Latin Squares in reduced form/Randomizing using Jacobson and Matthews’ Technique
You are encouraged to solve this task according to the task description, using any language you may know.
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
Julia
<lang julia>const Cube = Vector{Vector{Vector{Int}}} const Mat = Vector{Vector{Int}}
function reduced(m::Mat)
n = length(m) r = deepcopy(m) for j in 1:n-1 if r[1][j] != j for k in j+1:n if r[1][k] == j for i in 1:n r[i][j], r[i][k] = r[i][k], r[i][j] end break end end end end for i in 2:n-1 if r[i][1] != i for k in i+1:n if r[k][1] == i for j in 1:n r[i][j], r[k][j] = r[k][j], r[i][j] end break end end end end return r
end
""" print matrix as small integers, no punctuation """ function print_matrix(m::Mat)
n = length(m) padding = max(2, Int(ceil(log(10, n+1))) + 1) for i in 1:n for j in 1:n print(lpad(m[i][j], padding)) end println() end println()
end
function shuffle_cube(c::Cube)
n = length(c) proper = true rx, ry, rz = 0, 0, 0 while true rx, ry, rz = rand(1:n, 3) c[rx][ry][rz] == 0 && break end while true ox = something(findfirst(i -> c[i][ry][rz] == 1, 1:n), n) oy = something(findfirst(i -> c[rx][i][rz] == 1, 1:n), n) oz = something(findfirst(i -> c[rx][ry][i] == 1, 1:n), n) if !proper rand() < 1/2 && (ox = something(findlast(i -> c[i][ry][rz] == 1, 1:n), n)) rand() < 1/2 && (oy = something(findlast(i -> c[rx][i][rz] == 1, 1:n), n)) rand() < 1/2 && (oz = something(findlast(i -> c[rx][ry][i] == 1, 1:n), n)) end
c[rx][ry][rz] += 1 c[rx][oy][oz] += 1 c[ox][ry][oz] += 1 c[ox][oy][rz] += 1
c[rx][ry][oz] -= 1 c[rx][oy][rz] -= 1 c[ox][ry][rz] -= 1 c[ox][oy][oz] -= 1
if c[ox][oy][oz] < 0 rx, ry, rz = ox, oy, oz proper = false else break end end
end
function matrix(c::Cube)::Mat
n = length(c) m = [[0 for i in 1:n] for j in 1:n] for i in 1:n, j in 1:n for k in 1:n if c[i][j][k] != 0 m[i][j] = k break end end end return m
end
function cube(from, n)
c = [[[0 for i in 1:n] for j in 1:n] for k in 1:n] for i in 1:n, j in 1:n k = (from isa Nothing) ? mod1(i + j, n) : from[i][j] c[i][j][k] = 1 end return c
end
function testJacobsenMatthews()
# part 1 println("PART 1: 10,000 latin Squares of order 4 in reduced form:\n") from = [[1, 2, 3, 4], [2, 1, 4, 3], [3, 4, 1, 2], [4, 3, 2, 1]] freqs4 = Dict{Array, Int}() c = cube(from, 4) for i in 1:10000 shuffle_cube(c) m = matrix(c) rm = reduced(m) n = get!(freqs4, rm, 0) freqs4[rm] = n + 1 end for (a, freq) in freqs4 print_matrix(a) println("Occurs $freq times\n") end
# part 2 println("\nPART 2: 10,000 latin squares of order 5 in reduced form:\n") from = [[1, 2, 3, 4, 5], [2, 3, 4, 5, 1], [3, 4, 5, 1, 2], [4, 5, 1, 2, 3], [5, 1, 2, 3, 4]] freqs5 = Dict{Array, Int}() c = cube(from, 5) for i in 1:10000 shuffle_cube(c) m = matrix(c) rm = reduced(m) n = get!(freqs5, rm, 0) freqs5[rm] = n + 1 end for (i, freq) in enumerate(sort(collect(values(freqs5)))) i > 1 && (print(", "); (i - 1) % 8 == 0 && println()) print(lpad(i, 2), "(", lpad(freq, 3), ")") end println("\n")
# part 3 println("\nPART 3: 750 latin squares of order 42, showing the last one:\n") m42 = [[0 for i in 1:42] for j in 1:42] c = cube(nothing, 42) for i in 1:750 shuffle_cube(c) i == 750 && (m42 = matrix(c)) end print_matrix(m42)
# part 4 println("\nPART 4: 1000 latin squares of order 256:\n") @time begin c = cube(nothing, 256) for i in 1:1000 shuffle_cube(c) end end
end
testJacobsenMatthews()
</lang>
- 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
<lang Nim>import random, sequtils, strformat, tables, times
type
Vector = seq[int] Matrix = seq[Vector] Cube = seq[Matrix] Array16 = array[16, int] Array25 = array[25, int]
