 30 41 36 19  4 
31 38 36 21 16 26 28 30 15  3 32 41 18  1  6 29  9 17  5 35  7 40 27 37 13 20 23 22 11 19 12 42 34  8 10 14 25 39 24  4 33  2 
40  4 22 38 35 11 21 17 31  1 28 19 37  2 42 24 14 12 13 30 33 25 34 32 27 36 39  3  9 15 10 18  8  5  6 41 26 16 29  7 20 23 
 5 17 39  4 26 14 31 37 35 11 38  3  1 30 19 36 20 33 15 16 21 29  9  6 25 27  2 13 41 34 24 12 10 32 22  7 28 18 40 42 23  8 
 8 29 24 26 31 21 39 23 11 14 19 10 20 15  7 35 32 38  1 12 25 22 16  4  6 40 42 41 18 30 28  2 17 36  3 13 37 33 27  5 34  9 
11 25 14 17 18 24 19 32 33 31  7 26  2 21 20 30 15 27 23 41 29 35 39 28 34 12 10  4  8 42  5 13 37  9 16 40  1 36 38  3  6 22 
26 21 18 25 29 15  1 13 19  2 34 23 38 27 41  3 10 22 17  4 16 11 42 12  8  6  5 35 30 39 37 14  9 24 36 33 20  7 31 28 40 32 
25 27 12 33 17 35 24  9 28 10 42 21  8 13  2 15 34 16  3 18  5 31 41  7 23  4  1  6 22 14 19 36 40 37 26 38 30 32 20 11 39 29 
23 19 25  9 30 37 38 40 14 41 31 17  7  4 16 11  1  6 33  5 24  2  3  8 21 29 34 32 28 22 15 20 12 35 18 36 39 27 10 13 26 42 
34  9 10 13  2  6 22 31 26 40  1 14 41  3 11 12 37 32 27 29 35 19 30 33 28 38 21 25  7  5 16  8 36 15 20 42 23 17 39 18  4 24 
20 11 37 28 41  8 10 15 36 12 26 33 39 32 13  1 25  9 42 19  3  6 24 14  5 23  7 27 38  2 30  4 22 34 35 31 18 29 16 40 21 17 
28 30 21 23 24 29  3  1 10  6 33  2 27 40 14 34 31 15 19 37 18  9  4 13 35  8 12 20 36 16 17 32 41  7 25 39 42  5 26 22 11 38 
32 12  8 40 11 16 23 28 18 42 41 30  3 38 33  2 22 19  4 25 37  1 31 20 36  5  9  7 13 17 14  6 27 39 34 24 35 21 15 26 29 10 
18 37 41 10 36 28 11 42 13 34  2 35  5  7 22 40 39  3 30  1 38 27 20 17 19 33 26 15 25  6 21 29 23 31  4  9 32  8 12 14 24 16 
39 24 29 37 25 19 33 27 17 16 10 40 36 12 30 41 11  4 34 15  2  5 32  1 31 14 38 18 42  3  9  7  6 20 21 22  8 13 23 35 28 26 
19 14  5  8 40  3 29  6 21 26 23 15 16 33 28 31 38 13  9 17 27 12 10 11  7 24 20  1  4 41 39 25 30 22 32  2 36 42 35 34 18 37 
37  7 32 34  8 36 41  2 12 24 16 39 33 31  4 13  6 28 38 22 20 42 40 18  9 10 14 29 26  1 23 15 21 27 19 17 11  3  5 25 35 30 
 4 41 27  2 42 17 15 38 30 35 12 25 13 28 39 20  5  1 16 33 36 23  7 40 37 32 24 10 31  8  6 21 14 26 29 11  3  9 18 19 22 34 
38 35 23 36  4 10 12 11  5 21 27 32 17 25 24 18 28 40 20  6 42 14 22 30 26 39 33  8 37  7 13 34  1 29 15 19  2 41  9 31 16  3 
30 33 31 24 12 41 36 19 23 32  4 37 29 11 34 39 16 14 21 42  6 26  1 38  3 17 22  2 40 18 20  9 35 28 13  5 27 15 25 10  8  7 
42 28  3 14  1 25 16 22 34 23 39  9 35  5 40 26 36  7 10 31 32 21 13 41 30 18  4 38  6 37 29 17 33 12 11 20 19 24  8  2 15 27 
16  5 38  6 10 27  4  3 40 18 11 13 22 35  1 21  2 34 36  8 23 30 17 39 42  7 15 37 32 20 26 31 19 33 28 29  9 25 14 24 12 41 
24 23 33 18 14  2 25 39 29 19  9  5 28 20 27 38  7  8 31 11 15 10 35 34 12 16 32 17 21 36 40  3 26 30 42  1 22  4 13 37 41  6 
12 20  2 29  5 33 42  7 24  4 18 22 14 19 36  9 27 37 28 26 30 38 23 10 11 31 17 34 15 13 41 40  3  1  8 16  6 35 32 21 25 39 
13  8  9 27 37 42  6 20 25 39 40 29 32 18  5 28 30 24 41 14 22 33 21 35  1 15 16 36 10  4 34 26 38 11  2  3 12 31  7 23 17 19 
10 22  4 20  7 18 35  5 38 13 30 42  6 36 23 32 21  2 29 24  1  8 33 26 40 28 27 39 19  9 25 11 31 17 14 37 16 12 34 41  3 15 


PART 4: 1000 latin squares of order 256:

Generated in 6.581088256s
</pre>

=={{header|Julia}}==
{{trans|Go}}
<syntaxhighlight lang="julia">const Cube = Vector{Vector{Vector{Int}}}
const Mat = Vector{Vector{Int}}

function reduced(m::Mat)
    n = length(m)
    r = deepcopy(m)
    for j in 1:n-1
        if r[1][j] != j
            for k in j+1:n
                if r[1][k] == j
                    for i in 1:n
                        r[i][j], r[i][k] = r[i][k], r[i][j]
                    end
                    break
                end
            end
        end
    end
    for i in 2:n-1
        if r[i][1] != i
            for k in i+1:n
                if r[k][1] == i
                    for j in 1:n
                        r[i][j], r[k][j] = r[k][j], r[i][j]
                    end
                    break
                end
            end
        end
    end
    return r
end

""" print matrix as small integers, no punctuation """
function print_matrix(m::Mat)
    n = length(m)
    padding = max(2, Int(ceil(log(10, n+1))) + 1)
    for i in 1:n
        for j in 1:n
            print(lpad(m[i][j], padding))
        end
        println()
    end
    println()
end

function shuffle_cube(c::Cube)
    n = length(c)
    proper = true
    rx, ry, rz = 0, 0, 0
    while true
        rx, ry, rz = rand(1:n, 3)
        c[rx][ry][rz] == 0 && break
    end
    while true
        ox = something(findfirst(i -> c[i][ry][rz] == 1, 1:n), n)
        oy = something(findfirst(i -> c[rx][i][rz] == 1, 1:n), n)
        oz = something(findfirst(i -> c[rx][ry][i] == 1, 1:n), n)
        if !proper
            rand() < 1/2 && (ox = something(findlast(i -> c[i][ry][rz] == 1, 1:n), n))
            rand() < 1/2 && (oy = something(findlast(i -> c[rx][i][rz] == 1, 1:n), n))
            rand() < 1/2 && (oz = something(findlast(i -> c[rx][ry][i] == 1, 1:n), n))
        end

        c[rx][ry][rz] += 1
        c[rx][oy][oz] += 1
        c[ox][ry][oz] += 1
        c[ox][oy][rz] += 1

        c[rx][ry][oz] -= 1
        c[rx][oy][rz] -= 1
        c[ox][ry][rz] -= 1
        c[ox][oy][oz] -= 1

        if c[ox][oy][oz] < 0
            rx, ry, rz = ox, oy, oz
            proper = false
        else
            break
        end
    end
end

function matrix(c::Cube)::Mat
    n = length(c)
    m = [[0 for i in 1:n] for j in 1:n]
    for i in 1:n, j in 1:n
        for k in 1:n
            if c[i][j][k] != 0
                m[i][j] = k
                break
            end
        end
    end
    return m
end

function cube(from, n)
    c = [[[0 for i in 1:n] for j in 1:n] for k in 1:n]
    for i in 1:n, j in 1:n
        k = (from isa Nothing) ? mod1(i + j, n) : from[i][j]
        c[i][j][k] = 1
    end
    return c
end

function testJacobsenMatthews()
    # part 1
    println("PART 1: 10,000 latin Squares of order 4 in reduced form:\n")
    from = [[1, 2, 3, 4], [2, 1, 4, 3], [3, 4, 1, 2], [4, 3, 2, 1]]
    freqs4 = Dict{Array, Int}()
    c = cube(from, 4)
    for i in 1:10000
        shuffle_cube(c)
        m = matrix(c)
        rm = reduced(m)
        n = get!(freqs4, rm, 0)
        freqs4[rm] = n + 1
    end
    for (a, freq) in freqs4
        print_matrix(a)
        println("Occurs $freq times\n")
    end

    # part 2
    println("\nPART 2: 10,000 latin squares of order 5 in reduced form:\n")
    from = [[1, 2, 3, 4, 5], [2, 3, 4, 5, 1], [3, 4, 5, 1, 2], [4, 5, 1, 2, 3], [5, 1, 2, 3, 4]]
    freqs5 = Dict{Array, Int}()
    c = cube(from, 5)
    for i in 1:10000
        shuffle_cube(c)
        m = matrix(c)
        rm = reduced(m)
        n = get!(freqs5, rm, 0)
        freqs5[rm] = n + 1
    end
    for (i, freq) in enumerate(sort(collect(values(freqs5))))
        i > 1 && (print(", "); (i - 1) % 8 == 0 && println())
        print(lpad(i, 2), "(", lpad(freq, 3), ")")
    end
    println("\n")

    # part 3
    println("\nPART 3: 750 latin squares of order 42, showing the last one:\n")
    m42 = [[0 for i in 1:42] for j in 1:42]
    c = cube(nothing, 42)
    for i in 1:750
        shuffle_cube(c)
        i == 750 && (m42 = matrix(c))
    end
    print_matrix(m42)

    # part 4
    println("\nPART 4: 1000 latin squares of order 256:\n")
    @time begin
        c = cube(nothing, 256)
        for i in 1:1000
            shuffle_cube(c)
        end
    end
end

testJacobsenMatthews()
</syntaxhighlight>{{out}}
<pre>
PART 1: 10,000 latin Squares of order 4 in reduced form:

 1 2 3 4
 2 4 1 3
 3 1 4 2
 4 3 2 1

Occurs 2508 times

 1 2 3 4
 2 1 4 3
 3 4 2 1
 4 3 1 2

Occurs 2427 times

 1 2 3 4
 2 1 4 3
 3 4 1 2
 4 3 2 1

Occurs 2529 times

 1 2 3 4
 2 3 4 1
 3 4 1 2
 4 1 2 3

Occurs 2536 times


PART 2: 10,000 latin squares of order 5 in reduced form:

 1(152),  2(152),  3(153),  4(154),  5(158),  6(160),  7(160),  8(160), 
 9(162), 10(165), 11(166), 12(167), 13(168), 14(170), 15(170), 16(172),
17(172), 18(173), 19(174), 20(174), 21(175), 22(177), 23(177), 24(177),
25(177), 26(178), 27(179), 28(180), 29(180), 30(181), 31(181), 32(182),
33(182), 34(182), 35(183), 36(184), 37(185), 38(185), 39(185), 40(186), 
41(187), 42(187), 43(187), 44(188), 45(189), 46(189), 47(190), 48(195),
49(195), 50(197), 51(197), 52(199), 53(199), 54(199), 55(201), 56(203)


