A special solution grid

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A special solution grid

Postby udosuk » Fri Mar 02, 2007 2:37 am

How special is this solution grid?
Code: Select all
+---+---+---+
|168|924|573|
|924|573|168|
|573|168|924|
+---+---+---+
|816|492|357|
|492|357|816|
|357|816|492|
+---+---+---+
|681|249|735|
|249|735|681|
|735|681|249|
+---+---+---+

1. The middle block (nonet 5) is the 3x3 magic square.

2. All mini-rows and mini-columns sum to 15.

3. 6 of the mini-diagonals sum to 15.

4. It is a Diagonal Sudoku (Sudoku X): the 2 main diagonals contain 1..9.

5. Besides, 4 more "broken diagonals" contain 1..9.

6. It is an "Emerald": symmetrical pairs of cells across the centre always sum to 10.

7. It is a "Disjoint Sets" Sudoku: all the 9 "spots" (same-positioned cells) within each block (e.g. r147c147, r258c258) contain 1..9.

8. Besides, if you make the "Disjoint Sets" conversion (i.e. nonets become rows and spots become columns), the resulting grid is also a Diagonal Sudoku: r159c159 and r357c357 both contain 1..9. (In fact, the resulting grid is just identical to the original grid.)

9. There are many other "3x3 rectangles" across the grid containing 1..9:
r147c159, r258c159, r369c159, r159c147, r159c258, r159c369
r147c357, r258c357, r369c357, r357c147, r357c258, r357c369

10. The grid is isomorphic to the "most canonical grid" (one of its 648 isomorphisms).

Now, what's left is just to design a puzzle that makes use of all these properties...:)
udosuk
 
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Re: A special solution grid

Postby Mauricio » Fri Mar 02, 2007 4:15 am

udosuk wrote:How special is this solution grid?
10. The grid is isomorphic to the "most canonical grid" (one of its 648 isomorphisms).

That is not correct, the MC grid has 648 automorphisms, and then it has (Total number of sudoku morphisms)/648 different isomorphic sudokus.

Besides, I do not think that grid is so special, after all is an isomorph of the MC grid.
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Re: A special solution grid

Postby m_b_metcalf » Fri Mar 02, 2007 2:57 pm

udosuk wrote:Now, what's left is just to design a puzzle that makes use of all these properties...:)


Well, here are three with at least the X property:
Code: Select all
 . . . . . . . . .
 9 . . 5 . 3 . . 8
 . 7 . . 6 . . 2 .
 . 1 6 . . . 3 5 .
 . . . . . . . . .
 . 5 7 . . . 4 9 .
 . 8 . . 4 . . 3 .
 2 . . 7 . 5 . . 1
 . . . . . . . . .      Symmetric, 22 givens, none on main diagonals, empty central box

Code: Select all
 . . . . . . . . .
 . . 4 5 . . . . .
 5 . . 1 6 . . . .
 . . 6 . . . . . .
 . . . 3 . 7 8 . .
 . . . . . . 4 9 .
 . 8 . . . 9 . . .
 . 4 . . . . . . .
 . 3 . 6 . . . . 9   17 givens

Code: Select all
 . . . 9 2 4 . . .
 . . 4 5 . 3 1 . .
 . 7 . . . . . 2 .
 . 1 . . . . . 5 .
 4 . . . . . . . 6
 . 5 . . . . . 9 .
 . 8 . . . . . 3 .
 . . 9 7 . 5 6 . .
 . . . 6 8 1 . . .   Symmetric, 24 givens, none on diagonals, empty central box

Regards,

Mike Metcalf
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Re: A special solution grid

Postby udosuk » Fri Mar 02, 2007 4:56 pm

Mauricio wrote:That is not correct, the MC grid has 648 automorphisms, and then it has (Total number of sudoku morphisms)/648 different isomorphic sudokus.

Thanks... I mixed up the term...:)

Mauricio wrote:Besides, I do not think that grid is so special, after all is an isomorph of the MC grid.

It's true being an isomorph of the MC grid gives it most of it's properties, but all these properties (diagonals/broken diagonals, emerald, disjoint sets, magic square, mini-row/mini-columns etc) allow us to generate a variant puzzle with very few clues...

