A 4x4 Persian Sudoku grid occurs in this 16th century manuscript (in MS P 28, fol. 52a), although the marginalia are of an unknown date.

Question: Among all 4x4 Latin squares, what proportion are Sudoku grids?

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A 4x4 Persian Sudoku grid occurs in this 16th century manuscript (in MS P 28, fol. 52a), although the marginalia are of an unknown date.

Question: Among all 4x4 Latin squares, what proportion are Sudoku grids?

Question: Among all 4x4 Latin squares, what proportion are Sudoku grids?

- r.e.s.
**Posts:**337**Joined:**31 August 2005

r.e.s.'s original question is easily answered; according to the OEIS (see http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A002860 which counts the Latin squares of order n), there are 576 4x4 Latin squares. On another forum (see http://www.setbb.com/phpbb/viewtopic.php?t=27&mforum=sudoku - about the third or fourth message), the number of 4x4 sudokus was worked out at 288. So exactly half of all Latin squares of order 4 are 4x4 sudokus! Presumably there will be a simple explanation for this (?), but I'm going to be too busy this week at work to devote any thought to it! Perhaps at the weekend...

Frazer

Frazer

- frazer
**Posts:**46**Joined:**06 June 2005

frazer wrote:r.e.s.'s original question is easily answered; according to the OEIS (see http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A002860 which counts the Latin squares of order n), there are 576 4x4 Latin squares. On another forum (see http://www.setbb.com/phpbb/viewtopic.php?t=27&mforum=sudoku - about the third or fourth message), the number of 4x4 sudokus was worked out at 288. So exactly half of all Latin squares of order 4 are 4x4 sudokus! Presumably there will be a simple explanation for this (?), but I'm going to be too busy this week at work to devote any thought to it! Perhaps at the weekend...

Frazer

Right -- I'd found both numbers on OEIS and was also surprised, considering that the corresponding proportion in the 9x9 case is about 10**-6 (that is, about one in a million).

Thinking this might be entertaining for the denizens of the rec.puzzles newsgroup, I also posted the question there ... One person reported finding the answer by letting his sudoku solver count the solutions starting with an empty grid, then reconfiguring it to count Latin square solutions!

- r.e.s.
**Posts:**337**Joined:**31 August 2005

[quote="r.e.s."]A 4x4 Persian Sudoku grid occurs in this 16th century manuscript (in MS P 28, fol. 52a), although the marginalia are of an unknown date.

How do you know it is a Sudoku grid? It looks to me like a regular latin square.

How do you know it is a Sudoku grid? It looks to me like a regular latin square.

- giant
**Posts:**18**Joined:**09 August 2005

A Latin Square has only two constraints -- each symbol appears once in each row and column. The solution to a Sudoku has a third constraint -- the boxes.

[I] A simple 4x4 Latin Square:

[II] A 4x4 Latin Square that is also a solution to a 4x4 Sudoku:

[III] A 4x4 Latin Square that is also a solution to a 4x4 Sudoku with diagonal constraints:

(This is the Persion square written in the margin.)

[IV] A 4x4 Euler Square (aka Graeco-Latin Square) with diagonal constraints:

(This was formed by taking [III] and combining it with a copy of itself rotated 90 degrees. Each number is paired with each digit once. Each number and digit appears once in each row, column, 2x2 box and main diagonal.)

EDIT -- added for completeness:

Three mutually orthogonal (the maximum for a 4x4 array) Latin Squares.

No character appears twice in a row or column.

No character appears twice with any other symbol.

By the way, I think this illustrates that it's absurd to suggest that Euler -- or *anyone* -- invented or discovered the simple Latin Square -- it simply too simple! This is a 9x9 Latin Square:

I wonder what percentage of school children do NOT discover this on their own? I KNOW that I did, long before I'd heard of Euler, Latin Squares, etc.