PART 3: 750 latin squares of order 42, showing the last one:

 32 34 23 19  7 42 37  4 38  2 26 25 17 16 22 20 18  8 28 24 40 35  3 33  6  1 41 36 13 39 10 14  9 30 27 29 15  5 12 11 21 31
 19 16 27 14  4 15 31  8 36  3 34 18  2 10 30 42 22 35 41 21 13  5 11 29 37 39  9 12 32  7 33 17 28 40 25 26 23 24 38  6  1 20
 22  7 11 41 14 27  4  3 30 39 38 40 23 36 19  5 25 13 29 37 33  8 15 32 16 34  6 42 24  1 28 18 21 10  9 35 17 20  2 26 31 12
 18 26 38 24 25 14  6 39 40  5 13 21 20 34 29  4  3 22 30 42 12 19 23  8 32 17  7 27 35 28  2 31 15 41 10 36 11  9  1 37 16 33
 37 24 21 15 30 36  2 27  4 11  6 16 26 38 14 31  9 34 39  1  8 41 40 42 17  3 18 33 12 13 22 23 19 35  7  5 25 32 28 10 20 29
  8 38 22 21 26 28 12 37 10 41 35 34 13 24 27 16  2 17 20  6  7 30 42 39 40  4  1 18 36  3 15 33  5 29 19 11 32 25 14 31 23  9
 31 35  2  8 10 39 13 22 20 14 15 24 16 30 21 40 36  4  1  3 23 25 29 26  5 32 33 38  9 37 11 19 18 42 17 41 12  6 34  7 27 28
 29 10 35  4 39 13  5 12 21 18 37 14 40 17 33  7 30 25  2 11 34 22 41 15  3  9 38 31 26 16 32 36 27 19  8  1 20 28 23 24  6 42
  5 14 37 16 19 10 21 17 18 23 29 42 12 11  4 34 35 38  6 28  3 27  9 24  8 30 26  2 41 25  1 13  7 32 36 15 39 31 20 40 33 22
 13 28 31  2 37  7 34  9 24 38  1 30  3 14 40 35 20 12 23 19  4 11 27 25 26 10 15 17 42 36 18  5 22 39 16  6 29 41 21 33  8 32
  7 19 24 10  9 30 15 42 26 34 28 32 36  4 11 41 40 27 13 23 31 18 14  3 20 12  5  1 38 29 35 39 16  8  2 21 37 33 25 17 22  6
 28 22 36  6 21 20 16 34 32 29  8 27 18 42 26 24 12 30  5 39 14 10  4 19 15 31 25  9 40 41 38  2 33 23 35  7  1 37 11 13  3 17
 16 23 30 33 18 38 22  5 41  9  4 12 35 13 37 32 11  6 19 10 42 31 20  1  2  7 17 21 28 15 34 40 29 36  3 14  8 27 24 39 25 26
 20 12 41 13 38 21 23 29 17 10 30  2 25  8 42  3  4 24 18 35 11 40 33 36 22 26 32 19  1  6 14 28 37 31 34  9 27  7 39 16 15  5
 15 36 34 26  1 18 28  6 31 37 17 20 29 41 35 22 10 33 25 32 21  3  7 16 14 42 27 24  5 19  4 11 39 13 40 38 30 23  9 12  2  8
 36 42  1 31 13  3 39 32 27  4 23 28  7  2 18 11  6 19 26 16 22 15 12 41 21  5 34 40 25 38 37 20 30 17 29  8 33 14 10  9 24 35
 33  8  3  7 29  9 40 28  2 19  5 13 15 26 39 37 32  1 14 17 38  4 21 27 41  6 23 34 10 12 36 30 35 24 20 25 42 22 31 18 11 16
 21 37 12  9 17  2 29 16 34  7  3 19 42 40  5 33 31 28 36  8 18 23 22  6 10 41 30 39 20 11 24 25 38 15 13 27 26  4 35 32 14  1
  1 32 29 35 22 40  7 23 28 26 18 37  6 20 16 19 14  9 17 33 41 24 31 38 34 25 12  3 30  2  8 27 36 11 39 42  4 10  5 21 13 15
 42 30  4 18 24 33 27 35 12 16 32 15 41  5 10  6 13 11  7 29  9 14 25 23 19 37  3 20  2 22 31 21 40  1 26 28 36 38 17  8 39 34
 35  5  7 36 23  6 20 15 13 25 19 17 28 32  9  1 33 40 10 26  2 12 39  4 29 38  8 16  3 14 27 24 41 22 11 30 31 21 42 34 37 18
 24 29 28 17 20  4  9 33 19 40 31  6 22 12 25 30 38  5 32 41 39 34 36  2  1  8 37 15 21 26 16 42 10 27 18 23 35 11 13 14  7  3
 30 40 32 39 35 41 14 20  9  6 10 11  4 31 13 23 16  2  3 22 36 29 24  7 25 15 21 28 17 33 12 38 34  5  1 19 18  8 26 27 42 37
 27 25 18 11 15 35 30 41 42 24 33 29 32 23  3  8 21 10 22 31 28 20 19 12 39 36 14  6 34  9  5  1  2 38 37 40 16 26  7  4 17 13
 12 13 40 42 11 26 17 30 39 36 14  4 19 18 20  2 15  7 38 25  5 16 37 21 31 35 29  8  6 27 23 32 24  9 28 22  3  1 33 41 34 10
  2  3  5 23 12 31 33 14 15 21 25  8 24 39 28  9 41 29 27  7 30 13 16 34 38 40  4 37 19 42 17 26  1 20 22 32 10 18  6 36 35 11
 25  9 39 40 33 17 26  1 29 22 24 36 37  7 15 21  8 18 16 30  6 38 35 14 23 28 10 11 27 31 13  3 42 34  4 20 41  2 32  5 12 19
  9 41  6 22  8 12 10 21  7 15  2  5 34 25 31 14  1 16 33 20 37 39 32 17 27 29 19 35 11 18 30  4 13 26 38  3 24 42 36 28 40 23
  6 21 26 29  3 24  8 18 23 42 40 33 38  9 36 27 28 14 11  5 15  7 34 37 12  2 20  4 16 10 41 35 31 25 32 17 19 13 22  1 30 39
 14  4 16 30 31  8 11 25 22 28 41 10  1 19  2 39 37 26 35 15 24  6 38 20 13 27 42  5 23 34 21  9 32 18 33 12  7 17  3 29 36 40
 11  1 25  3  5 32 35  7  8 31 36 39 30 37 24 12 34 41 42 27 10 28  6  9 18 21  2 22 29 23 20 15 17 33 14 13 40 16  4 19 26 38
 34 27 42  1 40 16 32 36  3 30 39 22 33 29 23 17  5 21 24 18 35 26 10 28  4 19 13  7  8 20  9 37 11 12 31  2  6 15 41 25 38 14
 17 39 19 32 42 23 25  2 11  1 20 35 10  3  6 36 27 37  9 13 26 21  8 22 28 33 24 14 18 40  7 29  4 16 12 31 38 34 15 30  5 41
 26 31 17 37  2 29 42 40 14 12 27 23 11  1  7 15 24 32  8 34 25 33 30 10  9 16 22 41  4 35  3  6 20 28  5 39 13 36 19 38 18 21
  4 33 13 20 41 34 18 31  1 17 16 38 27 35  8 28 23 39 15 36 19  9 26 30 42 24 11 32 14 21 25 12  3  7  6 37  5 40 29 22 10  2
 41 20 15  5 36 37 19 38 25 13 42  7 39  6 32 10 26 31 34 12  1 17  2 11 35 18 28 29 22  4 40 16 23 21 24 33 14 30  8  3  9 27
 38 18 10 34  6 11 24 26 37 27 12  3 31 28 17 13 42 15 21 14 16 32  1 40 33 23 36 25 39  5 19 22  8  2 41  4  9 35 30 20 29  7
  3 11 33 12 34 25 38 24  5 35  7 26  9 27  1 18 39 20 37  4 29  2 17 13 30 22 40 23 31 32 42  8  6 14 21 10 28 19 16 15 41 36
 40 17 20 38 27  5  3 13 33  8 22 41 21 15 12 25 29 36 31  9 32  1 28 35 11 14 16 10 37 30  6  7 26  4 42 34  2 39 18 23 19 24
 39  2  9 28 16  1 41 19  6 32 21 31  8 33 34 29  7 23 40 38 20 37  5 18 36 11 35 13 15 17 26 10 14  3 30 24 22 12 27 42  4 25
 10 15  8 27 32 22 36 11 35 20  9  1 14 21 38 26 19  3 12 40 17 42 18  5  7 13 31 30 33 24 39 41 25  6 23 16 34 29 37  2 28  4
 23  6 14 25 28 19  1 10 16 33 11  9  5 22 41 38 17 42  4  2 27 36 13 31 24 20 39 26  7  8 29 34 12 37 15 18 21  3 40 35 32 30


PART 4: 1000 latin squares of order 256:

 10.811605 seconds (745.43 k allocations: 157.305 MiB, 0.30% gc time)
</pre>

=={{header|Nim}}==
{{trans|Go}}
<syntaxhighlight lang="nim">import random, sequtils, strformat, tables, times

type
  Vector = seq[int]
  Matrix = seq[Vector]
  Cube = seq[Matrix]
  Array16 = array[16, int]
  Array25 = array[25, int]


func newCube(m: Matrix; n: Positive): Cube =
  result.setLen(n)
  for i in 0..<n:
    result[i].setLen(n)
    for j in 0..<n:
      result[i][j].setLen(n)
      let k = if m.len == 0: (i + j) mod n else: m[i][j] - 1
      result[i][j][k] = 1


proc shuffle(c: var Cube) =
  let n = c[0].len
  var proper = true

  var rx, ry, rz: int
  while true:
    rx = rand(n - 1)
    ry = rand(n - 1)
    rz = rand(n - 1)
    if c[rx][ry][rz] == 0: break

  while true:
    var ox, oy, oz = 0

    while ox < n:
      if c[ox][ry][rz] == 1: break
      inc ox
    if not proper and rand(1) == 0:
      inc ox
      while ox < n:
        if c[ox][ry][rz] == 1: break
        inc ox

    while oy < n:
      if c[rx][oy][rz] == 1: break
      inc oy
    if not proper and rand(1) == 0:
      inc oy
      while oy < n:
        if c[rx][oy][rz] == 1: break
        inc oy

    while oz < n:
      if c[rx][ry][oz] == 1: break
      inc oz
    if not proper and rand(1) == 0:
      inc oz
      while oz < n:
        if c[rx][ry][oz] == 1: break
        inc oz

    inc c[rx][ry][rz]
    inc c[rx][oy][oz]
    inc c[ox][ry][oz]
    inc c[ox][oy][rz]

    dec c[rx][ry][oz]
    dec c[rx][oy][rz]
    dec c[ox][ry][rz]
    dec c[ox][oy][oz]

    if c[ox][oy][oz] < 0:
      (rx, ry, rz) = (ox, oy, oz)
      proper = false
    else:
      proper = true
      break


func toMatrix(c: Cube): Matrix =
  let n = c[0].len
  result = newSeqWith(n, newSeq[int](n))
  for i in 0..<n:
    for j in 0..<n:
      for k in 0..<n:
        if c[i][j][k] != 0:
          result[i][j] = k
          break


func toReduced(m: Matrix): Matrix =
  let n = m.len
  result = m

  for j in 0..n-2:
    if result[0][j] != j:
      for k in j+1..<n:
        if result[0][k] == j:
          for i in 0..<n:
            swap result[i][j], result[i][k]
          break

  for i in 1..n-2:
    if result[i][0] != i:
      for k in i+1..<n:
        if result[k][0] == i:
          for j in 0..<n:
            swap result[i][j], result[k][j]
          break


func asArray16(m: Matrix): Array16 =
  var k = 0
  for i in 0..3:
    for j in 0..3:
      result[k] = m[i][j]
      inc k


func asArray25(m: Matrix): Array25 =
  var k = 0
  for i in 0..4:
    for j in 0..4:
      result[k] = m[i][j]
      inc k


proc printArray16(a: Array16) =
  for i in 0..3:
    for j in 0..3:
      let k = i * 4 + j
      stdout.write &"{a[k]+1:2} "   # Back to 1 based.
    echo()
  echo()


proc printMatrix(m: Matrix) =
  let n = m.len
  for i in 0..<n:
    for j in 0..<n:
      stdout.write &"{m[i][j]+1:2} "  # Back to 1 based.
    echo()
  echo()


randomize()

