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
<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)) </lang>
The Task
- part 1
<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)) </lang>
- 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
<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 "" </lang>
- 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
<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 "" </lang>
- 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 <lang fsharp> printfn "%d" (Array2D.length1 (Seq.item 999 (randLS 256))) </lang>
- 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. <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)
}</lang>
- 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
Phix
<lang Phix>function shuffleCube(sequence c)
integer n = length(c), rx, ry, rz
bool proper = 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 ox=1 to n do
if c[ox][ry][rz] == 1 then exit end if
end for
if not proper and rand(2)==2 then
for ox=ox+1 to n do
if c[ox][ry][rz] == 1 then exit end if
end for
end if
for oy=1 to n do
if c[rx][oy][rz] == 1 then exit end if
end for
if not proper and rand(2)==2 then
for oy=oy+1 to n do
if c[rx][oy][rz] == 1 then exit end if
end for
end if
for oz=1 to n do
if c[rx][ry][oz] == 1 then exit end if
end for
if not proper and rand(2)==2 then
for oz=oz+1 to n do
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}
proper = false
else
proper = 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)
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
{m[i][j], m[i][k]} = {m[i][k], m[i][j]}
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
{m[i][j], m[k][j]} = {m[k][j], m[i][j]}
end for
exit
end if
end for
end if
end for
return m
end function
function makeCube(object from, 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(from==NULL?mod(i+j,n)+1:from[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 from = {{1, 2, 3, 4}, {2, 1, 4, 3}, {3, 4, 1, 2}, {4, 3, 2, 1}},
c := makeCube(from, 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")
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}}
c = makeCube(from, 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()</lang>
- 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.