[I] A simple 4x4 Latin Square:

- Code: Select all
`1 2 3 4`

2 3 4 1

3 4 1 2

4 1 2 3

[II] A 4x4 Latin Square that is also a solution to a 4x4 Sudoku:

- Code: Select all
`1 2 3 4`

3 4 1 2

2 3 4 1

4 1 2 3

[III] A 4x4 Latin Square that is also a solution to a 4x4 Sudoku with diagonal constraints:

(This is the Persion square written in the margin.)

- Code: Select all
`1 2 3 4`

4 3 2 1

2 1 4 3

3 4 1 2

[IV] A 4x4 Euler Square (aka Graeco-Latin Square) with diagonal constraints:

(This was formed by taking [III] and combining it with a copy of itself rotated 90 degrees. Each number is paired with each digit once. Each number and digit appears once in each row, column, 2x2 box and main diagonal.)

- Code: Select all
`1d 2a 3c 4b`

4c 3b 2d 1a

2b 1c 4a 3d

3a 4d 1b 2c

EDIT -- added for completeness:

Three mutually orthogonal (the maximum for a 4x4 array) Latin Squares.

No character appears twice in a row or column.

No character appears twice with any other symbol.

- Code: Select all
`4dδ 1aα 2bβ 3cγ`

1bγ 4cβ 3dα 2aδ

2cα 3bδ 4aγ 1dβ

3aβ 2dγ 1cδ 4bα

By the way, I think this illustrates that it's absurd to suggest that Euler -- or *anyone* -- invented or discovered the simple Latin Square -- it simply too simple! This is a 9x9 Latin Square:

- Code: Select all
`1 2 3 4 5 6 7 8 9`

2 3 4 5 6 7 8 9 1

3 4 5 6 7 8 9 1 2

4 5 6 7 8 9 1 2 3

5 6 7 8 9 1 2 3 4

6 7 8 9 1 2 3 4 5

7 8 9 1 2 3 4 5 6

8 9 1 2 3 4 5 6 7

9 1 2 3 4 5 6 7 8

I wonder what percentage of school children do NOT discover this on their own? I KNOW that I did, long before I'd heard of Euler, Latin Squares, etc.

Last edited by tso on Thu Sep 22, 2005 12:53 pm, edited 1 time in total.

- tso
**Posts:**798**Joined:**22 June 2005

giant wrote:Thnaks tso.

I wish I had a better copy of this folio than the one I get from the NLB site (I can hardly recognize the signs in the square). It would be interesting to translate this passage. If you have a better copy I would be able to get it translated.

I think you may not realise that the image on that page is a link to a higher-resolution image. Here's a snippet:

It would be great to have a translation of the four grid-symbols (or any of the margin text)!

EDIT: Having just looked at http://www.iranchamber.com/scripts/persian_alphabet.php

my uneducated guess is that the grid contains the Persian numerals for 2,4,6,8, as follows:

- Code: Select all
`8 6 4 2`

2 4 6 8

6 8 2 4

4 2 8 6

If this is correct, 20 is the sum of every row, column, diagonal, and of all nine 2x2 subsquares with adjacent cells, and of all four 2x2 subsquares with alternating cells.

Last edited by r.e.s. on Sat Sep 24, 2005 1:04 am, edited 2 times in total.

- r.e.s.
**Posts:**337**Joined:**31 August 2005

Your transklation of the square itself is correct. The text is written in the ancient variant caled dari (something which is still in use in parts of Tadjikistan), but, according to the guy I asked, is purely medical without association with the square. The text written next to the square is in yet another, very obscure tongue, and I haven't got any translation of that.

The square on the page 2851b contains only letters,

t z h j a

h j a t z

t a z h j

z h j a t

j a t z h

but is only a latin square of order 5.

I will se if I can find someone else who will be able to give more information.

Concerning the history of latin squares so the were known in the Islamic countries in 12th century, mostly used on amulets. Sometimes, on 4x4 squares one used the letters of the name of God.

In Europe latin squaers were known i 16th century (for example in Libro de Abaco of Tagliente - 1515), and Jacques Ozanam (Recreations Mathematiques - 1725) had an pair of orthogonal latin squares of order 4 using four suits of cards.

