⚠️ Warning: This is a draft ⚠️

This means it might contain formatting issues, incorrect code, conceptual problems, or other severe issues.

If you want to help to improve and eventually enable this page, please fork RosettaGit's repository and open a merge request on GitHub.

A Latin square of size `n` is an arrangement of `n` symbols in an `n-by-n` square in such a way that each row and column has each symbol appearing exactly once.

A randomised Latin square generates random configurations of the symbols for any given `n`.

;Example n=4 randomised Latin square:

```0 2 3 1
2 1 0 3
3 0 1 2
1 3 2 0
```

# Use the function to generate ''and show here'', two randomly generated squares of size 5.

;Note: Strict ''Uniformity'' in the random generation is a hard problem and '''not''' a requirement of the task.

;Reference:

• [[wp:Latin_square|Latin square]]
• [http://oeis.org/A002860 OEIS A002860]

=={{header|F_Sharp|F#}}== This solution uses functions from [[Factorial_base_numbers_indexing_permutations_of_a_collection#F.23]] and [[Latin_Squares_in_reduced_form#F.23]]. This solution generates completely random uniformly distributed Latin Squares from all possible Latin Squares of order 5. It takes 5 thousandths of a second can that really be called hard?

```
// Generate 2 Random Latin Squares of order 5. Nigel Galloway: July 136th., 2019
let N=let N=System.Random() in (fun n->N.Next(n))
let rc()=let β=lN2p [|0;N 4;N 3;N 2|] [|0..4|] in Seq.item (N 56) (normLS 5) |> List.map(lN2p [|N 5;N 4;N 3;N 2|]) |> List.permute(fun n->β.[n]) |> List.iter(printfn "%A")
rc(); printfn ""; rc()

```

{{out}}

```
[|5; 3; 1; 4; 2|]
[|1; 4; 5; 2; 3|]
[|4; 1; 2; 3; 5|]
[|2; 5; 3; 1; 4|]
[|3; 2; 4; 5; 1|]

[|4; 1; 2; 5; 3|]
[|3; 5; 1; 2; 4|]
[|2; 4; 5; 3; 1|]
[|1; 2; 3; 4; 5|]
[|5; 3; 4; 1; 2|]

```

I thought some statistics might be interesting so I generated 1 million Latin Squares of order 5. There are 161280 possible Latin Squares of which 3174 were not generated. The remainder were generated:

```
Times Generated    Number of Latin Squares
1                       1776
2                       5669
3                      11985
4                      19128
5                      24005
6                      25333
7                      22471
8                      18267
9                      12569
10                       7924
11                       4551
12                       2452
13                       1130
14                        483
15                        219
16                         93
17                         37
18                          5
19                          7
20                          2

```

## Factor

A brute force method for generating uniformly random Latin squares. Repeatedly select a random permutation of (0, 1,...n-1) and add it as the next row of the square. If at any point the rules for being a Latin square are violated, start the entire process over again from the beginning.

```USING: arrays combinators.extras fry io kernel math.matrices
prettyprint random sequences sets ;
IN: rosetta-code.random-latin-squares

: rand-permutation ( n -- seq ) <iota> >array randomize ;
: ls? ( n -- ? ) [ all-unique? ] column-map t [ and ] reduce ;
: (ls) ( n -- m ) dup '[ _ rand-permutation ] replicate ;
: ls ( n -- m ) dup (ls) dup ls? [ nip ] [ drop ls ] if ;
: random-latin-squares ( -- ) [ 5 ls simple-table. nl ] twice ;

MAIN: random-latin-squares
```

{{out}}

```
0 4 3 2 1
3 0 2 1 4
4 2 1 3 0
2 1 4 0 3
1 3 0 4 2

4 0 1 3 2
0 2 4 1 3
1 3 0 2 4
2 4 3 0 1
3 1 2 4 0