Anyway, you're welcome to suggest another particular solutional grid which is more special than this one...:)


And thanks heaps to Mike Metcalf for the nice puzzles!:)
udosuk
 
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Re: A special solution grid

Postby m_b_metcalf » Sat Mar 03, 2007 7:09 pm

udosuk wrote:
And thanks heaps to Mike Metcalf for the nice puzzles!:)


You're most welcome. Here's another with just 16 cells. Can you solve it?
Code: Select all
 . . . . 2 . . . .
 . . . . . . 1 . .
 . . . . . 8 . . .
 8 . . . . 2 . 5 .
 . . 2 3 . . . . .
 . . . . . . . 9 .
 . . . . . . . 3 .
 . . . . . 5 6 . 1
 7 . . . 8 . 2 . .

Regards,

Mike Metcalf
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Re: A special solution grid

Postby m_b_metcalf » Sun Mar 04, 2007 11:46 pm

m_b_metcalf wrote:
udosuk wrote:
And thanks heaps to Mike Metcalf for the nice puzzles!:)


You're most welcome. Here's another with just 16 cells. Can you solve it?

And now a symmetric puzzle with just 22 clues and with 2 empty rows and columns (and both diagonals empty).
Code: Select all
 . . . . . . . . .
 . . . 5 . 3 . . .
 . 7 . 1 . 8 . 2 .
 . . 6 . 9 . 3 . .
 . 9 2 . . . 8 1 .
 . . 7 . 1 . 4 . .
 . 8 . 2 . 9 . 3 .
 . . . 7 . 5 . . .
 . . . . . . . . .

Regards,

Mike Metcalf
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Postby udosuk » Mon Mar 05, 2007 3:22 am

Mike M, thanks for your puzzles... They're very hard for me, even with the diagonal and DS properties... Perhaps there're some advanced emerald moves that can help solving them, but I haven't looked too deeply... I don't suppose they would need ALS, xyzw-wings, ultimate fish etc involving the diagonals & disjoint sets?:!:
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Postby ravel » Mon Mar 05, 2007 12:34 pm

Using properties 6 (Emerald), 2 (mini-rows and mini-columns sum to 15) and 4 (Sudoku X) i did not have more than locked candidates and pairs for the 2 puzzles.
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Postby m_b_metcalf » Tue Mar 06, 2007 12:24 am

udosuk wrote:Mike M, thanks for your puzzles...

I've put some others here (not based on your grid). I've resuscitated my x-code and found that I could make it more aggressive at removing redundant cells.

Regards,

Mike Metcalf
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Postby m_b_metcalf » Tue Mar 06, 2007 2:29 am

ravel wrote:Using properties 6 (Emerald), 2 (mini-rows and mini-columns sum to 15) and 4 (Sudoku X) i did not have more than locked candidates and pairs for the 2 puzzles.

Ravel,
For your comments. I think these Xs are harder (but not based on Udosuk's grid!).
Code: Select all
 . . . . . 4 . 9 .
 8 . 6 1 . . . . .
 5 . . . . 6 1 . .
 . . . . . . . 5 .
 . . 3 . . . 4 . .
 2 5 . . 3 . 7 . .
 . . . . . . . . .
 . 2 8 . . . . 7 .
 . . . . 7 . . . .    19 clues

 . . . . . . . . .
 . . . 8 2 9 . . .
 . 4 . . . . . 3 .
 . . 8 . 5 . 7 . .
 . 9 7 . . . 6 5 .
 . . 1 . 3 . 4 . .
 . 6 . . . . . 8 .
 . . . 5 8 1 . . .
 . . . . . . . . .    20 clues, symmetric

 . . . . . . . . .
 . . 2 . 7 . 1 . .
 . 7 . . 9 . . 4 .
 . . 7 2 . 8 4 . .
 . . . . . . . . .
 . . 1 9 . 6 2 . .
 . 6 . . 2 . . 5 .
 . . 8 . 3 . 6 . .
 . . . . . . . . .   20 symmetric


 . . 7 6 . 2 8 . .
 . . . . 3 . . . .
 . . . 5 . 9 . . .
 9 . . . . . . . 2
 . 4 . . . . . 1 .
 8 . . . . . . . 7
 . . . 3 . 4 . . .
 . . . . 7 . . . .
 . . 6 1 . 8 4 . .    20 symmetric

 . . . . . . . . .
 . . 5 . 1 . 8 . .
 . . . 4 8 9 . . .
 . . 4 9 . 8 6 . .
 . 8 . . . . . 9 .
 . . 3 7 . 6 4 . .
 . . . 3 6 5 . . .
 . . 2 . 9 . 3 . .
 . . . . . . . . .   22 symmetric

