Latin Squares in reduced form/Randomizing using Jacobson and Matthews' technique
Latin Squares in reduced form/Randomizing using Jacobson and Matthews' technique is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
Section 3.3 of [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
F#
The Functions
// 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))
The Task
- part 1
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))
- Output:
[[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
- part 2
randLS 5 |> Seq.take 10000 |> Seq.map asNormLS |> Seq.countBy id |> Seq.iteri(fun n g->printf "%d(%d) " (n+1) (snd g)); printfn ""
- Output:
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)
- part 3
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 ""
- Output:
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
- part 4
Generating 1000 Latin Squares of order 256 takes about 1.5secs
printfn "%d" (Array2D.length1 (Seq.item 999 (randLS 256)))
- Output:
256 Real: 00:00:01.512, CPU: 00:00:01.970, GC gen0: 10, gen1: 10
Go
The J & M implementation is based on the C code 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.
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)
}
- Output:
Sample run:
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
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()
- Output:
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)
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."
- Output:
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.
Phix
function shuffleCube(sequence c) integer n = length(c), rx, ry, rz bool bProper = true while true do rx = rand(n) ry = rand(n) rz = rand(n) if c[rx][ry][rz] == 0 then exit end if end while while true do integer ox, oy, oz for i=1 to n do ox = i if c[ox][ry][rz] == 1 then exit end if end for if not bProper and rand(2)==2 then for i=ox+1 to n do ox = i if c[ox][ry][rz] == 1 then exit end if end for end if for i=1 to n do oy = i if c[rx][oy][rz] == 1 then exit end if end for if not bProper and rand(2)==2 then for i=oy+1 to n do oy = i if c[rx][oy][rz] == 1 then exit end if end for end if for i=1 to n do oz = i if c[rx][ry][oz] == 1 then exit end if end for if not bProper and rand(2)==2 then for i=oz+1 to n do oz = i if c[rx][ry][oz] == 1 then exit end if end for end if 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 then {rx, ry, rz} = {ox, oy, oz} bProper = false else bProper = true exit end if end while return c end function function toMatrix(sequence c) integer n = length(c) sequence m = repeat(repeat(0,n),n) for i=1 to n do for j=1 to n do for k=1 to n do if c[i][j][k] != 0 then m[i][j] = k exit end if end for end for end for return m end function function toReduced(sequence m) integer n := length(m) m = deep_copy(m) for j=1 to n-1 do if m[1][j]!=j then for k=j+1 to n do if m[1][k]==j then for i=1 to n do atom mij = m[i][j], mik = m[i][k] m[i][j] = mik m[i][k] = mij end for exit end if end for end if end for for i=2 to n-1 do if m[i][1]!=i then for k=i+1 to n do if m[k][1]==i then for j=1 to n do atom mij = m[i][j], mkj = m[k][j] m[i][j] = mkj m[k][j] = mij end for exit end if end for end if end for return m end function function makeCube(object orig, integer n) sequence c = repeat(repeat(repeat(0,n),n),n) for i=1 to n do for j=1 to n do integer k = iff(orig==NULL?mod(i+j,n)+1:orig[i][j]) c[i][j][k] = 1 end for end for return c end function procedure main() printf(1,"Part 1: 10,000 latin Squares of order 4 in reduced form:\n\n") sequence orig = {{1, 2, 3, 4}, {2, 1, 4, 3}, {3, 4, 1, 2}, {4, 3, 2, 1}}, c := makeCube(orig, 4), m, rm, fk integer freq = new_dict() for i=1 to 10000 do c = shuffleCube(c) m = toMatrix(c) rm = toReduced(m) setd(rm,getd(rm,freq)+1,freq) end for fk = getd_all_keys(freq) for i=1 to length(fk) do printf(1,"%v occurs %d times\n", {fk[i],getd(fk[i],freq)}) end for printf(1,"\nPart 2: 10,000 latin squares of order 5 in reduced form:\n\n") orig = {{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}} c = makeCube(orig, 5) destroy_dict(freq, justclear:=true) for i=1 to 10000 do c = shuffleCube(c) m = toMatrix(c) rm = toReduced(m) setd(rm,getd(rm,freq)+1,freq) end for fk = getd_all_keys(freq) for i=1 to length(fk) do fk[i] = sprintf("%2d(%3d)", {i,getd(fk[i],freq)}) end for puts(1,join_by(fk,8,7," ","\n")) destroy_dict(freq) -- part 3 printf(1,"\nPart 3: 750 latin squares of order 42, showing the last one:\n\n") c = makeCube(NULL, 42) for i=1 to 750 do c = shuffleCube(c) end for m = toMatrix(c) integer n := length(m) for i=1 to n do for j=1 to n do m[i,j] = sprintf("%2d",m[i,j]) end for m[i] = join(m[i]," ") end for printf(1,"%s\n",join(m,"\n")) -- part 4 printf(1,"\nPART 4: 1000 latin squares of order 256:\n\n") atom t0 = time() c = makeCube(NULL, 256) for i=1 to 1000 do c = shuffleCube(c) end for printf(1,"Generated in %s\n", elapsed(time()-t0)) end procedure main()
- Output:
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
Unfortunately the last part of this task exposes the relatively poor performance of subscripting in phix.
Raku
# 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."
- Output:
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.
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)
- Output:
Sample run:
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