```

## Go

### Restarting Row method

As the task is not asking for large squares to be generated and even n = 10 is virtually instant, we use a simple brute force approach here known as the 'Restarting Row' method (see Talk page). However, whilst easy to understand, this method does not produce uniformly random squares.

```package main

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

type matrix [][]int

func shuffle(row []int, n int) {
rand.Shuffle(n, func(i, j int) {
row[i], row[j] = row[j], row[i]
})
}

func latinSquare(n int) {
if n <= 0 {
fmt.Println("[]\n")
return
}
latin := make(matrix, n)
for i := 0; i < n; i++ {
latin[i] = make([]int, n)
if i == n-1 {
break
}
for j := 0; j < n; j++ {
latin[i][j] = j
}
}
// first row
shuffle(latin[0], n)

// middle row(s)
for i := 1; i < n-1; i++ {
shuffled := false
shuffling:
for !shuffled {
shuffle(latin[i], n)
for k := 0; k < i; k++ {
for j := 0; j < n; j++ {
if latin[k][j] == latin[i][j] {
continue shuffling
}
}
}
shuffled = true
}
}

// last row
for j := 0; j < n; j++ {
used := make([]bool, n)
for i := 0; i < n-1; i++ {
used[latin[i][j]] = true
}
for k := 0; k < n; k++ {
if !used[k] {
latin[n-1][j] = k
break
}
}
}
printSquare(latin, n)
}

func printSquare(latin matrix, n int) {
for i := 0; i < n; i++ {
fmt.Println(latin[i])
}
fmt.Println()
}

func main() {
rand.Seed(time.Now().UnixNano())
latinSquare(5)
latinSquare(5)
latinSquare(10) // for good measure
}
```

{{out}} Sample run:

```
[3 2 1 0 4]
[0 3 2 4 1]
[4 1 0 3 2]
[2 4 3 1 0]
[1 0 4 2 3]

[3 1 0 4 2]
[1 0 2 3 4]
[2 4 3 0 1]
[4 3 1 2 0]
[0 2 4 1 3]

[9 2 8 4 6 1 7 5 0 3]
[4 3 7 6 0 8 5 9 2 1]
[2 1 9 7 3 4 6 0 5 8]
[8 6 0 5 7 2 3 1 9 4]
[5 0 6 8 1 3 9 2 4 7]
[7 5 4 9 2 0 1 3 8 6]
[3 9 2 1 5 6 8 4 7 0]
[1 4 5 2 8 7 0 6 3 9]
[6 8 3 0 4 9 2 7 1 5]
[0 7 1 3 9 5 4 8 6 2]