 . . . 8 . 3 . . .
 . . . . . . . . .
 6 5 4 . . . 3 8 2
 . . 5 . . . 6 . .
 . 1 . . . . . 2 .
 . . 7 . . . 9 . .
 1 7 9 . . . 2 4 6
 . . . . . . . . .
 . . . 4 . 1 . . .     22 symmetric
 
 . . . . . . . . .
 9 . . 5 . 6 . . 3
 . . 1 7 . 3 5 . .
 . . 6 . . . 2 . .
 4 . . . . . . . 8
 . . 8 . . . 9 . .
 . . 9 3 . 5 7 . .
 2 . . 4 . 7 . . 6   
 . . . . . . . . .    22 symmetric

 . 6 . . . . . 4 .
 9 . . 6 . 4 . . 7
 . . . 9 2 3 . . .
 8 . . . . . . . 4
 . . . . . . . . .
 6 . . . . . . . 1
 . . . 7 5 8 . . .
 2 . . 1 . 9 . . 3
 . 5 . . . . . 1 .    22 symmetric

Regards,

Mike Metcalf
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Postby Scott H » Tue Mar 06, 2007 3:24 am

m_b_metcalf wrote:
ravel wrote:Using properties 6 (Emerald), 2 (mini-rows and mini-columns sum to 15) and 4 (Sudoku X) i did not have more than locked candidates and pairs for the 2 puzzles.

Ravel,
For your comments. I think these Xs are harder (but not based on Udosuk's grid!).
Code: Select all
 . . . . . 4 . 9 .
 8 . 6 1 . . . . .
 5 . . . . 6 1 . .
 . . . . . . . 5 .
 . . 3 . . . 4 . .
 2 5 . . 3 . 7 . .
 . . . . . . . . .
 . 2 8 . . . . 7 .
 . . . . 7 . . . .    19 clues

[snip]
 

Regards,

Mike Metcalf

Mike, What are the constraints on these puzzles, other than X? The 19 clue one clearly isn't Emerald (constraint 6) or mini-magic (constraint 3). Thanks.
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Postby udosuk » Tue Mar 06, 2007 7:23 am

ravel wrote:Using properties 6 (Emerald), 2 (mini-rows and mini-columns sum to 15) and 4 (Sudoku X) i did not have more than locked candidates and pairs for the 2 puzzles.

Thanks ravel... I didn't try to use the "mini-rows/mini-columns sum to 15" property, which I guess is too powerful... Being a keen Killer Sudoku player should allow me to deal with those puzzles without any trouble using that rule...

I thought Mike M implied that those puzzles were solvable using the main diagonal (X) property only...
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Postby m_b_metcalf » Tue Mar 06, 2007 2:04 pm

Scott H wrote:Mike, What are the constraints on these puzzles, other than X?


udosuk wrote:I thought Mike M implied that those puzzles were solvable using the main diagonal (X) property only...


All my X puzzles use only the X constraint. Sorry for any confusion.

Regards,

Mike
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Re: A special solution grid

Postby Scott H » Thu Mar 08, 2007 8:36 am

udosuk wrote:[snip]
Now, what's left is just to design a puzzle that makes use of all these properties...:)

Here's a unique solution puzzle designed using just 3 of the 10 constraints: #2 (minimagic), #4 (diagonal), and #6 (Emerald). Constraint #1 follows automatically from #2 and #6. Enjoy!
Code: Select all
 + --- + --- + --- +
 | ... | ... | 2.. |
 | ... | ... | ... |
 | ... | ... | ... |
 + --- + --- + --- +
 | ... | ... | ... |
 | ... | ... | ... |
 | ... | ... | 4.. |
 + --- + --- + --- +
 | ... | .3. | ... |
 | ... | ... | ... |
 | ... | ... | ... |
 + --- + --- + --- +
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Postby Scott H » Thu Mar 08, 2007 9:01 am

Here's another, using constraints 2, 4, 6, and 7.
Code: Select all
 + --- + --- + --- +
 | 4.. | ... | ... |
 | ... | ... | ... |
 | ... | ... | ... |
 + --- + --- + --- +
 | ... | ... | ... |
 | ... | ... | ... |
 | ... | ... | ... |
 + --- + --- + --- +
 | ... | ... | ... |
 | ... | 2.. | ... |
 | ... | ... | ... |
 + --- + --- + --- +
Scott H
 
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