Euler's Recherches sur une nouvelle espèce de Quarrés Magiques came first 1782 and was account of his work on the problem of 36 officers.

The square on the page 2851b contains only letters,

t z h j a

h j a t z

t a z h j

z h j a t

j a t z h

but is only a latin square of order 5.

I will se if I can find someone else who will be able to give more information.

Concerning the history of latin squares so the were known in the Islamic countries in 12th century, mostly used on amulets. Sometimes, on 4x4 squares one used the letters of the name of God.

In Europe latin squaers were known i 16th century (for example in Libro de Abaco of Tagliente - 1515), and Jacques Ozanam (Recreations Mathematiques - 1725) had an pair of orthogonal latin squares of order 4 using four suits of cards.

Euler's Recherches sur une nouvelle espèce de Quarrés Magiques came first 1782 and was account of his work on the problem of 36 officers.

- giant
**Posts:**18**Joined:**09 August 2005

giant wrote:Your transklation of the square itself is correct. [...] The text written next to the square is in yet another, very obscure tongue, and I haven't got any translation of that.

Thanks to both you and your translator-friend. Of course it's the marginal text in the snippet above that I expect to be most-relevant to the grid.

Considering the many summations to 20 in this 4x4 Persian Sudoku grid, I can't help but wonder whether there exists an analogous 9x9 Sudoku grid that sums to 45 in both main diagonals and in every extended diagonal, every 3x3 subsquare of adjacent cells, and every 3x3 subsquare of alternating cells. (By "extended diagonal" I mean the composite of the pair of parallel diagonals comprising 9 cells.)

EDIT: Oops ... I forgot to require that, analogous to the 4x4 Persian Sudoku, each of the main diagonals should contain all nine digits.

Last edited by r.e.s. on Thu Sep 22, 2005 7:37 pm, edited 2 times in total.

- r.e.s.
**Posts:**337**Joined:**31 August 2005

- Code: Select all
`1 2 3 4 5 6 7 8 9`

4 5 6 7 8 9 1 2 3

7 8 9 1 2 3 4 5 6

2 3 1 5 6 4 8 9 7

5 6 4 8 9 7 2 3 1

8 9 7 2 3 1 5 6 4

3 1 2 6 4 5 9 7 8

6 4 5 9 7 8 3 1 2

9 7 8 3 1 2 6 4 5

I believe all of the following sum to 45:

1) Rows and columns.

2) Main and broken diagonals.

3) All orthoganally oriented 3x3 squares, including those that wrap around.

4) Cells in this pattern:

- Code: Select all
`x . x . x`

. . . . .

x . x . x

. . . . .

x . x . x

... including those that wrap around.

5) Cells in this pattern:

- Code: Select all
`x . . x . . x`

. . . . . . .

. . . . . . .

x . . x . . x

. . . . . . .

. . . . . . .

x . . x . . x

... including those that wrap around.

6) Cells in this pattern:

- Code: Select all
`x . . . x . . . x`

. . . . . . . . .

. . . . . . . . .

x . . . x . . . x

. . . . . . . . .

. . . . . . . . .

x . . . x . . . x

... including those that wrap around.

Also, cells in this pattern:

- Code: Select all
`. . x . .`

. x . x .

x . x . x

. x . x .

. . x . .

... ususally, but not always, sum to 45 (including those that wrap around). Someone else can check which ones do and under what circumstances. Same goes for:

- Code: Select all
`. . . . x . . . .`

. . . . . . . . .

. . x . . . x . .

. . . . . . . . .

x . . . x . . . x

. . . . . . . . .

. . x . . . x . .

. . . . . . . . .

. . . . x . . . .

Did I miss any?

(The "magic square" term for diagonals that 'wrap around' as it were a manifold or a torus is "broken diagonal". A magic square in which all the broken diagonals work is a "pan-diagonal" square.)

- tso
**Posts:**798**Joined:**22 June 2005

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