```

### Latin Squares in Reduced Form method

Unlike the "Restarting Row" method, this method does produce uniformly random Latin squares for n <= 6 (see Talk page) but is more involved and therefore slower. It reuses some (suitably adjusted) code from the [https://rosettacode.org/wiki/Latin_Squares_in_reduced_form#Go Latin Squares in Reduced Form] and [https://rosettacode.org/wiki/Permutations#non-recursive.2C_lexicographical_order Permutations] tasks.

```package main

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

type matrix [][]int

// generate derangements of first n numbers, with 'start' in first place.
func dList(n, start int) (r matrix) {
start-- // use 0 basing
a := make([]int, n)
for i := range a {
a[i] = i
}
a[0], a[start] = start, a[0]
sort.Ints(a[1:])
first := a[1]
// recursive closure permutes a[1:]
var recurse func(last int)
recurse = func(last int) {
if last == first {
// bottom of recursion.  you get here once for each permutation.
// test if permutation is deranged.
for j, v := range a[1:] { // j starts from 0, not 1
if j+1 == v {
return // no, ignore it
}
}
// yes, save a copy
b := make([]int, n)
copy(b, a)
for i := range b {
b[i]++ // change back to 1 basing
}
r = append(r, b)
return
}
for i := last; i >= 1; i-- {
a[i], a[last] = a[last], a[i]
recurse(last - 1)
a[i], a[last] = a[last], a[i]
}
}
recurse(n - 1)
return
}

func reducedLatinSquares(n int) []matrix {
var rls []matrix
if n < 0 {
n = 0
}
rlatin := make(matrix, n)
for i := 0; i < n; i++ {
rlatin[i] = make([]int, n)
}
if n <= 1 {
return append(rls, rlatin)
}
// first row
for j := 0; j < n; j++ {
rlatin[0][j] = j + 1
}
// recursive closure to compute reduced latin squares
var recurse func(i int)
recurse = func(i int) {
rows := dList(n, i) // get derangements of first n numbers, with 'i' first.
outer:
for r := 0; r < len(rows); r++ {
copy(rlatin[i-1], rows[r])
for k := 0; k < i-1; k++ {
for j := 1; j < n; j++ {
if rlatin[k][j] == rlatin[i-1][j] {
if r < len(rows)-1 {
continue outer
} else if i > 2 {
return
}
}
}
}
if i < n {
recurse(i + 1)
} else {
rl := copyMatrix(rlatin)
rls = append(rls, rl)
}
}
return
}

// remaining rows
recurse(2)
return rls
}

func copyMatrix(m matrix) matrix {
le := len(m)
cpy := make(matrix, le)
for i := 0; i < le; i++ {
cpy[i] = make([]int, le)
copy(cpy[i], m[i])
}
return cpy
}

func printSquare(latin matrix, n int) {
for i := 0; i < n; i++ {
for j := 0; j < n; j++ {
fmt.Printf("%d ", latin[i][j]-1)
}
fmt.Println()
}
fmt.Println()
}

func factorial(n uint64) uint64 {
if n == 0 {
return 1
}
prod := uint64(1)
for i := uint64(2); i <= n; i++ {
prod *= i
}
return prod
}

// generate permutations of first n numbers, starting from 0.
func pList(n int) matrix {
fact := factorial(uint64(n))
perms := make(matrix, fact)
a := make([]int, n)
for i := 0; i < n; i++ {
a[i] = i
}
t := make([]int, n)
copy(t, a)
perms[0] = t
n--
var i, j int
for c := uint64(1); c < fact; c++ {
i = n - 1
j = n
for a[i] > a[i+1] {
i--
}
for a[j] < a[i] {
j--
}
a[i], a[j] = a[j], a[i]
j = n
i++
for i < j {
a[i], a[j] = a[j], a[i]
i++
j--
}
t := make([]int, n+1)
copy(t, a)
perms[c] = t
}
return perms
}

func generateLatinSquares(n, tests, echo int) {
rls := reducedLatinSquares(n)
perms := pList(n)
perms2 := pList(n - 1)
for test := 0; test < tests; test++ {
rn := rand.Intn(len(rls))
rl := rls[rn] // select reduced random square at random
rn = rand.Intn(len(perms))
rp := perms[rn] // select a random permuation of 'rl's columns
// permute columns
t := make(matrix, n)
for i := 0; i < n; i++ {
t[i] = make([]int, n)
}
for i := 0; i < n; i++ {
for j := 0; j < n; j++ {
t[i][j] = rl[i][rp[j]]
}
}
rn = rand.Intn(len(perms2))
rp = perms2[rn] // select a random permutation of 't's rows 2 to n
// permute rows 2 to n
u := make(matrix, n)
for i := 0; i < n; i++ {
u[i] = make([]int, n)
}
for i := 0; i < n; i++ {
for j := 0; j < n; j++ {
if i == 0 {
u[i][j] = t[i][j]
} else {
u[i][j] = t[rp[i-1]+1][j]
}
}
}
if test < echo {
printSquare(u, n)
}
if n == 4 {
for i := 0; i < 4; i++ {
for j := 0; j < 4; j++ {
u[i][j]--
}
}
for i := 0; i < 4; i++ {
copy(a[4*i:], u[i])
}
for i := 0; i < 4; i++ {
if testSquares[i] == a {
counts[i]++
break
}
}
}
}
}

var testSquares = [4][16]int{
{0, 1, 2, 3, 1, 0, 3, 2, 2, 3, 0, 1, 3, 2, 1, 0},
{0, 1, 2, 3, 1, 0, 3, 2, 2, 3, 1, 0, 3, 2, 0, 1},
{0, 1, 2, 3, 1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2},
{0, 1, 2, 3, 1, 3, 0, 2, 2, 0, 3, 1, 3, 2, 1, 0},
}

var (
counts [4]int
a      [16]int
)

func main() {
rand.Seed(time.Now().UnixNano())
fmt.Println("Two randomly generated latin squares of order 5 are:\n")
generateLatinSquares(5, 2, 2)

fmt.Println("Out of 1,000,000 randomly generated latin squares of order 4, ")
fmt.Println("of which there are 576 instances ( => expected 1736 per instance),")
fmt.Println("the following squares occurred the number of times shown:\n")
generateLatinSquares(4, 1e6, 0)
for i := 0; i < 4; i++ {
fmt.Println(testSquares[i][:], ":", counts[i])
}

fmt.Println("\nA randomly generated latin square of order 6 is:\n")
generateLatinSquares(6, 1, 1)
}
```

