- Code: Select all
`. . . . . 1 . . .`

. . . . 3 4 . 2 1

. . . 6 2 . 5 3 .

. . 3 . 7 2 . 6 .

. 6 4 1 8 3 2 7 .

7 2 . 4 6 . 1 . .

. . 5 . 4 7 . 1 6

. 4 6 3 1 . 7 5 .

. 7 . . . . 3 . .

14 posts
• Page **1** of **1**

- Code: Select all
`. . . . . 1 . . .`

. . . . 3 4 . 2 1

. . . 6 2 . 5 3 .

. . 3 . 7 2 . 6 .

. 6 4 1 8 3 2 7 .

7 2 . 4 6 . 1 . .

. . 5 . 4 7 . 1 6

. 4 6 3 1 . 7 5 .

. 7 . . . . 3 . .

- dobrichev
- 2016 Supporter
**Posts:**1621**Joined:**24 May 2010

4 Singles : 3r6c9, 6r9c6, 5r6c6, 9r4c4.

The solutions of the puzzle must have the same symmetries.

Therefore along the main diagonal : r3c3 <> 1 and 7; r1c1 <> 2 and 4, 3 and 6; r9c9 <> 2 and 4; 18 Singles.

- Code: Select all
`+---------------------+-------------+-----------------+`

| 2345689 3589 2789 | 578 59 1 | 4689 489 4789 |

| 5689 589 789 | 578 3 4 | 689 2 1 |

| 1489 189 1789 | 6 2 89 | 5 3 4789 |

+---------------------+-------------+-----------------+

| 158 158 3 | 9 7 2 | 48 6 458 |

| 59 6 4 | 1 8 3 | 2 7 59 |

| 7 2 89 | 4 6 5 | 1 89 3 |

+---------------------+-------------+-----------------+

| 2389 389 5 | 28 4 7 | 89 1 6 |

| 289 4 6 | 3 1 89 | 7 5 289 |

| 1289 7 1289 | 258 59 6 | 3 489 2489 |

+---------------------+-------------+-----------------+

The solutions of the puzzle must have the same symmetries.

Therefore along the main diagonal : r3c3 <> 1 and 7; r1c1 <> 2 and 4, 3 and 6; r9c9 <> 2 and 4; 18 Singles.

- Code: Select all
`+-----------------+--------------+------------+`

| 59-8 3 2 | 7 59 1 | 6 89 4 |

| 6 589 7 | 58 3 4 | 89 2 1 |

| 4 1 9(8) | 6 2 9(8) | 5 3 7 |

+-----------------+--------------+------------+

| 1 58 3 | 9 7 2 | 4 6 58 |

| 59 6 4 | 1 8 3 | 2 7 59 |

| 7 2 89 | 4 6 5 | 1 89 3 |

+-----------------+--------------+------------+

| 3 89 5 | 2 4 7 | 89 1 6 |

| 9(8) 4 6 | 3 1 9(8) | 7 5 2 |

| 2 7 1 | 58 59 6 | 3 4 89 |

+-----------------+--------------+------------+

- JC Van Hay
**Posts:**719**Joined:**22 May 2010

JC that's a brilliant solution but I have a couple of comments/questions, neither of which affects the solution.

1. I think you meant a reflection, or transposition, about the top left to bottom right diagonal. Strictly speaking a 180 rotation of the puzzle doesn't give you the symmetries.

2. I get the symmetries 1 and 7, 2 and 4, 3 and 6, 5 and 5 but I can't see how you can infer 8 and 8, 9 and 9 symmetries, as there are no solved cells for these digits off the diagonal.

3. You've just taught me how to apply the symmetries - if, on the diagonal, two cells in a box don''t hold a pair of symmetry digits then the 3rd cell on the diagonal in that box can't hold them either - thanks for the lesson.

Leren

1. I think you meant a reflection, or transposition, about the top left to bottom right diagonal. Strictly speaking a 180 rotation of the puzzle doesn't give you the symmetries.