# Part 1.
echo "Part 1: 10_000 latin Squares of order 4 in reduced form:\n"
const From1: Matrix = @[@[1, 2, 3, 4], @[2, 1, 4, 3], @[3, 4, 1, 2], @[4, 3, 2, 1]]
var freqs4: CountTable[Array16]
var c = newCube(From1, 4)
for _ in 1..10_000:
  c.shuffle()
  let m = c.toMatrix
  let rm = m.toReduced
  let a16 = rm.asArray16
  freqs4.inc(a16)

for a, freq in freqs4.pairs:
  printArray16(a)
  echo &"Occurs {freq} times\n"

# Part 2.
echo "\nPart 2: 10_000 latin squares of order 5 in reduced form:"
const From2: Matrix = @[@[1, 2, 3, 4, 5], @[2, 3, 4, 5, 1], @[3, 4, 5, 1, 2],
                        @[4, 5, 1, 2, 3], @[5, 1, 2, 3, 4]]
var freqs5: CountTable[Array25]
c = newCube(From2, 5)
for _ in 1..10_000:
  c.shuffle()
  let m = c.toMatrix
  let rm = m.toReduced
  let a25 = rm.asArray25
  freqs5.inc(a25)

var count = 0
for freq in freqs5.values:
  inc count
  if count > 1: stdout.write ", "
  if (count - 1) mod 8 == 0: echo()
  stdout.write &"{count:2}({freq:3})"
echo '\n'

# Part 3.
echo "\nPart 3: 750 latin squares of order 42, showing the last one:\n"
var m42: Matrix
c = newCube(@[], 42)
for i in 1..750:
  c.shuffle()
  if i == 750:
    m42 = c.toMatrix
printMatrix(m42)

# Part 4.
echo "\nPart 4: 1000 latin squares of order 256:\n"
let start = cpuTime()
c = newCube(@[], 256)
for _ in 1..1000:
  c.shuffle()
echo &"Generated in {cpuTime() - start:.3f} s."</syntaxhighlight>

{{out}}
<pre>Part 1: 10_000 latin Squares of order 4 in reduced form:

 1  2  3  4 
 2  1  4  3 
 3  4  2  1 
 4  3  1  2 

Occurs 2469 times

 1  2  3  4 
 2  4  1  3 
 3  1  4  2 
 4  3  2  1 

Occurs 2430 times

 1  2  3  4 
 2  3  4  1 
 3  4  1  2 
 4  1  2  3 

Occurs 2561 times

 1  2  3  4 
 2  1  4  3 
 3  4  1  2 
 4  3  2  1 

Occurs 2540 times


Part 2: 10_000 latin squares of order 5 in reduced form:

 1(175),  2(201),  3(209),  4(158),  5(175),  6(190),  7(178),  8(173), 
 9(172), 10(179), 11(199), 12(170), 13(167), 14(199), 15(166), 16(164), 
17(180), 18(189), 19(170), 20(183), 21(165), 22(172), 23(199), 24(190), 
25(165), 26(178), 27(178), 28(190), 29(173), 30(187), 31(163), 32(157), 
33(179), 34(190), 35(168), 36(209), 37(185), 38(181), 39(161), 40(180), 
41(162), 42(161), 43(184), 44(157), 45(195), 46(179), 47(181), 48(161), 
49(198), 50(190), 51(176), 52(184), 53(148), 54(209), 55(167), 56(181)


Part 3: 750 latin squares of order 42, showing the last one:

31 27  9 29 25 19 40 35 16 34 32 33 10 13  1 36 17 24 39 14 18 26 38 15 21 42  3  6 12  2 41 28 37  7  5 11 23 20 22  4 30  8 
13 33 39 32 16 26 21 25  3 37 22 11 27 41 42 10  7  2 35 18 14 31  5 17 19 29 20 28 38  9 12 40  8  6 23 30 34 36 24 15  4  1 
40  7  6  5 21 25  3  4  8 20  1 23 19 35  2 29 24 37 32 39 28 11  9 14 15 26 41 18 36 12 33 16 17 27 38 13 22 34 30 31 42 10 
17 40 32 42 15 35 18 41  9 25  6  3 30  1 37 34 19 14  4 31 24 28 21 20 38 13 26 27 10 16 36 22  5 12 39 29 33 11 23  8  7  2 
12  9 29 23  8 22 42 32 33 14 30 31 20 18 10 24 36 13 15 16  3 25 17 26 27 19  7 39 41 21 28  5  2 11  4  1 40 35 37 34  6 38 
11  2 41 30  3  1 17 22 40 32  5  8  4 14 26  7 10 16 29  9 31 37 25 19 36 27 34 35 24 33 23 21 12 15 20 18 38 39  6 42 13 28 
24 26 40  4  9 37 36 13 10 28 15 21 22 27 14 20 34  5  6 23 16 38 29  7  8 12 35 41 33 18 19 17 11  1 30 25  2 32 42  3 31 39 
37 18  4 24 42 38 30 31 20 36 39  2  6 11 41  3 15 33 26 13  9 29 40 32 10 28 23 25 22 34 14 35  1 16 27 12 19 21  8 17  5  7 
29 39 18 19 37  6 14  1 31 30  2  5 28  8 33 25 23  9 17 21  7 35 20  3 11 16 22 38 15 36 10 32 13 34 26  4 27 40 12 24 41 42 
 4 16  8 10 19  5  6 24  2 12 40 32 15 29 13 41 20 38 36 28 42 39 23 34 17 11 30 14 21  1 27 31  7  3 37 26 35 33 18 22 25  9 
15 29 12 26  5  9  1 23 32  2 36 16 11 30 21 33 13  3 31 41  8  7 14 37 22 10 39 40  6 20 24 27 42 18 17 28  4 38 19 25 35 34 
41 22 35 34 28  3 31 17 13  7 23 30  1  4 11 37 29 40 38 10 15 14 24 27  6 21 33 16  2 42  9  8 25 32 18 19 12  5 20 39 36 26 
25 37 23 33 26 32 22  6 38 13 11 29  5 31 19 27 40 20 14  7  2 12 36 10 34 39 18 24  1  8  4 42 15 30 41  9  3 17 28 35 16 21 
 5 20 21 13 23 15 34 27 25 11 19 12 24 16 17  1 26 28 18 36 30 32 42 22 40  3 37  9 35 10 31 41  4 33 29 38  8 14  2  7 39  6 
 1 12 10 41 14 20 16 33 39  5 27  6 17 23 24 13 38 25 19 15 34 21 28 36  4 30  8 29 32 37  2  9 26 42  7 35 18 22 11 40  3 31 
 6 24 36  7  2 10 28 29 18 41 12  4 25 40  3 17 35 23 11  5 22 42 31 16  9 33 32 21 37 30 38 13 19  8 34 39 26  1 15 20 27 14 
16 30 25 39  4 33 38  9 15 42 24 36 35  3 31 12 32 18 27  8 41 23 22 13 20  7 21  2 11  6  1 26 10 40 28 14  5 29 34 19 37 17 
38  3 37 15 11 14  7  8  5  9 41  1  2 24 16  6 12 42 33 27 21 30 32 35 23 25  4 20 18 26 40 36 31 10 22 17 39 28 13 29 34 19 
26 21  2  1 12 18 25 40  4 16 17 13 33  9 22 30 27 29  3 34 35 36 15 23 24 37 14 19  7 41  5 39  6 38 11 42 10  8 31 28 32 20 
19 31 20  3  1 42 11 36 30 10 13  7 14 26  5  8 22 35 34 12 25 15 33 38 28 23 17 32 27 39 21 29 41  9  6  2 24 16  4 37 18 40 
42 10  3 22 35  7 13  5 41 33 34 18 39 28 15 38 14 30  1 37 26  4 16  6 32 40  2  8 31 11 25 12 27 17 19 20 29 24 21  9 23 36 
10  1 14  6 13 31 23 15 37 18 28 41 12 42 34  4 39  7 22 17 40  5 19  9 29  8 16 33 26 25 32 24  3 20 35 36 11  2 38 27 21 30 
23 25 11 28 39 24  5 18 35 27 10 15 16 22  4 40 42 21 12 30 38 20 37  2  3  1 31 13 14 17  8 19 29 36  9 34  7  6 26 32 33 41 
39 36 31 38 20 12  9 37 24 15  7 25 34 21 28 19 41 26  5 22 27 18  8  4 16 17 11 30 23 35  6  1 32  2 13 40 14 42 33 10 29  3 
14  5 27 40 34 41 19 21 22 31  8 37 42 17 29 23  9 12 20 25  1  3 26 33 13 38  6  4 30 28 39 10 35 24  2  7 15 18 36 16 11 32 
27 17 34  2 22 11  4 42 12 19 26  9 41 37 36 28 31 15 21  6 32 40 18 30 33 20 29  3 39 38 35 14 23 13 16  8 25  7 10  5  1 24 
21 14 13 25 27 34 15 20 42  3  4 28 18  5 23  2 30 22 16 26 29 24 41 12 37  6 40  7  9 32 17 38 36 35 10 31  1 19 39 33  8 11 
28 23 33  9 30  8 12  3 17  6 18 39 38 36  7 42 16 19 24 32 11  1 10 25 35  2 27 31  5 14 13 34 40 37 15 22 20 41 29 21 26  4 
34 19 42 27 31 21  8 16  7 24 14 35 36 38 20 39 33 41 37 40 13  2  4  1 26 15 12 10 28 23 29  6  9 22  3 32 30 25  5 11 17 18 
18 11 15 20 36 28 32 26 19 39 42 40 31 25 35 22  8 34  7 38 33 17  6 21 14  9  5 23  4  3 16 30 24 29  1 41 37 13 27  2 10 12 
33 41  5 35 18 13 10 19 23  8 31 24 32 20 30 15 28 11 42  1 17  6 27 29  7  4 38 26  3 22 37  2 34 39 25 21 36  9 40 12 14 16 
 8 38 26 21 24 30 35 11 14 29  3 10 40 32  6 16  5 31 23  2 39 33 12 42 41 34  1 17 25  4 15 18 20 19 36 37 28 27  7 13  9 22 
22  6 38 16 33 27 29 28 26  4 21 20 23 34 40  9  1 36 10 11 19  8 35 39 25 31 15 42 17  7  3 37 30  5 14 24 41 12 32 18  2 13 
20 34 24 12 32 17 37 14  1 22 25 27 26  7 18 21 11  8  9 29 10 13 39 28  2 35 36  5 19 31 42 15 33  4 40  3 16 30 41  6 38 23 
 2 28 16 37 29 40 39 34  6 17 35 38  7 33 32 14  4 27  8 19 20 10  3 41 31 36 24 12 42  5 30 11 18 26 21 15 13 23  9  1 22 25 
30 42  1 31 41 36  2 39 21 38  9 19 13  6 27 35 37 17 40  3  4 22 11  8 12 18 25 15 34 29  7 23 14 28 24 33 32 10 16 26 20  5 
36 35 17  8  7  4 41 30 28 26 33 22 29 39 38 31 18 10  2 20 37 19 13 24 42  5  9 34 16 40 11 25 21 14 32 27  6  3  1 23 12 15 
 9 13  7 18  6  2 26 38 27  1 29 34 37 12 25 32  3  4 28 35 23 41 30 40 39 24 10 11  8 15 22 20 16 21 42  5 17 31 14 36 19 33 
32 15 22 36 38 23 24 10 11 35 20 17 21 19  8 26  2  6 30 42 12  9 34  5  1 14 28 37 29 13 18  7 39 25 33 16 31  4  3 41 40 27 
 7  8 28 17 10 29 20 12 34 21 16 14  3  2  9  5  6 39 25  4 36 27  1 11 18 32 13 22 40 19 26 33 38 41 31 23 42 15 35 30 24 37 
35  4 19 11 17 16 33  2 36 40 38 26  8 10 39 18 21 32 41 24  5 34  7 31 30 22 42  1 13 27 20  3 28 23 12  6  9 37 25 14 15 29 
 3 32 30 14 40 39 27  7 29 23 37 42  9 15 12 11 25  1 13 33  6 16  2 18  5 41 19 36 20 24 34  4 22 31  8 10 21 26 17 38 28 35 