{{out}} Sample run:

```
Two randomly generated latin squares of order 5 are:

2 1 3 4 0
4 3 0 1 2
1 0 2 3 4
0 4 1 2 3
3 2 4 0 1

1 2 3 4 0
0 3 4 2 1
2 4 0 1 3
4 0 1 3 2
3 1 2 0 4

Out of 1,000,000 randomly generated latin squares of order 4,
of which there are 576 instances ( => expected 1736 per instance),
the following squares occurred the number of times shown:

[0 1 2 3 1 0 3 2 2 3 0 1 3 2 1 0] : 1737
[0 1 2 3 1 0 3 2 2 3 1 0 3 2 0 1] : 1736
[0 1 2 3 1 2 3 0 2 3 0 1 3 0 1 2] : 1726
[0 1 2 3 1 3 0 2 2 0 3 1 3 2 1 0] : 1799

A randomly generated latin square of order 6 is:

3 5 1 0 4 2
2 0 5 4 1 3
0 4 2 5 3 1
1 3 4 2 0 5
5 1 0 3 2 4
4 2 3 1 5 0

```

## Javascript

```
class Latin {
constructor(size = 3) {
this.size = size;
this.mst = [...Array(this.size)].map((v, i) => i + 1);
this.square = Array(this.size).fill(0).map(() => Array(this.size).fill(0));

if (this.create(0, 0)) {
console.table(this.square);
}
}

create(c, r) {
const d = [...this.mst];
let s;
while (true) {
do {
s = d.splice(Math.floor(Math.random() * d.length), 1)[0];
if (!s) return false;
} while (this.check(s, c, r));

this.square[c][r] = s;
if (++c >= this.size) {
c = 0;
if (++r >= this.size) {
return true;
}
}
if (this.create(c, r)) return true;
if (--c < 0) {
c = this.size - 1;
if (--r < 0) {
return false;
}
}
}
}

check(d, c, r) {
for (let a = 0; a < this.size; a++) {
if (c - a > -1) {
if (this.square[c - a][r] === d)
return true;
}
if (r - a > -1) {
if (this.square[c][r - a] === d)
return true;
}
}
return false;
}
}
new Latin(5);

```

{{out}}

```
3 5 4 1 2
4 3 1 2 5
1 2 3 5 4
5 1 2 4 3
2 4 5 3 1

4 5 1 3 2
3 1 4 2 5
5 4 2 1 3
1 2 3 5 4
2 3 5 4 1

```

## Julia

Using the Python algorithm as described in the discussion section.

```using Random

shufflerows(mat) = mat[shuffle(1:end), :]
shufflecols(mat) = mat[:, shuffle(1:end)]

n = size(mat)[1] + 1
newmat = similar(mat, size(mat) .+ 1)
for j in 1:n, i in 1:n
newmat[i, j] = (i == n && j < n) ? mat[1, j] : (i == j) ? n - 1 :
(i < j) ? mat[i, j - 1] : mat[i, j]
end
newmat
end

function makelatinsquare(N)
mat = [0 1; 1 0]
for i in 3:N
end
shufflecols(shufflerows(mat))
end

function printlatinsquare(N)
mat = makelatinsquare(N)
for i in 1:N, j in 1:N
print(rpad(mat[i, j], 3), j == N ? "\n" : "")
end
end

printlatinsquare(5), println("\n"), printlatinsquare(5)

```

{{out}}

```
1  3  0  4  2
3  0  4  2  1
0  4  2  1  3
2  1  3  0  4
4  2  1  3  0

2  0  1  3  4
4  3  2  1  0
3  2  0  4  1
1  4  3  0  2
0  1  4  2  3

```

## M2000 Interpreter

### Easy Way

One row shuffled to be used as the destination row. One more shuffled and then n times rotated by one and stored to array

for 40x40 need 2~3 sec, including displaying to screen

We use the stack of values, a linked list, for pushing to top (Push) or to bottom (Data), and we can pop from top using Number or by using Read A to read A from stack. Also we can shift from a chosen position to top using Shift, or using shiftback to move an item from top to chosen position. So we shuffle items by shifting them.

```
Module FastLatinSquare {
n=5
For k=1 To 2
latin()
Next
n=40
latin()
Sub latin()
Local i,a, a(1 To n), b, k
Profiler
flush
Print "latin square ";n;" by ";n
For i=1 To n
Push i
Next i
For i=1 To n div 2
Shiftback random(2, n)
Next i
a=[]
Push ! stack(a)
a=array(a)  ' change a from stack to array
For i=1 To n*10
Shiftback random(2, n)
Next i
For i=0 To n-1
Data number  ' rotate by one the stack items
b=[]    ' move stack To b, leave empty stack
a(a#val(i))=b
Push ! stack(b)  ' Push from a copy of b all items To stack
Next i
flush
For k=1 To  n div 2
z=random(2, n)
For i=1 To n
a=a(i)
stack a {
shift z
}
Next
Next
For i=1 To n
a=a(i)
a(i)=array(a)  ' change To array from stack
Next i
For i=1 To n
Print a(i)
Next i
Print TimeCount
End Sub
}
FastLatinSquare
```

