## A special solution grid

Everything about Sudoku that doesn't fit in one of the other sections

### A special solution grid

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

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.
Mauricio

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Joined: 22 March 2006

### Re: A special solution grid

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:
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` . . . . . . . . .  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

m_b_metcalf
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### Re: A special solution grid

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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Joined: 17 July 2005

### Re: A special solution grid

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

m_b_metcalf
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### Re: A special solution grid

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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Location: Berlin

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?
udosuk

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Joined: 17 July 2005

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

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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

m_b_metcalf
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Posts: 10415
Joined: 15 May 2006
Location: Berlin

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

m_b_metcalf
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Joined: 15 May 2006
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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.
Scott H

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Joined: 28 July 2005

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...
udosuk

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Joined: 17 July 2005

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

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. | ... | | ... | ... | ... | | ... | ... | ... | + --- + --- + --- +`
Scott H

Posts: 73
Joined: 28 July 2005

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

Posts: 73
Joined: 28 July 2005

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