Part 4: 1000 latin squares of order 256:

Generated in 1.306 s.</pre>

=={{header|Phix}}==
{{trans|Go}}
<!--<syntaxhighlight lang="phix">(phixonline)-->
 <span style="color: #008080;">function</span> <span style="color: #000000;">shuffleCube</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">integer</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">rx</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">ry</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">rz</span>
     <span style="color: #004080;">bool</span> <span style="color: #000000;">bProper</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">true</span>
     <span style="color: #008080;">while</span> <span style="color: #004600;">true</span> <span style="color: #008080;">do</span>
         <span style="color: #000000;">rx</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">rand</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
         <span style="color: #000000;">ry</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">rand</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
         <span style="color: #000000;">rz</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">rand</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
         <span style="color: #008080;">if</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">rx</span><span style="color: #0000FF;">][</span><span style="color: #000000;">ry</span><span style="color: #0000FF;">][</span><span style="color: #000000;">rz</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">==</span> <span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
     <span style="color: #008080;">while</span> <span style="color: #004600;">true</span> <span style="color: #008080;">do</span>
         <span style="color: #004080;">integer</span> <span style="color: #000000;">ox</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">oy</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">oz</span>
         <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
             <span style="color: #000000;">ox</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span>
             <span style="color: #008080;">if</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">ox</span><span style="color: #0000FF;">][</span><span style="color: #000000;">ry</span><span style="color: #0000FF;">][</span><span style="color: #000000;">rz</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">==</span> <span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
         <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
         <span style="color: #008080;">if</span> <span style="color: #008080;">not</span> <span style="color: #000000;">bProper</span> <span style="color: #008080;">and</span> <span style="color: #7060A8;">rand</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)==</span><span style="color: #000000;">2</span> <span style="color: #008080;">then</span>
             <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">ox</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
                 <span style="color: #000000;">ox</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span>
                 <span style="color: #008080;">if</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">ox</span><span style="color: #0000FF;">][</span><span style="color: #000000;">ry</span><span style="color: #0000FF;">][</span><span style="color: #000000;">rz</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">==</span> <span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
             <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
         <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
         <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
             <span style="color: #000000;">oy</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span>
             <span style="color: #008080;">if</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">rx</span><span style="color: #0000FF;">][</span><span style="color: #000000;">oy</span><span style="color: #0000FF;">][</span><span style="color: #000000;">rz</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">==</span> <span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
         <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
         <span style="color: #008080;">if</span> <span style="color: #008080;">not</span> <span style="color: #000000;">bProper</span> <span style="color: #008080;">and</span> <span style="color: #7060A8;">rand</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)==</span><span style="color: #000000;">2</span> <span style="color: #008080;">then</span>
             <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">oy</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
                 <span style="color: #000000;">oy</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span>
                 <span style="color: #008080;">if</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">rx</span><span style="color: #0000FF;">][</span><span style="color: #000000;">oy</span><span style="color: #0000FF;">][</span><span style="color: #000000;">rz</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">==</span> <span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> 
             <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
         <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
         <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
             <span style="color: #000000;">oz</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span>
             <span style="color: #008080;">if</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">rx</span><span style="color: #0000FF;">][</span><span style="color: #000000;">ry</span><span style="color: #0000FF;">][</span><span style="color: #000000;">oz</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">==</span> <span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
         <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
         <span style="color: #008080;">if</span> <span style="color: #008080;">not</span> <span style="color: #000000;">bProper</span> <span style="color: #008080;">and</span> <span style="color: #7060A8;">rand</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)==</span><span style="color: #000000;">2</span> <span style="color: #008080;">then</span>
             <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">oz</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
                 <span style="color: #000000;">oz</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span>
                 <span style="color: #008080;">if</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">rx</span><span style="color: #0000FF;">][</span><span style="color: #000000;">ry</span><span style="color: #0000FF;">][</span><span style="color: #000000;">oz</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">==</span> <span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
             <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
         <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
  
         <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">rx</span><span style="color: #0000FF;">][</span><span style="color: #000000;">ry</span><span style="color: #0000FF;">][</span><span style="color: #000000;">rz</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
         <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">rx</span><span style="color: #0000FF;">][</span><span style="color: #000000;">oy</span><span style="color: #0000FF;">][</span><span style="color: #000000;">oz</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
         <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">ox</span><span style="color: #0000FF;">][</span><span style="color: #000000;">ry</span><span style="color: #0000FF;">][</span><span style="color: #000000;">oz</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
         <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">ox</span><span style="color: #0000FF;">][</span><span style="color: #000000;">oy</span><span style="color: #0000FF;">][</span><span style="color: #000000;">rz</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
  
         <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">rx</span><span style="color: #0000FF;">][</span><span style="color: #000000;">ry</span><span style="color: #0000FF;">][</span><span style="color: #000000;">oz</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">1</span>
         <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">rx</span><span style="color: #0000FF;">][</span><span style="color: #000000;">oy</span><span style="color: #0000FF;">][</span><span style="color: #000000;">rz</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">1</span>
         <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">ox</span><span style="color: #0000FF;">][</span><span style="color: #000000;">ry</span><span style="color: #0000FF;">][</span><span style="color: #000000;">rz</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">1</span>
         <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">ox</span><span style="color: #0000FF;">][</span><span style="color: #000000;">oy</span><span style="color: #0000FF;">][</span><span style="color: #000000;">oz</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">1</span>
  
         <span style="color: #008080;">if</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">ox</span><span style="color: #0000FF;">][</span><span style="color: #000000;">oy</span><span style="color: #0000FF;">][</span><span style="color: #000000;">oz</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;"><</span> <span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
             <span style="color: #0000FF;">{</span><span style="color: #000000;">rx</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">ry</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">rz</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">ox</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">oy</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">oz</span><span style="color: #0000FF;">}</span>
             <span style="color: #000000;">bProper</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">false</span>
         <span style="color: #008080;">else</span>
             <span style="color: #000000;">bProper</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">true</span>
             <span style="color: #008080;">exit</span>
         <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">c</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #008080;">function</span> <span style="color: #000000;">toMatrix</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">integer</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">sequence</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">),</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
         <span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
             <span style="color: #008080;">for</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
                 <span style="color: #008080;">if</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">][</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">!=</span> <span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
                     <span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">k</span>
                     <span style="color: #008080;">exit</span>
                 <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
             <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
         <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">m</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #008080;">function</span> <span style="color: #000000;">toReduced</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">integer</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">:=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
     <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">deep_copy</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
         <span style="color: #008080;">if</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]!=</span><span style="color: #000000;">j</span> <span style="color: #008080;">then</span>
             <span style="color: #008080;">for</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">=</span><span style="color: #000000;">j</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
                 <span style="color: #008080;">if</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">][</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]==</span><span style="color: #000000;">j</span> <span style="color: #008080;">then</span>
                     <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
                         <span style="color: #004080;">atom</span> <span style="color: #000000;">mij</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">],</span>
                              <span style="color: #000000;">mik</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]</span>
                         <span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">mik</span>
                         <span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">mij</span>
 
                     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
                     <span style="color: #008080;">exit</span>
                 <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
             <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
         <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
         <span style="color: #008080;">if</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]!=</span><span style="color: #000000;">i</span> <span style="color: #008080;">then</span>
             <span style="color: #008080;">for</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">=</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
                 <span style="color: #008080;">if</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">][</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]==</span><span style="color: #000000;">i</span> <span style="color: #008080;">then</span>
                     <span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
                         <span style="color: #004080;">atom</span> <span style="color: #000000;">mij</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">],</span>
                              <span style="color: #000000;">mkj</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span>
                         <span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">mkj</span>
                         <span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">mij</span>
                     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
                     <span style="color: #008080;">exit</span>
                 <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
             <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
         <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">m</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #008080;">function</span> <span style="color: #000000;">makeCube</span><span style="color: #0000FF;">(</span><span style="color: #004080;">object</span> <span style="color: #000000;">orig</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">sequence</span> <span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">),</span><span style="color: #000000;">n</span><span style="color: #0000FF;">),</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
         <span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
             <span style="color: #004080;">integer</span> <span style="color: #000000;">k</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">orig</span><span style="color: #0000FF;">==</span><span style="color: #004600;">NULL</span><span style="color: #0000FF;">?</span><span style="color: #7060A8;">mod</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">j</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">:</span><span style="color: #000000;">orig</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">])</span>
             <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">][</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span>
         <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">c</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #008080;">procedure</span> <span style="color: #000000;">main</span><span style="color: #0000FF;">()</span>
  
     <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Part 1: 10,000 latin Squares of order 4 in reduced form:\n\n"</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">sequence</span> <span style="color: #000000;">orig</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">},</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">},</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">},</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">4</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">}},</span>
              <span style="color: #000000;">c</span> <span style="color: #0000FF;">:=</span> <span style="color: #000000;">makeCube</span><span style="color: #0000FF;">(</span><span style="color: #000000;">orig</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">rm</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">fk</span>
     <span style="color: #004080;">integer</span> <span style="color: #000000;">freq</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">new_dict</span><span style="color: #0000FF;">()</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">10000</span> <span style="color: #008080;">do</span>
         <span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">shuffleCube</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">)</span>
         <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">toMatrix</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">)</span>
         <span style="color: #000000;">rm</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">toReduced</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
         <span style="color: #7060A8;">setd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rm</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">getd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">freq</span><span style="color: #0000FF;">)+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">freq</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #000000;">fk</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">getd_all_keys</span><span style="color: #0000FF;">(</span><span style="color: #000000;">freq</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fk</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
         <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%v occurs %d times\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">fk</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #7060A8;">getd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fk</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">freq</span><span style="color: #0000FF;">)})</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
  