### Hard Way

for 5x5 need some miliseconds

for 16X16 need 56 seconds

for 20X20 need 22 min (as for 9.8 version)

```
Module LatinSquare (n, z=1, f\$="latin.dat", NewFile As Boolean=False) {
If Not Exist(f\$)  Or NewFile Then
Open f\$ For Wide Output As f
Else
Open f\$ For Wide Append As f
End If
ArrayToString=Lambda -> {
Shift 2  ' swap two top values in stack
Push Letter\$+Str\$(Number)
}
Dim line(1 to n)
flush   ' erase current stack of value
z=if(z<1->1, z)
newColumn()
For j=1 To z
Profiler
ResetColumns()
For i=1 To n
placeColumn()
Next
Print "A latin square of ";n;" by ";n
For i=1 To n
Print line(i)
Print #f, line(i)#Fold\$(ArrayToString)
Next
Print TimeCount
Refresh
Next
close #f
Flush  ' empty stack again
End
Sub ResetColumns()
Local i
For i=1 To n:line(i)=(,):Next
End Sub
Sub newColumn()
Local i
For i=1 To n : Push i: Next
End Sub
Sub shuffle()
Local i
For i=1 To n div 2: Shift Random(2, n): Next
End Sub
Sub shuffleLocal(x)
If Stack.size<=x Then Exit Sub
Shift Random(x+1, Stack.size)
Shiftback x
End Sub
Sub PlaceColumn()
Local i, a, b, k
shuffle()
Do
data number   ' rotate one position
k=0
For i=1 To n
a=line(i)  ' get the pointer
Do
If a#Pos(Stackitem(i))=-1 Then k=0 :Exit Do
shuffleLocal(i)
k++
Until k>Stack.size-i
If k>0 Then Exit For
Next
Until k=0
For i=1 To n
a=line(i)
Append a, (Stackitem(i),)
Next
End Sub
}
Form 100,50
LatinSquare 5, 2, True
LatinSquare 16

