"Graeco-Latin Soduku" challenge

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"Graeco-Latin Soduku" challenge

Postby tinfoil » Fri Jun 17, 2005 6:29 pm

For reference, please see http://en.wikipedia.org/wiki/Graeco-Latin_square

A Graeco-Latin square (GLS) is one which contains TWO sets of symbols in each cell, where no row or column contains any symbol more than one, and yet every particular combination of symbols is represented in a cell (This can also be stated as 'no particular pairing of symbols is allowed to be repeated thoughout the puzzle).

For example, the following is a GLS of order 5 (or 5x5):

A1 B4 C2 D5 E3
B2 C5 D3 E1 A4
C3 D1 E4 A2 B5
D4 E2 A5 B3 C1
E5 A3 B1 C4 D2

Note that A-E appears one in each row and column, and so does 1-5, but A1 and B4 only appears once altogether.


The challenge: Create a 9x9 "Graeco-Latin Soduku" where each group of symbols respect all of the rules of Soduku, including the 3x3 box rule {For consistency, please use 1-9 as one group, and ABCDEFGHJ (skipped 'I' to avoid confusion with '1') as the other}, and no particular pairing of symbols to be repeated anywhere in the puzzle.

I have no I idea if this is even possible.
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Postby johnw » Fri Jun 17, 2005 10:19 pm

If I understand the proposition correctly, the following unimaginative grid should satisfy.

Code: Select all
A1 B2 C3 | D4 E5 F6 | G7 H8 J9
G4 H5 J6 | A7 B8 C9 | D1 E2 F3
D7 E8 F9 | G1 H2 J3 | A4 B5 C6
---------|----------|---------
C2 A3 B1 | F5 D6 E4 | J8 G9 H7
J5 G6 H4 | C8 A9 B7 | F2 D3 E1
F8 D9 E7 | J2 G3 H1 | C5 A6 B4
---------|----------|---------
B3 C1 A2 | E6 F4 D5 | H9 J7 G8
H6 J4 G5 | B9 C7 A8 | E3 F1 D2
E9 F7 D8 | H3 J1 G2 | B6 C4 A5


No doubt someone can proceed from there.
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Further thoughts...

Postby johnw » Fri Jun 17, 2005 10:43 pm

Look at the posting "Common sudoku variation in Japanese magazines" which is in the "non-pappocom" forum.

The grid I have posted above satisfies the extra rule too! so it satisfies nine rules:
normal sudoku rules (row, column, box) in 1-9
additional (position-in-box) rule in 1-9
normal sudoku rules (row, column, box) in A-J
additional (position-in-box) rule in A-J
Graeco-Latin uniqueness rule in A1-J9

Perhaps someone can test what the link is between the (position-in-box) rule and the Graeco-Latin uniqueness rule. I suspect there is a link...
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Postby tinfoil » Wed Jun 22, 2005 12:24 am

Good one johnw.

It looks so obvious once someone else does it for you!!

Next step: anyone feel like tackling the actual construction of one of these as a puzzle, where you would actually have to rely on the 'unique pair in puzzle' rule to solve it properly?

It would be most elegant if Pappocon's 'no T&E, unique solution' convenstions were followed as well.
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Challenge met

Postby pab » Fri Nov 28, 2008 8:46 am

Sorry about the delay
I suspect the original poster may not still be checking in.:)

Here they are --- http://www.boorer.freeuk.com

These are not too difficult, but still take longer than solving a pair
of sudoku of that level.
If people want some real stinkers I could look into that.


johnw wrote:Perhaps someone can test what the link is between the (position-in-box) rule and the Graeco-Latin uniqueness rule. I suspect there is a link...

There did not seem to be.


Paul
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Postby tinfoil » Sun Jan 04, 2009 3:47 pm

On the related link, paul boorer seems to be claiming to have 'invented' these, even though johnw's puzzle above predates paul's webpage.
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Postby coloin » Mon Jan 05, 2009 7:48 am

Of only a little relevance, but the extra constraint has been looked at in the past - for a single grid.

Buried in the maths section here is the relevant page for a single grid

http://forum.enjoysudoku.com/viewtopic.php?t=44&postdays=0&postorder=asc&start=405

number of grids = 554191840696*9! = 201105135151764480

The varient puzzles from such grids were termed "Sudoku DG"

http://forum.enjoysudoku.com/viewtopic.php?t=3284&start=0

min clues =11

To creat a difficult puzzle - It probably is possible to "combine" 2 moderate DG-puzzles from the above MC grid [or other valid GLS grid] - and maybe remove an unecessary clue.

C
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Postby David P Bird » Wed Jan 07, 2009 10:32 am

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Postby coloin » Wed Jan 07, 2009 10:39 pm

Thanks for that.

That would indicate that there arnt many grid combinations which fulfil the criteria.

And no non canonical grid pairs give a valid grid.

I can believe Red Ed when he says these are the only representative grids

Code: Select all
123456789456789123789123456231564897564897231897231564312645978645978312978312645
123456789564897231978312645456789123897231564312645978789123456231564897645978312
123456789645978312897231564978312645231564897456789123564897231789123456312645978
123456789789123456456789123645978312312645978978312645897231564564897231231564897
123456789897231564645978312312645978789123456564897231231564897978312645456789123
123456789978312645564897231789123456645978312231564897456789123312645978897231564


I am not sure about all the combinations:
6 self-paired grids [as first-posted] - probably equivalent
5+4+3+2+1 = 15 others are possibly not essentially different

The puzzles will all be uninspiring, given that the minirows are always repeated.

This also indicates that the puzzles from http://www.boorer.freeuk.com/ dont fulfil all the criteria.

C
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Postby David P Bird » Wed Jan 07, 2009 11:44 pm

coloin wrote:The puzzles will all be uninspiring, given that the minirows are always repeated.
I don't have time to follow up on the other items you've mentioned, but that statement isn't quite true! This is the grid I gave in that thread on 9th October:
Code: Select all
33 65 78 | 12 44 87 | 21 56 99
52 19 94 | 61 28 76 | 43 37 85
47 81 26 | 59 93 35 | 68 72 14
------------------------------
69 92 15 | 48 71 24 | 57 83 36
88 46 31 | 97 55 13 | 79 64 22
74 27 53 | 86 39 62 | 95 18 41
------------------------------
96 38 42 | 75 17 51 | 84 29 63
25 73 67 | 34 82 49 | 16 91 58
11 54 89 | 23 66 98 | 32 45 77
However, it's highly symmetrical and also has trellis 1-to-9 sets (the 9 cells with the same row/column coordinates in the boxes).
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Postby Red Ed » Thu Jan 08, 2009 7:36 am

coloin wrote:I can believe Red Ed when he says these are the only representative grids ...

I didn't say that. I only said that it was impossible to get more than six mutually orthogonal sudoku grids (e.g. those shown).
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Similarities with Setdoku & Sudoku^2

Postby tarek » Sun Jul 31, 2011 7:39 pm

I discovered this thread by chance ... The idea behind this puzzle has some similarities with both Setdoku (81 symbols) and with Sudoku^2 which I have been generating as well.

simon_blow_snow's symbols would fit this variant nicely :D

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