     <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\nPart 2: 10,000 latin squares of order 5 in reduced form:\n\n"</span><span style="color: #0000FF;">)</span>
     <span style="color: #000000;">orig</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">},</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">},</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">},</span>
             <span style="color: #0000FF;">{</span><span style="color: #000000;">4</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">},</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">5</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">}}</span>
     <span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">makeCube</span><span style="color: #0000FF;">(</span><span style="color: #000000;">orig</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">)</span>
     <span style="color: #7060A8;">destroy_dict</span><span style="color: #0000FF;">(</span><span style="color: #000000;">freq</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">justclear</span><span style="color: #0000FF;">:=</span><span style="color: #004600;">true</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">10000</span> <span style="color: #008080;">do</span>
         <span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">shuffleCube</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">)</span>
         <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">toMatrix</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">)</span>
         <span style="color: #000000;">rm</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">toReduced</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
         <span style="color: #7060A8;">setd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rm</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">getd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">freq</span><span style="color: #0000FF;">)+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">freq</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #000000;">fk</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">getd_all_keys</span><span style="color: #0000FF;">(</span><span style="color: #000000;">freq</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fk</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
         <span style="color: #000000;">fk</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"%2d(%3d)"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">i</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">getd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fk</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">freq</span><span style="color: #0000FF;">)})</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #7060A8;">puts</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">join_by</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fk</span><span style="color: #0000FF;">,</span><span style="color: #000000;">8</span><span style="color: #0000FF;">,</span><span style="color: #000000;">7</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" "</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\n"</span><span style="color: #0000FF;">))</span>
     <span style="color: #7060A8;">destroy_dict</span><span style="color: #0000FF;">(</span><span style="color: #000000;">freq</span><span style="color: #0000FF;">)</span>
  
     <span style="color: #000080;font-style:italic;">-- part 3</span>
     <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\nPart 3: 750 latin squares of order 42, showing the last one:\n\n"</span><span style="color: #0000FF;">)</span>
     <span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">makeCube</span><span style="color: #0000FF;">(</span><span style="color: #004600;">NULL</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">42</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">750</span> <span style="color: #008080;">do</span>
         <span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">shuffleCube</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">toMatrix</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">integer</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">:=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
         <span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
             <span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"%2d"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j</span><span style="color: #0000FF;">])</span>
         <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
         <span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">join</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #008000;">" "</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%s\n"</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">join</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\n"</span><span style="color: #0000FF;">))</span>
  
     <span style="color: #000080;font-style:italic;">-- part 4</span>
     <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\nPART 4: 1000 latin squares of order 256:\n\n"</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">atom</span> <span style="color: #000000;">t0</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()</span>
     <span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">makeCube</span><span style="color: #0000FF;">(</span><span style="color: #004600;">NULL</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">256</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">1000</span> <span style="color: #008080;">do</span>
         <span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">shuffleCube</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Generated in %s\n"</span><span style="color: #0000FF;">,</span> <span style="color: #7060A8;">elapsed</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t0</span><span style="color: #0000FF;">))</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
 <span style="color: #000000;">main</span><span style="color: #0000FF;">()</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
Part 1: 10,000 latin Squares of order 4 in reduced form:

{{1,2,3,4},{2,1,4,3},{3,4,1,2},{4,3,2,1}} occurs 2503 times
{{1,2,3,4},{2,1,4,3},{3,4,2,1},{4,3,1,2}} occurs 2560 times
{{1,2,3,4},{2,3,4,1},{3,4,1,2},{4,1,2,3}} occurs 2510 times
{{1,2,3,4},{2,4,1,3},{3,1,4,2},{4,3,2,1}} occurs 2427 times

Part 2: 10,000 latin squares of order 5 in reduced form:

 1(172)  9(197) 17(228) 25(166) 33(171) 41(224) 49(171)
 2(168) 10(162) 18(216) 26(227) 34(172) 42(155) 50(226)
 3(159) 11(198) 19(206) 27(165) 35(189) 43(190) 51(174)
 4(170) 12(207) 20(159) 28(166) 36(177) 44(171) 52(196)
 5(211) 13(148) 21(172) 29(173) 37(183) 45(189) 53(197)
 6(169) 14(163) 22(128) 30(179) 38(184) 46(138) 54(173)
 7(168) 15(155) 23(146) 31(170) 39(187) 47(170) 55(206)
 8(193) 16(177) 24(146) 32(176) 40(157) 48(183) 56(177)

Part 3: 750 latin squares of order 42, showing the last one:

 5 29 15  7 25 26  2 35 21 39  8 12 17 31  3 20 23 22 40 34 13 32 27 38  9  6 36 41 11 19  4 42 10 28 33 18 30 16  1 14 37 24
34 17 22 12 38 28 20 42 15 10  4  3 30 16 35 23 11 19 31  8 32  1 33 36 24  2 18 39  9 41 40 26 25 27 29  5  7 37 21 13  6 14
23 14 41 38  2 36  4 34 29 16 11 10 24 13 26 31 30 12 28 18  7 21 40 42 27  9 37 35  1  3 17 22 20  5  6 33 32 39 25 19 15  8
29 21 27 41  3 10 12 23  4 18 39  1 11  6 20 34  2 35 36 37 40  5 14 26 17 42 24 33 32 16 28  8 13 30 15  9 25 19 38  7 31 22
 8 32 10 17 30 15 18 13 19  6 26 29 34 42 28 40 24 23 33  7  3  4 12 37 38 36  1 21 41 20 16 25  5 11  2 39 14 22 31 35  9 27
27 40 39 16 11 23 14 20  6  4 19 28 36 12 31 24 42 10 35 33 17 18 30  3 21  5 38 15  7  1  9 34  8 32 37 13  2 26 29 22 25 41
31 39 29 22 20  6 11 17 16 19 41 36 35 33 30 14  4  2 15 24 21 10 25  1 18 12 40 28  5 37 32 27  3 13 42 38  9 34 26  8  7 23
11 33 42 28 14  7  6 24 37 26 13 35  9  5 19 18 15 20 25 41 30 17  3 12 22  8 21 27 39 10 34 40 32 36  4 31 23 29  2 38 16  1
20 11  7  8 32 31 40 37 42 13 21 22 26  2 12 29  1 27  6 14 19 41 38 17 36 25  4  5 30 15 24 35 16 34 39  3 28 23  9 18 33 10
24  9 28 40 33 29  3  7 34 11 16 27  2 30 42 25 21 13 41 10 38  8 39 35 12 26 19 20 23 31  5 32  1 22 14  6  4 15 37 36 17 18
 2  8 23 37 27  9 38 36 13 24 31 14 29  7  6 42  3 34 18 32  1 20 22 41 25 30 33 16 15  4 11 10 26 39 21 28 17 40 19  5 12 35
13  2 26 15 10 40 39  6 33 29 42 34 12 17 11 28 22 32 14 25 24 37 21  5  8 23 30  9 18  7 41 31  4  3 27 19 16 35 20  1 38 36
22 23 34 31 28 25 36 38  9 32 30  8  3 11 17 41 26 39 24  6  2 35 13  4  7 21 29 18 14 27 19 37 15 20 16 12 10 33 40 42  1  5
36 28 20 11 29 39 22 41 35  7  5 15 31 24  8 19 27 37  1 38 16 13  6  2 32 40 14 25 33 17 21  4 34 23 30 10 18 42 12  9 26  3
 6 25  8  2 17 33 19 12  1 38 40 39  5 32 18  7 34 30  9 11 15  3 31 23 37 24 27 14 20 28 36 16 21 42 13 29 41  4 35 10 22 26
14 24 38 32 12  3 15  2 17 28 36 40 19 26  1 27 29 41  8  5 23 42 20 13 10 34  6 31 16 35 30  7 11 18 22 21 33 25  4 37 39  9
39 30  5 20  1 22  9 40 36 27  7 33 37 18 29 38 25 42  4 21 14 31 10 28 26 15 16  8  3 13 35 19 41  2 32 24 12 11 17 23 34  6
35 18 17 14 13 41 25 31  2  3 32 24 10 19 22 33  6  1 16 23  9 15  8 39  5  7 11 12 42 34 37 28 38  4 26 20 40 36 27 21 30 29
 9 19 24 26 42 16  7 30 10 40 29  4 33  8 38 22 14 25 37 28  5 27 41 32  1 13 17 36 34 39 23 11 31  6 35  2 20 21 18 15  3 12
12 22 37  1  4 20 32  3 30 25 28 26  6 14 36 11 39 21 38 29 27 24  7 16 15 31  9 34 10 33 13 18 40 35  5 17 19  8 42 41 23  2
 3 16 31 42  7 17 37 25 23 36 15 18 27 22  5 21 40  9 10 39  4 26 29  6  2 33 41 19 35  8 12 20 28 38 24 32 11  1 34 30 14 13
19 41 36 34 21 18 26 29 27 20 14 16 38 40  7 15 32  3 17  4 10 28 35 33 13 22  8  6 25 42 31 23  2 37  9 30  1 12  5 24 11 39
25  4 12 29 26 37 16  9 22 30  6 23 40 21 15 35 20 38 19 42 11  2  1 18  3 41  5 10 28 36 33 39 27 24 34  8 31 32 14 17 13  7
41 12 14 33 40 35 28 15  7  9  1  5 13 23 27 32  8 17 26 31 42 34 37 19 30 38 20 22  2  6 39 21 36 29 18 16  3 24 11  4 10 25
26 20  3 19 16 30  5 14  8 41 10  7 25 15 21 13 38 36 39 22 28 23 17 27 33 37 34 32  4  2 29 12  9 31  1 42 24 18  6 40 35 11
10  1 25 36 37 24  8 26  3 12 34 42 18 38 41 16  9 14 32 35 31 30  5 22 39 27  7  4 13 29  6 15 23 19 28 11 21  2 33 20 40 17
37 35 40 13 39  8 31 33 38 15 12 32 16 41 34  6  5 11 30 27 20 22 26 14 29 18 28 23 36 21 25  2  7  1 17  4 42  9 10  3 24 19
30 34  2 24 35  1 23 10 20 42 22 37 15 39  9 17 12  4  5 26 18 38 16 29 31  3 25 11 21 14  8 41  6 40 19  7 13 27 28 32 36 33
16  7 19 21 18 27 29 22 39 35  2 38 28 20 40  9 36  8 12  1 41 33 15 31 11 10 42 24  6 32 26 17 37 14 25 23  5 13  3 34  4 30
32  3 11 25  5 12  1  4 18 31 33 19 41  9 37 10  7 24 13 40  6 16 42 21 34 20 26  2 38 22 15 14 35 17 23 36  8 30 39 27 29 28
 1 13 30 39 36  4 34 32 12 14 17  6 23 27 24  3 41 40 11 20 22  9 28 15 42 16  2 29 31  5  7 33 19 21 10 35 26 38  8 25 18 37
18 38  4 23 41 19 35 21 26 33 37 20 42 28 13  5 10  7  3 15 25 39 32  9 14 17 31 40 29 24  1 36 30  8 12 34 27  6 22 11  2 16
 4 27 21  3  8 42 41 16 40 37 18  2 22 25 32 36 17  5 23 30 29  6  9 34 19 35 15 13 24 11 14  1 12 10 38 26 39 20  7 33 28 31
40 26  9 30  6 21 42 19  5  2  3 31  4 35 23 37 28 15 20 13 34 12 11  8 16 14 39 17 22 25 27 38 18 33  7  1 36 10 24 29 41 32
28 15  1  4 19 11 24  5 31  8 23 17 21 34 14 26 37 18  7  2 35 29 36 10  6 39 32 30 27 38  3  9 33 16 20 25 22 41 13 12 42 40
21 10 35 27 31  2 13 39 28  5  9 41  1 36  4  8 19 29 34 16 33 40 24 25 20 11 22  7 12 18 42 30 14 26  3 37 15 17 23  6 32 38
17 42 18  6 23  5 33  1 24 34 35 30  7 37 16 12 31 26 21 19 39 14  4 11 41 32 10  3 40  9 38 13 22 25 36 27 29 28 15  2  8 20
42  6 13 35 22 32 10  8 14 21 24 11 39  1  2  4 18 33 27  9 12 25 23 40 28 29  3 26 37 30 20  5 17 41 31 15 38  7 36 16 19 34
 7 36 16  5  9 34 21 11 32 22 20 25  8 10 33 30 35 31 29 12 26 19  2 24  4  1 13 38 17 23 18  6 39 15 40 14 37  3 41 28 27 42
15 37 32  9 24 38 27 28 41 17 25 13 20 29 10 39 33  6  2 36  8  7 18 30 35  4 23  1 19 26 22  3 42 12 11 40 34 14 16 31  5 21
33 31  6 18 34 14 17 27 25  1 38 21 32  4 39  2 13 16 42  3 36 11 19  7 23 28 12 37  8 40 10 29 24  9 41 22 35  5 30 26 20 15
38  5 33 10 15 13 30 18 11 23 27  9 14  3 25  1 16 28 22 17 37 36 34 20 40 19 35 42 26 12  2 24 29  7  8 41  6 31 32 39 21  4