```

{{out}}

```A latin square of 5 by 5
4 5 3 1 2
5 4 2 3 1
2 1 5 4 3
1 3 4 2 5
3 2 1 5 4
A latin square of 5 by 5
4 3 5 1 2
2 4 3 5 1
1 2 4 3 5
5 1 2 4 3
3 5 1 2 4
A latin square of 16 by 16
12 14 5 16 1 2 7 15 9 11 10 8 13 3 6 4
3 13 16 12 7 4 1 11 5 6 15 2 8 14 10 9
13 2 8 3 4 12 5 9 14 7 16 10 6 1 15 11
8 3 13 9 2 10 16 1 15 14 5 4 11 7 12 6
4 12 2 7 5 3 6 10 1 9 11 16 14 8 13 15
16 8 3 4 14 6 13 7 11 10 9 15 1 12 2 5
15 4 14 1 16 8 2 13 6 12 7 9 10 11 5 3
11 16 12 10 15 9 4 5 7 1 8 6 3 13 14 2
10 15 4 5 12 16 3 6 8 13 1 11 7 2 9 14
9 11 15 8 3 1 14 12 13 4 6 5 2 16 7 10
7 10 11 13 9 14 15 4 3 5 2 12 16 6 1 8
6 7 10 2 8 13 9 16 12 15 14 3 5 4 11 1
5 6 1 14 13 11 8 2 10 3 12 7 15 9 4 16
2 5 6 15 11 7 12 14 4 8 3 1 9 10 16 13
1 9 7 11 6 15 10 8 2 16 13 14 4 5 3 12
14 1 9 6 10 5 11 3 16 2 4 13 12 15 8 7
```
## Perl 6 {{works with|Rakudo|2019.03}} {{trans|Python}} ```perl6 sub latin-square { [[0],] }; sub random ( @ls, :\$size = 5 ) { # Build for 1 ..^ \$size -> \$i { @ls[\$i] = @ls[0].clone; @ls[\$_].splice(\$_, 0, \$i) for 0 .. \$i; } # Shuffle @ls = @ls[^\$size .pick(*)]; my @cols = ^\$size .pick(*); @ls[\$_] = @ls[\$_][@cols] for ^@ls; # Some random Latin glyphs my @symbols = ('A' .. 'Z').pick(\$size); @ls.deepmap: { \$_ = @symbols[\$_] }; } sub display ( @array ) { \$_.fmt("%2s ").put for |@array, '' } # The Task # Default size 5 display random latin-square; # Specified size display random :size(\$_), latin-square for 5, 3, 9; # Or, if you'd prefer: display random latin-square, :size(\$_) for 12, 2, 1; ``` {{out|Sample output}} ```txt V Z M J U Z M U V J U J V M Z J V Z U M M U J Z V B H K U D H D U B K K U H D B U B D K H D K B H U I P Y P Y I Y I P Y J K E Z B I W H E Y B W K H J Z I B K Y H J E Z I W I H W J E Z B Y K J I Z Y W K H E B W E H Z B I Y K J H B E I Y W K J Z K Z J B I Y W H E Z W I K H J E B Y L Q E M A T Z C N Y R D Q R Y L N D C E M T A Z E Y M C D Q A N Z L T R M L C N R Y D Z A E Q T N M Z A Q E T D R C L Y T D Q Y C A M L E R Z N R A T Q M Z E Y L D N C D Z R T E N L Q Y A C M Y T L E Z R N M C Q D A A N D R L C Y T Q Z M E Z C A D Y M Q R T N E L C E N Z T L R A D M Y Q Y G G Y I ``` ## Phix Brute force, begins to struggle above 42. ```Phix string aleph = "123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz" function ls(integer n) if n>length(aleph) then ?9/0 end if -- too big... atom t1 = time()+1 sequence tn = tagset(n), -- {1..n} vcs = repeat(tn,n), -- valid for cols res = {} integer clashes = 0 while length(res)t1 then printf(1,"rows completed:%d/%d, clashes:%d\n", {length(res),n,clashes}) t1 = time()+1 end if end if end while for i=1 to n do string line = "" for j=1 to n do line &= aleph[res[i][j]] end for res[i] = line end for return res end function procedure latin_square(integer n) atom t0 = time() string res = join(ls(n),"\n"), e = elapsed(time()-t0) printf(1,"Latin square of order %d (%s):\n%s\n",{n,e,res}) end procedure latin_square(3) latin_square(5) latin_square(5) latin_square(10) latin_square(42) ``` {{out}} ```txt Latin square of order 3 (0s): 231 123 312 Latin square of order 5 (0s): 15423 53142 24315 42531 31254 Latin square of order 5 (0s): 32514 21453 43125 15342 54231 Latin square of order 10 (0s): 3258A69417 9314275A86 586312479A 19A2753864 61294873A5 267194A538 473A618952 85973A6241 7A45831629 A486592173 rows completed:40/42, clashes:49854 rows