As a varient, some say forerunner of Sudoku, It desrves a recent thread.

Recently a version of Sudoku Explainer for Latin Squares has been developed by workers

A more up to date version of Sudoku Explainer which can also process Latin Square puzzles is available here as SukakuExplainer

Relevant discussions regarding the historical origins etc of Latin squares are leonard Euler and there is probably more written on the forum in the depths.

Latin squares can be size n, and for these discussion purposes Latin Squares refers to the 9*9 Latin Square grid [solution grid]case.

There are significantly more 9*9 LS solution grids

Latin square puzzles can be termed LS or LSQ puzzles or historically LQ [Q also for quasi]

Sudoku solution grids, ie 9*9/3*3 are all 9*9 LS solution grids

Two grids [actually the same LS grid] of note emerged recently, generated using a method suggested by qiuyanzhe and it was an easy way to get low clue puzzles

- Code: Select all
`123456789 1234.....`

234567891 234......

345678912 34.......

456789123 4........

567891234 .........

678912345 ........5

789123456 .......56

891234567 ......567

912345678 .....5678 Valid 20 Clue LS puzzle

1234.....234......34.......4.........................5.......56......567.....5678

This one was originally found as valid sudoku puzzle which coincidentally was a valid LS puzzle

- Code: Select all
`123456789 1234.....`

456789123 4........

789123456 .......56

234567891 234......

567891234 .........

891234567 ......567

345678912 34.......

678912345 ........5

912345678 .....5678 Valid 20 clue Sudoku and LS puzzle - grid named "Back Circulant" BC grid

1234.....4...............56234.....................56734...............5.....5678

using qiuyanzhe's method here these are easily made ... but it is not proven that this is the miniumum

Overall there are more Essentially Different LSQ grid solutions [11e11 isotopes [In wikipedia isotopes can be read as isomorphs]] .... and probably more minimal puzzles per grid solution

And paradoxically these 20C latin square puzzles have been found / constructed, even before 21C and larger puzzles !!

From initial searches its probably a safe bet that this will be the minimum.

The upper bound for the minimum number of clues for a latin square of sise n as stated above was n^2 / 4 e.g 9^2 / 4 ~ 20

Looking at the examples below this is the same as [n/2 ]^2 and this would seem to hold for larger n - albeit a little more difficult to prove !

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`8*8`

1 2 3 4 . . . .

2 3 4 . . . . .

3 4 . . . . . .

4 . . . . . . .

. . . . . . . .

. . . . . . . 5

. . . . . . 5 6

. . . . . 5 6 7 16 clues

7*7

1 2 3 . . . .

2 3 . . . . .

3 . . . . . .

. . . . . . .

. . . . . . 4

. . . . . 4 5

. . . . 4 5 6 12 clues

6*6

1 2 3 . . .

2 3 . . . .

3 . . . . .

. . . . . .

. . . . . 4

. . . . 4 5 9 clues

5*5

1 2 . . .

2 . . . .

. . . . .

. . . . 3

. . . 3 4 6 clues

4*4

1 2 . .

2 . . .

. . . .

. . . 3 4 clues

3*3

1 . .

. . .

. . 2 2 clues