PART 4: 1000 latin squares of order 256:

Generated in 19.5s
</pre>
Unfortunately the last part of this task exposes the relatively poor performance of subscripting in phix.

=={{header|Raku}}==
{{trans|Go}}
<syntaxhighlight lang="raku" line># 20210729 Raku programming solution 

#!/usr/bin/env raku

sub makeCube(\from, Int \n) {  
   my @c = [[[ 0 xx n ] xx n ] xx n ]; 
   from.Bool ?? do race for ^n X ^n -> (\i,\j) { @c[i;j; { from[i;j]-1 } ] = 1 }
             !! do race for ^n X ^n -> (\i,\j) { @c[i;j; {   (i+j)%n   } ] = 1 }
   return @c                   
}

sub shuffleCube(@c) {
   my ($rx, $ry, $rz); my \n = +@c; my Bool \proper = $ = True;

   repeat { ($rx ,$ry, $rz) = (^n).roll: 3 } until @c[$rx;$ry;$rz] == 0;
   loop { 
      my ($ox, $oy, $oz); 
      for ^n { last if @c[ $ox = $_ ;$ry;$rz] == 1 }
      if !proper and (^3).roll==0 { 
         for $ox^…^n { last if @c[ $ox = $_ ;$ry;$rz] == 1 } 
      }
      for ^n { last if @c[$rx; $oy = $_ ;$rz] == 1 }
      if !proper and (^3).roll==0 {
         for $oy^…^n { last if @c[$rx; $oy = $_ ;$rz] == 1 }
      } 
      for ^n { last if @c[$rx;$ry;  $oz = $_ ] == 1 }
      if !proper and (^3).roll==0 {
         for $oz^…^n { last if @c[$rx;$ry; $oz = $_ ] == 1 }
      }

      (@c[$rx;$ry;$rz],@c[$rx;$oy;$oz],@c[$ox;$ry;$oz],@c[$ox;$oy;$rz]) »+=»1;
      (@c[$rx;$ry;$oz],@c[$rx;$oy;$rz],@c[$ox;$ry;$rz],@c[$ox;$oy;$oz]) »-=»1; 
      
      @c[$ox;$oy;$oz] < 0 ?? (($rx,$ry,$rz) = ($ox,$oy,$oz)) !! last ;
      proper = False 
   }        
}

sub toMatrix(@c) {
   my \n = +@c;  
   my @m = [[0 xx n] xx n]; 
   for ^n X ^n -> (\i,\j) { 
      for ^n -> \k { if @c[i;j;k] != 0 { @m[i;j] = k and last } }
   }
   return @m
}

sub toReduced(@m is copy) { 
   my \n = +@m; 
   for 0…(n-2) -> \j {
      if ( @m[0;j] != j ) { 
         for j^…^n -> \k {
            if ( @m[0;k] == j ) { 
               for 0…^n -> \i { (@m[i;j], @m[i;k]) = (@m[i;k], @m[i;j]) }
               last 
            } 
         } 
      } 
   }   
   for 1…(n-2) -> \i {
      if ( @m[i;0] != i ) {
         for i^…^n -> \k { 
            if ( @m[k;0] == i ) { 
               for 0…^n -> \j { (@m[i;j], @m[k;j]) = (@m[k;j], @m[i;j]) }     
               last      
            }
         }
      } 
   } 
   return @m
}

sub printAs1based { say ($_ »+» 1).Str for @_.rotor: @_.elems.sqrt }

my (%freq, @c, @in);

say "Part 1: 10,000 latin Squares of order 4 in reduced form:\n";
@in = [[1, 2, 3, 4], [2, 1, 4, 3], [3, 4, 1, 2], [4, 3, 2, 1]];
@c = makeCube(@in, 4); 
for ^10_000 {
   shuffleCube @c ; 
   %freq{@c.&toMatrix.&toReduced».List.flat.Str}++
}    
for %freq.kv -> $k, $v { 
   printAs1based $k.split(' ');   
   say "\nOccurs $v times.\n"
}

say "Part 2: 10,000 latin Squares of order 5 in reduced form:\n";
@in = [ [1,2,3,4,5], [2,3,4,5,1], [3,4,5,1,2], [4,5,1,2,3], [5,1,2,3,4] ];
%freq = ();
@c = makeCube(@in, 5);
for ^10_000 { 
   shuffleCube @c ; 
   %freq{@c.&toMatrix.&toReduced».List.flat.Str}++ 
}
for %freq.values.kv -> $i, $j { printf "%2d(%3d)%s", $i+1, $j, ' ' }

say "\n\nPart 3: 750 latin squares of order 42, showing the last one:\n";
@c = makeCube([], 42); # (1..42).pick(*)
( for ^750 { shuffleCube @c } ) and printAs1based @c.&toMatrix».List.flat ;

say "\nPart 4: 100 latin squares of order 256:\n";
my $snapshot = now;
@c = makeCube([], 256);
for ^100 { shuffleCube @c } # without hyper, will do only 100 cycles 
say "Generated in { now - $snapshot } seconds."
</syntaxhighlight>
{{out}}
<pre>
Part 1: 10,000 latin Squares of order 4 in reduced form:

1 2 3 4
2 1 4 3
3 4 1 2
4 3 2 1

Occurs 2442 times.

1 2 3 4
2 4 1 3
3 1 4 2
4 3 2 1

Occurs 2705 times.

1 2 3 4
2 3 4 1
3 4 1 2
4 1 2 3

Occurs 2548 times.

1 2 3 4
2 1 4 3
3 4 2 1
4 3 1 2

Occurs 2305 times.

Part 2: 10,000 latin Squares of order 5 in reduced form:

 1(210)  2(191)  3(186)  4(158)  5(219)  6(164)  7(147)  8(160)  9(196) 10(188) 11(193) 12(168) 13(195) 14(173) 15(151) 16(184) 17(211) 18(171) 19(185) 20(155) 21(157) 22(191) 23(195) 24(177) 25(157) 26(191) 27(165) 28(178) 29(191) 30(180) 31(193) 32(176) 33(196) 34(178) 35(156) 36(168) 37(155) 38(152) 39(155) 40(223) 41(159) 42(165) 43(210) 44(175) 45(195) 46(188) 47(178) 48(154) 49(172) 50(176) 51(178) 52(164) 53(175) 54(196) 55(187) 56(189) 

Part 3: 750 latin squares of order 42, showing the last one:

26 27 4 41 31 10 28 13 3 29 21 42 14 39 19 8 5 7 6 40 17 37 35 33 1 38 22 34 18 11 12 2 32 36 9 24 25 23 16 15 30 20
16 40 6 11 22 14 41 42 20 7 24 8 29 27 28 4 26 19 21 39 37 9 2 32 36 15 12 13 38 5 31 33 34 3 17 1 23 35 30 18 10 25
31 15 11 3 5 22 13 10 14 24 41 18 6 16 17 29 36 37 12 28 23 39 34 8 27 40 4 35 7 19 32 26 21 1 2 33 20 9 38 30 25 42
13 37 31 38 42 9 11 2 10 36 22 34 18 23 12 39 25 20 19 26 8 41 16 6 21 32 5 4 33 17 40 24 30 29 27 3 28 14 1 7 35 15
41 7 17 28 24 12 29 8 31 22 1 3 11 42 23 26 39 13 25 19 36 18 33 37 6 5 20 15 2 16 14 4 27 35 38 10 30 32 9 21 34 40
5 19 12 32 20 30 6 33 17 16 27 13 8 9 2 37 21 18 39 4 7 40 1 15 3 35 11 24 10 31 36 23 29 41 22 25 26 28 34 14 42 38
11 14 20 5 3 32 19 27 42 18 12 41 34 6 24 21 38 26 10 35 33 28 9 22 23 39 17 40 31 15 30 37 16 2 36 4 8 1 13 25 29 7
20 6 22 16 34 26 33 23 2 11 29 7 4 13 21 42 9 24 14 41 15 31 5 3 25 18 19 30 1 10 37 36 17 12 39 27 40 38 28 32 8 35
4 25 32 8 28 17 40 19 22 15 23 2 35 5 34 16 18 39 38 24 30 13 41 1 20 3 36 33 42 12 26 29 10 31 6 21 14 37 7 9 11 27
21 2 35 39 26 28 36 16 7 37 19 32 33 1 38 27 4 8 23 6 5 34 40 30 29 42 14 10 9 22 11 12 25 15 13 31 17 18 41 24 20 3
30 28 3 12 6 8 31 32 40 42 25 38 9 33 16 14 13 27 26 17 4 19 20 24 15 36 37 18 11 23 10 22 1 39 21 5 34 41 2 35 7 29
8 34 26 6 16 33 15 28 41 38 40 24 30 21 22 17 20 35 32 23 42 25 11 5 12 1 31 2 13 3 39 27 14 9 18 7 19 29 36 4 37 10
1 22 2 7 39 23 14 18 11 30 15 17 26 10 6 28 16 12 24 31 35 36 37 21 40 29 13 8 3 41 34 38 4 32 25 42 5 27 20 19 33 9
25 32 1 9 35 6 42 34 37 10 13 20 5 19 30 41 17 36 7 15 40 38 26 31 2 23 18 28 24 33 21 11 22 4 14 39 27 8 3 29 16 12
17 12 18 30 11 4 10 5 28 8 33 31 19 22 36 13 6 9 34 42 29 1 27 39 38 21 25 23 26 24 7 15 2 16 41 14 32 40 35 20 3 37
27 36 39 37 18 13 34 14 19 5 32 26 38 12 3 23 1 30 17 11 6 35 21 16 24 41 7 9 28 20 2 31 40 10 4 29 22 25 33 42 15 8
36 24 28 2 32 11 37 12 29 33 16 9 40 3 10 34 7 15 4 27 22 20 25 18 13 26 42 39 17 14 5 8 41 21 23 30 35 31 6 38 19 1
32 30 36 34 14 7 22 9 35 23 6 21 37 2 5 15 31 33 3 16 25 17 4 27 19 13 24 29 40 39 28 1 20 38 10 11 42 12 8 41 26 18
2 41 8 20 29 35 25 4 6 1 17 19 3 18 42 33 12 34 5 32 11 15 30 38 39 14 23 31 21 37 22 9 26 27 16 28 36 7 40 10 13 24
39 1 27 14 2 20 9 26 4 25 18 33 41 28 29 32 34 38 13 7 21 8 10 35 31 17 16 19 23 30 15 5 6 40 37 12 3 36 42 22 24 11
3 13 9 24 15 21 8 41 33 32 20 1 25 40 27 22 29 31 18 36 10 11 17 2 37 28 39 42 14 4 16 35 7 30 34 26 38 6 12 23 5 19
6 33 37 26 30 15 20 35 21 39 14 27 7 4 32 36 2 5 9 18 34 23 22 17 42 24 28 11 8 1 38 3 12 25 31 41 29 10 19 13 40 16
33 16 21 29 40 38 24 7 30 27 11 25 2 32 37 5 35 4 22 9 31 42 18 36 10 34 1 14 12 13 8 41 23 17 3 19 15 20 39 6 28 26
14 17 24 31 19 27 26 6 38 3 9 36 12 41 15 18 37 22 40 33 16 32 29 42 11 4 10 7 34 25 23 28 35 13 20 8 2 5 21 1 39 30
18 31 29 22 37 25 3 24 26 28 8 4 20 36 9 30 33 42 27 38 1 7 13 10 32 11 2 16 19 21 35 14 5 23 40 15 41 39 17 12 6 34
34 9 15 19 23 41 5 39 24 31 26 30 13 25 11 10 40 1 16 22 2 33 28 12 14 37 38 32 29 8 3 7 42 20 35 6 18 4 27 17 21 36
24 42 5 23 10 37 35 17 18 13 7 39 21 29 8 1 32 40 20 14 12 4 15 26 22 25 9 6 41 36 27 34 3 19 28 16 11 30 31 2 38 33
7 23 41 10 17 24 21 22 36 14 30 16 42 34 18 19 11 3 1 37 39 12 38 40 8 33 35 25 4 32 20 6 9 5 15 13 31 26 29 28 27 2
23 18 25 42 27 5 1 38 34 12 31 15 32 20 40 6 19 10 28 30 13 3 8 7 4 2 29 26 36 9 17 21 24 37 33 35 39 22 11 16 14 41
28 11 40 35 36 42 16 25 13 19 4 5 39 26 20 12 15 41 37 34 38 14 31 9 17 7 30 27 6 2 29 10 8 24 1 18 21 33 32 3 22 23
40 38 23 1 21 19 32 29 12 4 2 35 31 11 26 7 28 16 41 10 3 22 24 20 33 27 8 37 15 42 25 39 13 14 30 36 6 34 18 5 9 17
19 26 34 27 8 29 7 20 16 41 36 14 10 15 4 25 3 6 33 5 9 21 23 13 35 30 32 22 37 38 18 17 31 11 12 40 1 42 24 39 2 28
35 29 10 33 13 31 39 1 9 21 38 11 36 30 14 40 42 17 2 20 41 27 6 19 18 22 3 12 5 26 24 16 28 7 8 23 37 15 25 34 4 32
10 35 33 21 4 3 2 30 25 40 39 12 1 37 31 20 24 28 42 8 14 26 32 23 9 16 6 38 22 34 41 19 11 18 29 17 7 13 15 27 36 5
15 5 13 25 9 2 17 40 27 35 42 37 16 8 39 31 41 23 36 1 32 10 7 28 30 19 33 21 20 29 4 18 38 6 11 34 24 3 22 26 12 14
22 39 16 4 41 1 38 11 5 26 10 23 15 14 35 3 8 29 30 13 28 6 12 25 7 20 40 17 32 27 9 42 18 34 19 2 33 24 37 36 31 21
9 3 38 15 25 34 12 21 1 20 5 40 17 24 33 2 27 14 8 29 18 30 19 41 16 10 26 36 35 28 6 32 37 42 7 22 13 11 4 31 23 39
37 10 14 40 12 36 18 3 39 6 35 29 24 38 41 9 23 25 31 21 19 2 42 34 26 8 27 1 30 7 33 13 15 28 32 20 4 16 5 11 17 22
29 20 7 13 38 18 30 37 23 17 34 22 28 31 25 11 14 2 15 12 24 16 36 4 5 9 21 3 27 35 1 40 39 8 42 32 10 19 26 33 41 6
12 21 42 18 7 39 27 36 8 34 37 28 23 17 13 24 30 32 35 3 20 29 14 11 41 31 15 5 16 6 19 25 33 22 26 38 9 2 10 40 1 4
42 4 30 36 1 16 23 15 32 9 3 10 27 35 7 38 22 21 11 2 26 24 39 29 28 6 34 41 25 40 13 20 19 33 5 37 12 17 14 8 18 31
38 8 19 17 33 40 4 31 15 2 28 6 22 7 1 35 10 11 29 25 27 5 3 14 34 12 41 20 39 18 42 30 36 26 24 9 16 21 23 37 32 13

Part 4: 100 latin squares of order 256:

Generated in 76.816295878 seconds.
</pre>

=={{header|Wren}}==
{{trans|Go}}
{{libheader|Wren-fmt}}
{{libheader|Wren-seq}}
<syntaxhighlight lang="wren">import "random" for Random
import "./fmt" for Fmt
import "./seq" for Lst

var rand = Random.new()

var toReduced = Fn.new { |m|
    var n = m.count
    var r = List.filled(n, null)
    for (i in 0...n) r[i] = m[i].toList
    for (j in 0...n-1) {
        if (r[0][j] != j) {
            for (k in j+1...n) {
                if (r[0][k] == j) {
                    for (i in 0...n) r[i].swap(j, k)
                    break
                }
            }
        }
    }
    for (i in 1...n-1) {
        if (r[i][0] != i) {
            for (k in i+1...n) {
                if (r[k][0] == i) {
                    for (j in 0...n) {
                        var t = r[i][j]
                        r[i][j] = r[k][j]
                        r[k][j] = t
                    }
                    break
                }
            }
        }
    }
    return r
}

// 'm' is assumed to be 0 based
var printMatrix = Fn.new { |m|
    var n = m.count
    for (i in 0...n) {
        for (j in 0...n) Fmt.write("$2d ", m[i][j]+1) // back to 1 based
        System.print()
    }
    System.print()
}

// converts 4 x 4 matrix to 'flat' list
var asList16 = Fn.new { |m| Lst.flatten(m) }

// converts 5 x 5 matrix to 'flat' list
var asList25 = Fn.new { |m| Lst.flatten(m) }

// 'a' is assumed to be 0 based
var printList16 = Fn.new { |a|
    for (i in 0...4) {
        for (j in 0...4) {
            var k = i*4 + j
            Fmt.write("$2d ", a[k]+1) // back to 1 based
        }
        System.print()
    }
    System.print()
}

var shuffleCube = Fn.new { |c|
    var n = c[0].count
    var proper = true
    var rx
    var ry
    var rz
    while (true) {
        rx = rand.int(n)
        ry = rand.int(n)
        rz = rand.int(n)
        if (c[rx][ry][rz] == 0) break
    }
    while (true) {
        var ox = 0
        var oy = 0
        var oz = 0
        while (ox < n) {
            if (c[ox][ry][rz] == 1) break
            ox = ox + 1
        }
        if (!proper && rand.int(2) == 0) {
            ox = ox + 1
            while (ox < n) {
                if (c[ox][ry][rz] == 1) break
                ox = ox + 1
            }
        }
        while (oy < n) {
            if (c[rx][oy][rz] == 1) break
            oy = oy + 1
        }
        if (!proper && rand.int(2) == 0) {
            oy = oy + 1
            while (oy < n) {
                if (c[rx][oy][rz] == 1) break
                oy = oy + 1
            }
        }
        while (oz < n) {
            if (c[rx][ry][oz] == 1) break
            oz = oz + 1
        }
        if (!proper && rand.int(2) == 0) {
            oz = oz + 1
            while (oz < n) {
                if (c[rx][ry][oz] == 1) break
                oz = oz + 1
            }
        }

        c[rx][ry][rz] = c[rx][ry][rz] + 1
        c[rx][oy][oz] = c[rx][oy][oz] + 1
        c[ox][ry][oz] = c[ox][ry][oz] + 1
        c[ox][oy][rz] = c[ox][oy][rz] + 1

        c[rx][ry][oz] = c[rx][ry][oz] - 1
        c[rx][oy][rz] = c[rx][oy][rz] - 1
        c[ox][ry][rz] = c[ox][ry][rz] - 1
        c[ox][oy][oz] = c[ox][oy][oz] - 1

        if (c[ox][oy][oz] < 0) {
            rx = ox
            ry = oy
            rz = oz
            proper = false
        } else {
            proper = true
            break
        }
    }
}

var toMatrix = Fn.new { |c|
    var n = c[0].count
    var m = List.filled(n, null)
    for (i in 0...n) m[i] = List.filled(n, 0)
    for (i in 0...n) {
        for (j in 0...n) {
            for (k in 0...n) {
                if (c[i][j][k] != 0) {
                    m[i][j] = k
                    break
                }
            }
        }
    }
    return m
}

// 'from' matrix is assumed to be 1 based
var makeCube = Fn.new { |from, n|
    var c = List.filled(n, null)
    for (i in 0...n) {
        c[i] = List.filled(n, null)
        for (j in 0...n) {
            c[i][j] = List.filled(n, 0)
            var k = (!from) ? (i + j) % n : from[i][j] - 1
            c[i][j][k] = 1
        }
    }
    return c
}

// part 1
System.print("PART 1: 10,000 latin Squares of order 4 in reduced form:\n")
var from = [ [1, 2, 3, 4], [2, 1, 4, 3], [3, 4, 1, 2], [4, 3, 2, 1] ]
var freqs4 = {}
var c = makeCube.call(from, 4)
for (i in 1..10000) {
    shuffleCube.call(c)
    var m = toMatrix.call(c)
    var rm = toReduced.call(m)
    var a16 = asList16.call(rm)
    var a16s = a16.toString // can't use a list as a map key so convert it to string
    freqs4[a16s] = freqs4[a16s] ? freqs4[a16s] + 1 : 1
}
for (me in freqs4) {
    printList16.call(me.key[1..-2].split(", ").map { |n| Num.fromString(n) }.toList)
    Fmt.print("Occurs $d times\n", me.value)
}