completed:40/42, clashes:104051 Latin square of order 42 (2.1s): 5CMOgPTHbDBKGLU9d1aIREFNV8cQYeZ62AJXW3Sf74 VQEaBOIL2GefUYXbZWKP5cRDd3C4HS89MF7Jg6A1NT LIRc1AXPWKJH4bTNFC2VS935g6ZEDfaeOGdYQ8B7MU QPZ1LcN8O4I96dKRATfEWHaJUSM5e37GXBbDYFCV2g ATIZD2VY8gGRNHBO6P35QaEM1fS79dKFbCX4JWcUeL eE9Q7CJZ3VP254YMLHGOIBcdf1AbgXFKUR8WaTDNS6 7FfJdWGg4ZDC1UaVIcH9A5XLSb63MBTNPOQEKe2Y8R DMa53QWdB9bPSeEZJg6GK2A7YR1C4VLcIXTUFO8HfN G8d2eaARUPNEI6HCKQZSO4b1BM5g3L9VcTYFfXWJD7 9OHSGIRDEMf3YKb2cU1WVdNeTL8XA64gQPC57JZaBF 634YR52FINaeDQPKCXBAEZ8S7dgUG9OHJcfLMVTWb1 aVgXP7EO5f8JdFQ1RY9BHW42eTNGSUb3LMKZIC6AcD YBWLbFUAP1T8JSZ6N2gMeIQfHc3DC75d9EVOG4aKRX TXUBHe62JSCNA7WPfGLdYbOc9VRZ1IgD3aMQ5KEF48 dU1IS9D362V5cR8WTZCKfNPYA4OebQXJ7HBMLEFGga 3aDTUf5SCY6g8GOJbVIHZ1KWMNXBFPd7AQRc924LEe 1e2bJBHUDa9dTCRLYAWNXOMZFIK6QgfPE5S38G74Vc F786NJP1Gb4BLXVdUaAfcTgQC2eWKHYZ53D9ERISOM U1cRQGOeFBXWag6IPdVJ2Mf8L7YSNCHETK4bDZ539A 2DbgW6MfVAQY9acUX35ZNRBPIOJL81CSe4EHd7GTFK 8bNE2U7XTWHVBZJ5QMcePY9g6Ad1OaIf4DFKRSLC3G g5L8AbKN7JF6MD1SBEQU3fTXWHVdR2P4ZY9aCceIGO W2K9FHa5dT1OZ8C3gLScGVI4JB7YPDUbfe6RNAXMQE X6QdY49JZ7E1fB5THFD2MeGAPaI8VRSWNgUCcLbOK3 EASPTVLWRe7I35Mc24FX9gHO8KbJdN1aB6GfZQUDCY cSFM9ECbY65DW3GA17RTBQUVOg4K2ZNXadLPeIH8Jf O9GFf8QEXdAMVN47SRUD13ZB5PH26cWTCJaebYKgLI RcJUI319KFMXH2fea8dCL6WTG5EO74DYSNZAVgQBPb M4CHOZdIQUgGE9DB7KJaTLeF2WfN5A38V1c6XbPRYS IJBCad3cSQ27OEegM6XFb8LK4YDfTWGR1ZHNA59PUV SRXA4D8GeHUQ7c2aVbNgJP1E3FTILOMBKf5d69YZWC fNAeKMS7gR34bWIF5BEYDX69QCUTaGV28LO1HdJcZP PfOD6SeKaXcF2Id4W9bQCU73RZLMBYA5g81GTNVEHJ JZ5VMTcQfLRaP1SG8DOb4CYHNEFAXKeIW7g2BU36d9 4HVNEgbTLcKUCA7Y95eRFGJIaXBPZ8QMDW3SO1fd62 bgT7VKfBH5dLQJ3E4N86aSCRZUGFcM21Y9eIPDOXAW CW3fXRg6NOSTKV9HEJYL8D5GbQPcIFBUd2A74M1eaZ KGY3ZX4CA8OcFPNfeSM17JDbE92aW56LRVIgUHdQTB NYP4C1ZM9EWSROLDGI786KVaXeQHfJcAFU2T3Bgb5d HdeGcYB413LZgTFQDfP7UASCKJWVEbRO6IN82aM9X5 BK7W8LYacIZAeMgX3OT4dF26DG9RUEJCHbPVSfN51Q ZL6K5NFVMCYbXfA8Oe43g7dUcDa9JTEQGSWB1PR2IH ``` ## Python ```python from random import choice, shuffle from copy import deepcopy def rls(n): if n <= 0: return [] else: symbols = list(range(n)) square = _rls(symbols) return _shuffle_transpose_shuffle(square) def _shuffle_transpose_shuffle(matrix): square = deepcopy(matrix) shuffle(square) trans = list(zip(*square)) shuffle(trans) return trans def _rls(symbols): n = len(symbols) if n == 1: return [symbols] else: sym = choice(symbols) symbols.remove(sym) square = _rls(symbols) square.append(square[0].copy()) for i in range(n): square[i].insert(i, sym) return square def _to_text(square): if square: width = max(len(str(sym)) for row in square for sym in row) txt = '\n'.join(' '.join(f"{sym:>{width}}" for sym in row) for row in square) else: txt = '' return txt def _check(square): transpose = list(zip(*square)) assert _check_rows(square) and _check_rows(transpose), \ "Not a Latin square" def _check_rows(square): if not square: return True set_row0 = set(square[0]) return all(len(row) == len(set(row)) and set(row) == set_row0 for row in square) if __name__ == '__main__': for i in [3, 3, 5, 5, 12]: square = rls(i) print(_to_text(square)) _check(square) print() ``` {{out}} ```txt 2 1 0 0 2 1 1 0 2 1 0 2 0 2 1 2 1 0 1 0 3 2 4 3 4 2 0 1 4 2 1 3 0 2 1 0 4 3 0 3 4 1 2 2 1 0 4 3 0 4 3 2 1 3 2 1 0 4 4 3 2 1 0 1 0 4 3 2 6 2 4 8 11 9 3 1 7 0 5 10 1 11 5 2 8 6 0 9 4 10 7 3 2 7 10 5 4 8 9 11 0 6 3 1 8 5 0 4 7 11 1 2 3 9 10 6 11 4 3 7 5 2 6 8 10 1 0 9 10 1 8 6 9 0 7 3 11 4 2 5 7 0 1 3 10 5 8 4 6 2 9 11 9 8 7 11 2 1 10 6 5 3 4 0 3 9 2 1 6 10 4 0 8 5 11 7 5 3 6 10 0 4 11 7 9 8 1 2 4 10 9 0 3 7 2 5 1 11 6 8 0 6 11 9 1 3 5 10 2 7 8 4 ``` ## REXX This REXX version produces a randomized Latin square similar to the '''Julia''' program. The symbols could be any characters (except those that contain a blank), but the numbers from '''0''' ──► '''N-1''' are used. ```rexx /*REXX program generates and displays a randomized Latin square. */ parse arg N seed . /*obtain the optional argument from CL.