func newCube(m: Matrix; n: Positive): Cube =
result.setLen(n) for i in 0..<n: result[i].setLen(n) for j in 0..<n: result[i][j].setLen(n) let k = if m.len == 0: (i + j) mod n else: m[i][j] - 1 result[i][j][k] = 1
proc shuffle(c: var Cube) =
let n = c[0].len var proper = true
var rx, ry, rz: int while true: rx = rand(n - 1) ry = rand(n - 1) rz = rand(n - 1) if c[rx][ry][rz] == 0: break
while true: var ox, oy, oz = 0
while ox < n: if c[ox][ry][rz] == 1: break inc ox if not proper and rand(1) == 0: inc ox while ox < n: if c[ox][ry][rz] == 1: break inc ox
while oy < n: if c[rx][oy][rz] == 1: break inc oy if not proper and rand(1) == 0: inc oy while oy < n: if c[rx][oy][rz] == 1: break inc oy
while oz < n: if c[rx][ry][oz] == 1: break inc oz if not proper and rand(1) == 0: inc oz while oz < n: if c[rx][ry][oz] == 1: break inc oz
inc c[rx][ry][rz] inc c[rx][oy][oz] inc c[ox][ry][oz] inc c[ox][oy][rz]
dec c[rx][ry][oz] dec c[rx][oy][rz] dec c[ox][ry][rz] dec c[ox][oy][oz]
if c[ox][oy][oz] < 0: (rx, ry, rz) = (ox, oy, oz) proper = false else: proper = true break
func toMatrix(c: Cube): Matrix =
let n = c[0].len result = newSeqWith(n, newSeq[int](n)) for i in 0..<n: for j in 0..<n: for k in 0..<n: if c[i][j][k] != 0: result[i][j] = k break
func toReduced(m: Matrix): Matrix =
let n = m.len result = m
for j in 0..n-2: if result[0][j] != j: for k in j+1..<n: if result[0][k] == j: for i in 0..<n: swap result[i][j], result[i][k] break
for i in 1..n-2: if result[i][0] != i: for k in i+1..<n: if result[k][0] == i: for j in 0..<n: swap result[i][j], result[k][j] break
func asArray16(m: Matrix): Array16 =
var k = 0 for i in 0..3: for j in 0..3: result[k] = m[i][j] inc k
func asArray25(m: Matrix): Array25 =
var k = 0 for i in 0..4: for j in 0..4: result[k] = m[i][j] inc k
proc printArray16(a: Array16) =
for i in 0..3: for j in 0..3: let k = i * 4 + j stdout.write &"{a[k]+1:2} " # Back to 1 based. echo() echo()
proc printMatrix(m: Matrix) =
let n = m.len for i in 0..<n: for j in 0..<n: stdout.write &"{m[i][j]+1:2} " # Back to 1 based. echo() echo()
randomize()
- Part 1.
echo "Part 1: 10_000 latin Squares of order 4 in reduced form:\n" const From1: Matrix = @[@[1, 2, 3, 4], @[2, 1, 4, 3], @[3, 4, 1, 2], @[4, 3, 2, 1]] var freqs4: CountTable[Array16] var c = newCube(From1, 4) for _ in 1..10_000:
c.shuffle() let m = c.toMatrix let rm = m.toReduced let a16 = rm.asArray16 freqs4.inc(a16)
for a, freq in freqs4.pairs:
printArray16(a) echo &"Occurs {freq} times\n"
- Part 2.
echo "\nPart 2: 10_000 latin squares of order 5 in reduced form:" const From2: Matrix = @[@[1, 2, 3, 4, 5], @[2, 3, 4, 5, 1], @[3, 4, 5, 1, 2],
@[4, 5, 1, 2, 3], @[5, 1, 2, 3, 4]]
var freqs5: CountTable[Array25] c = newCube(From2, 5) for _ in 1..10_000:
c.shuffle() let m = c.toMatrix let rm = m.toReduced let a25 = rm.asArray25 freqs5.inc(a25)
var count = 0 for freq in freqs5.values:
inc count if count > 1: stdout.write ", " if (count - 1) mod 8 == 0: echo() stdout.write &"{count:2}({freq:3})"
echo '\n'
- Part 3.
echo "\nPart 3: 750 latin squares of order 42, showing the last one:\n" var m42: Matrix c = newCube(@[], 42) for i in 1..750:
c.shuffle() if i == 750: m42 = c.toMatrix
printMatrix(m42)
- Part 4.
echo "\nPart 4: 1000 latin squares of order 256:\n" let start = cpuTime() c = newCube(@[], 256) for _ in 1..1000:
c.shuffle()
echo &"Generated in {cpuTime() - start:.3f} s."</lang>
- 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.
Wren
<lang ecmascript>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)</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 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