// part 2
System.print("\nPART 2: 10,000 latin squares of order 5 in reduced form:")
from = [ [1, 2, 3, 4, 5], [2, 3, 4, 5, 1], [3, 4, 5, 1, 2],
         [4, 5, 1, 2, 3], [5, 1, 2, 3, 4] ]
var freqs5 = {}
c = makeCube.call(from, 5)
for (i in 1..10000) {
    shuffleCube.call(c)
    var m = toMatrix.call(c)
    var rm = toReduced.call(m)
    var a25 = asList25.call(rm)
    var a25s = a25.toString // can't use a list as a map key so convert it to string
    freqs5[a25s] = freqs5[a25s] ? freqs5[a25s] + 1 : 1
}
var count = 0
for (freq in freqs5.values) {
    count = count + 1
    if (count > 1) System.write(", ")
    if ((count-1) % 8 == 0) System.print()
    Fmt.write("$2d($3d)", count, freq)
}
System.print("\n")

// part 3
System.print("\nPART 3: 750 latin squares of order 42, showing the last one:\n")
var m42
c = makeCube.call(null, 42)
for (i in 1..750) {
    shuffleCube.call(c)
    if (i == 750) m42 = toMatrix.call(c)
}
printMatrix.call(m42)

// part 4
System.print("\nPART 4: 1,000 latin squares of order 256:\n")
var start = System.clock
c = makeCube.call(null, 256)
for (i in 1..1000) shuffleCube.call(c)
var elapsed = System.clock - start
Fmt.print("Generated in $s seconds", elapsed)</syntaxhighlight>

{{out}}
Sample run:
<pre>
PART 1: 10,000 latin Squares of order 4 in reduced form:

 1  2  3  4 
 2  1  4  3 
 3  4  2  1 
 4  3  1  2 

Occurs 2510 times

 1  2  3  4 
 2  3  4  1 
 3  4  1  2 
 4  1  2  3 

Occurs 2498 times

 1  2  3  4 
 2  1  4  3 
 3  4  1  2 
 4  3  2  1 

Occurs 2506 times

 1  2  3  4 
 2  4  1  3 
 3  1  4  2 
 4  3  2  1 

Occurs 2486 times


PART 2: 10,000 latin squares of order 5 in reduced form:

 1(187),  2(179),  3(186),  4(176),  5(177),  6(189),  7(193),  8(182), 
 9(168), 10(169), 11(147), 12(200), 13(198), 14(169), 15(200), 16(173), 
17(184), 18(179), 19(151), 20(174), 21(160), 22(198), 23(153), 24(184), 
25(170), 26(180), 27(171), 28(180), 29(184), 30(178), 31(197), 32(173), 
33(185), 34(181), 35(200), 36(188), 37(176), 38(196), 39(193), 40(183), 
41(163), 42(163), 43(173), 44(178), 45(177), 46(160), 47(155), 48(165), 
49(181), 50(188), 51(187), 52(182), 53(162), 54(192), 55(183), 56(180)


PART 3: 750 latin squares of order 42, showing the last one:

22 31 16 23 19 21  1 17  9 20 15  3 13 25  2 38 34 14 41  5 26 24 11 12  6 27  4 33 39  8 40 30 29 36 42 32  7 10 37 18 28 35 
 7 19 17 16 13 20 32 27 39 34 22 26  5 23 31  4 15  3 29 42 10 12  9  6 38  2 24  1 40 21 36 37 30 35 41 28 25 33 18  8 11 14 
21 15 11 40 25 18  4 14 38 13 32  8  2 39 37 10 35 24 31  7  1 16 20 42 34  6  9 28 26 30 23 41 36 33 19 27 29  3  5 17 12 22 
14 32 36 21  7  2 22 41 13 23 27  6 34 12 42 31 37 38 16 19  5 28 17  4 39  9 40  3 11 18 15  1 24 20 29 33 10 35 25 30 26  8 
 9 10 20 37 35 38 40 33 32 12 18 27 19 42  4  1 17 11  7 13 15 14 36  8 25 39 28  6 16  5  2  3 21 23 34 31 41 29 30 26 22 24 
 3 14  5 28 30  6  7 25 10  1 41 36 42  8 40 33 39 32  2 16 27 18 29 15 24 20 17 23 37 11 26  9 38 12 13 19  4 31 22 35 34 21 
33 28  6 18 22 27 30 38 36 39 14 21 37 26 15 12 10 23 32  3 17 29 24 35 41 16  1  8 20 34  9 13 25 11  5 40 31  7 42 19  4  2 
23 17  4 36 11 32 29 20 35  5 16 14 10 31 39  9 40 37 42  1 30 15  6  2 12  8 33 13  7 28 18 38 22 21 26 34 19 24 27 25 41  3 
40 25 13  1 14 24 17 16 42  6 11 28 15 20  8  7 27 35 26 21 41 39 18 31 19  3 34 30 33  9 37 29 12 22 36  5  2 38  4 32 23 10 
26  4  7 10 41 39 16 19  8 24 40 23  1  2  6 34 14  5  3 15 12 13 21 25 30 29 36 18 38 32 28 42  9 31 35 17 37 27 11 22 20 33 
27  5 23 17 21  4  6  9  3  7 25 18 32 14 34 19 33 31 28 12 20 38 15 36  2 22 13 35 41 10 11  8  1 39 30 37 24 26 29 42 16 40 
19  1 21 11 34 22  3 10 26 42 38 12 39 37 14 32 36  9 13  2  7 31 30 24 20 17 29 40 18 41  4 27 23 16  6 35 28 15 33  5  8 25 
18  3 33  7 27 14  9  8 15  2 34 38  4  6 13 40 41 28 17 31 24 26 37 32 29 19 11  5 10 36 42 35 20  1 22 30 12 16 21 23 25 39 
 4  9 41  5 36 34 39 24 25 11 29 31 40  3 32 18  8 17 21 26 14 27 38 10 35 30 19 15  1 23  7 33 28  2 16  6 22 42 13 20 37 12 
15 39 25 19 10  3  8  6 11 14 21  4 20 17 41 30 28 12 37 18 32 40 31 26 13 33  7 42 34 16 27 23  5 38 24 36 35 22  1  2  9 29 
36  8 19 13 16  9 27 11 17 32  1  5 21 41 28 24  3 26 39 33 42  7 10 18 22 25 37  4  6 35 34 20 15 30 40 12 38 14  2 31 29 23 
31 12 27  3 23 13 25 30  2  9  4 35 22 18 24 26 20  6 36 34 33 11 19 17 14  5 42 29  8 37 16 28 41 10 38 21  1 39 32  7 40 15 
35 33 28 20  1 37 26  3 21 27 13 40 38 19 25  2 22  4 12 32  8 23 39 29 11 18  6  7  5 17 30 24 16 41 14 42 36  9 15 34 10 31 
32 13  1 14  9  8 23 34 19 30 35 29 11 33 16 36  2 21 18 37  4 20 12 22  3 38 31 17 42 24  5 25 10 26 39  7 40 28  6 15 27 41 
24  7 26 30 39  5 41 21 12 22  3 15 36 40 33  6 42 29 10 28 11 34 35 19  4  1  8 37 27 14 38 17 31 25  2 13 32 23 20 16 18  9 
30 34 38 35  6 25 15 22 40 16 10 32  3 27 11 20 18 33 19 24 39  8  2  5 23 21 26 36 13  1 41  4 17 42 31 29 14 37 12  9  7 28 
34 21 18 39  8 26 10 32 37 33 12  2 28  4 38 17 24 36 30 25  9 35 40 16 42  7 41 22  3 29  1 11 27  5 15 20 13 19 23 14 31  6 
37 35  8 25 29 10 38 31 23 21 36 13 30 34 26 27 19  2  4  9  6 42 22 11 16 28 32 20 15  7 24  5 40 14 12 41 18  1 39  3 33 17 
10 36 40  4  2  7  5 35  1 41  9 20 31 16 29 23 38 34 33  6 28 21 32 39 27 37 22 26 25 15 13 14  8  3 18 11 30 17 24 12 19 42 
 5 29 37 12 31 16 18 13 33 38 39 22  6 30 19 28 11 40 20 27 35  2 42 14 15 10 23 21  9  4 25  7 34 17 32 24  8 36 41  1  3 26 
29 41 31  9 40  1 20  2  6 28 30 17 23 38 21 42  7 16 11 39 25  4 26 33 18 15 27 14 35 19 22 12 37 24 10  3  5  8 34 13 36 32 
 1 26 32  8 18 17 11 12 29 35  7 42 24  9 20 13  6 22  5 36 31 37 16 41 40 23 10 38  4 33 21 19  2 27 25 14 15 34  3 28 39 30 
12 11 15 41 42 35 36 29 18 40 33 10 17 21 22 39  5  1 34 30 37  3  7 23 31 24  2 19 28 26 14 32  4  8 20 25 16  6  9 27 38 13 
38 22 10 33  4 28 24 36 20 18  8  9 35 13 12 15 32  7 23 17 29  5 25 27  1 41 30 16 14  6 31 26 11 19  3  2 34 21 40 39 42 37 
42 27 39 24  3 11 13 26 41 25 28 37 18 29 35 14  4 19 38 23  2 22  1 20  9 36  5 31 30 12 33  6 32 15  7  8 21 40 16 10 17 34 
28 20  9 26 38 33 19  1 30 36 42 11  7 15 23 37 25 27 24 35 13  6 41 21  5  4 12 34 32 31  8 22 14 29 17 39  3 18 10 40  2 16 
13 37 35 42 33 15 31 28 14  3 19 34 16 32 30 25 23  8 27 11 36  9  5  7 10 12 39 24 29 22  6  2 26 40 21 18 20 41 17  4  1 38 
11  2 34 38 26 23 28  5 24 15 37 30  8 10  7 29 21 41 22 14 40 25 33 13 17 31 16  9 12 42 19 18  3 32 27  4 39 20 36  6 35  1 
17 42 30  2 28 29 33 37 22 31 20  7 14  5  9 35 12 13 40 38 23  1 27 34 32 26 15 41 19  3 39 10 18  4 11 16  6 25  8 21 24 36 
 2 16 42  6 12 36 14 23 34 37 17 25  9 22 27 41 31 20 15  4  3 10 28  1  7 13 35 11 21 39 29 40 19 18  8 26 33 32 38 24 30  5 
 8  6 14 31 24 30 21  7  5 19 26 16 25  1 36  3 29 42 35 10 18 17  4 40 37 34 38 32 22  2 12 39 33 13 23  9 27 11 28 41 15 20 
20 40 12 29 37 42 34  4 27 26  2 33 41  7 10 22  1 15 25  8 38 36  3 30 21 32 18 39 24 13 35 16  6  9 28 23 17  5 31 11 14 19 
16 18 24 22 20 41 42 40 28  8 23  1 12 11  3 21 26 39  6 29 34 32 14 37 33 35 25 10  2 38 17 31 13  7  4 15  9 30 19 36  5 27 
41 30  2 34  5 12 37 15 16 17 24 39 33 36 18  8 13 10 14 20 22 19 23 38 26 40  3 25 31 27 32 21 35 28  9  1 42  4  7 29  6 11 
39 23  3 15 32 40  2 18  7 29 31 24 27 35  5 16  9 30  1 22 19 41  8 28 36 14 20 12 17 25 10 34 42  6 33 38 11 13 26 37 21  4 
25 24 29 27 17 31 35 42  4 10  6 19 26 28  1  5 30 18  9 41 16 33 13  3  8 11 21  2 36 40 20 15 39 34 37 22 23 12 14 38 32  7 
 6 38 22 32 15 19 12 39 31  4  5 41 29 24 17 11 16 25  8 40 21 30 34  9 28 42 14 27 23 20  3 36  7 37  1 10 26  2 35 33 13 18 


PART 4: 1,000 latin squares of order 256:

Generated in 10.828862 seconds
</pre>