*/ if N=='' | N=="," then N= 5 /*Not specified? Then use the default.*/ if datatype(seed, 'W') then call random ,,seed /*Seed numeric? Then use it for seed.*/ w= length(N - 1) /*get the length of the largest number.*/ \$= /*initialize \$ string to null. */ do i=0 for N; \$= \$ right(i, w, '_') /*build a string of numbers (from zero)*/ end /*i*/ /* [↑] \$ string is (so far) in order.*/ z= /*Z: will be the 1st row of the square*/ do N; ?= random(1,words(\$)) /*gen a random number from the \$ string*/ z= z word(\$, ?); \$= delword(\$, ?, 1) /*add the number to string; del from \$.*/ end /*r*/ zz= z||z /*build a double-length string of Z. */ do j=1 for N /* [↓] display rows of random Latin sq*/ say translate(subword(zz, j, N), , '_') /*translate leading underbar to blank. */ end /*j*/ /*stick a fork in it, we're all done. */ ``` {{out|output|text= for 1st run when using the default inputs:}} ```txt 4 1 3 0 2 1 3 0 2 4 3 0 2 4 1 0 2 4 1 3 2 4 1 3 0 ``` {{out|output|text= for 2nd run when using the default inputs:}} ```txt 2 1 0 4 3 1 0 4 3 2 0 4 3 2 1 4 3 2 1 0 3 2 1 0 4 ``` ## Ruby This crude algorithm works fine up to a square size of 10; higher values take too much time and memory. It creates an array of all possible permutations, picks a random one as first row an weeds out all permutations which cannot appear in the remaining square. Repeat picking and weeding until there is a square. ```ruby N = 5 def generate_square perms = (1..N).to_a.permutation(N).to_a.shuffle square = [] N.times do square << perms.pop perms.reject!{|perm| perm.zip(square.last).any?{|el1, el2| el1 == el2} } end square end def print_square(square) cell_size = N.digits.size + 1 strings = square.map!{|row| row.map!{|el| el.to_s.rjust(cell_size)}.join } puts strings, "\n" end 2.times{print_square( generate_square)} ``` {{out}} ```txt 3 4 2 1 5 2 3 4 5 1 1 2 5 3 4 5 1 3 4 2 4 5 1 2 3 1 2 5 4 3 2 3 4 1 5 5 4 2 3 1 3 5 1 2 4 4 1 3 5 2 ``` ## zkl ```zkl fcn randomLatinSquare(n,symbols=[1..]){ //--> list of lists if(n<=0) return(T); square,syms := List(), symbols.walker().walk(n); do(n){ syms=syms.copy(); square.append(syms.append(syms.pop(0))) } // shuffle rows, transpose & shuffle columns T.zip(square.shuffle().xplode()).shuffle(); } fcn rls2String(square){ square.apply("concat"," ").concat("\n") } ``` ```zkl foreach n in (T(1,2,5)){ randomLatinSquare(n) : rls2String(_).println("\n") } randomLatinSquare(5, ["A".."Z"]) : rls2String(_).println("\n"); randomLatinSquare(10,"!@#\$%^&*()") : rls2String(_).println("\n"); ``` {{out}} ```txt 1 1 2 2 1 3 1 4 5 2 4 2 5 1 3 1 4 2 3 5 5 3 1 2 4 2 5 3 4 1 E D A B C D C E A B B A C D E A E B C D C B D E A & % # ! * @ ) \$ ( ^ @ ) * ^ # & % ( \$ ! ( & % # ) \$ @ ^ ! * ! ( & % @ ^ \$ * # ) % # ! ( ^ ) * @ & \$ ^ \$ @ ) & ! ( # * % # ! ( & \$ * ^ ) % @ \$ @ ) * % ( & ! ^ # ) * ^ \$ ! % # & @ ( * ^ \$ @ ( # ! % ) & ```