2. I get the symmetries 1 and 7, 2 and 4, 3 and 6, 5 and 5 but I can't see how you can infer 8 and 8, 9 and 9 symmetries, as there are no solved cells for these digits off the diagonal.

3. You've just taught me how to apply the symmetries - if, on the diagonal, two cells in a box don''t hold a pair of symmetry digits then the 3rd cell on the diagonal in that box can't hold them either - thanks for the lesson.

Leren

- Leren
**Posts:**3315**Joined:**03 June 2012

JC, thank you for this catastrophically compact simplification.

Leren, rotational symmetry resolves the central cell as in the following example

Pairing (4,6), (1,7), (2,5), (3,8) leaves only (9,9) which must occupy the central cell, ~stte.

Note that both puzzles are not only symmetric in pattern geometry, but also have automorphism.

I suppose horizontal/vertical mirror symmetry resolves the central column/row in a similar way that diagonal symmetry resolves the respective diagonal.

I am still curious whether rotational symmetries can do more, less obvious eliminations in the non-central cells. Thinking about the "symmetry of the symmetries" there should be either more eliminations or the central cell should be very special from the solver's perspective.

Cheers,

MD

Leren, rotational symmetry resolves the central cell as in the following example

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

. . 1 . 2 3 5 . 4

. 2 3 4 . 5 1 6 .

. . 4 . 6 2 3 8 .

2 . 6 3 . 8 4 . 5

. 3 8 5 4 . 6 . .

. 4 7 2 . 6 8 5 .

6 . 2 8 5 . 7 . .

. . . . . . . . . 38 givens, 180 degrees rotational symmetry

Pairing (4,6), (1,7), (2,5), (3,8) leaves only (9,9) which must occupy the central cell, ~stte.

Note that both puzzles are not only symmetric in pattern geometry, but also have automorphism.

I suppose horizontal/vertical mirror symmetry resolves the central column/row in a similar way that diagonal symmetry resolves the respective diagonal.

I am still curious whether rotational symmetries can do more, less obvious eliminations in the non-central cells. Thinking about the "symmetry of the symmetries" there should be either more eliminations or the central cell should be very special from the solver's perspective.

Cheers,

MD

- dobrichev
- 2016 Supporter
**Posts:**1621**Joined:**24 May 2010

Leren wrote:JC that's a brilliant solution but I have a couple of comments/questions, neither of which affects the solution.

1. I think you meant a reflection, or transposition, about the top left to bottom right diagonal. Strictly speaking a 180 rotation of the puzzle doesn't give you the symmetries.

2. I get the symmetries 1 and 7, 2 and 4, 3 and 6, 5 and 5 but I can't see how you can infer 8 and 8, 9 and 9 symmetries, as there are no solved cells for these digits off the diagonal.

3. You've just taught me how to apply the symmetries - if, on the diagonal, two cells in a box don''t hold a pair of symmetry digits then the 3rd cell on the diagonal in that box can't hold them either - thanks for the lesson.

Leren

leren, this is a classical main diagonal "symmetry of given".

having 3 paired groups, the 3 other digits occupy the main diagonal.

digit "9" is unpaired (or auto paired) and must be on the main diagonal.

Last edited by champagne on Fri Apr 24, 2015 7:04 am, edited 1 time in total.

- champagne
- 2017 Supporter
**Posts:**6609**Joined:**02 August 2007**Location:**France Brittany

dobrichev wrote:JC, thank you for this catastrophically compact simplification.

Leren, rotational symmetry resolves the central cell as in the following example

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

. . 1 . 2 3 5 . 4

. 2 3 4 . 5 1 6 .

. . 4 . 6 2 3 8 .

2 . 6 3 . 8 4 . 5

. 3 8 5 4 . 6 . .

. 4 7 2 . 6 8 5 .

6 . 2 8 5 . 7 . .

. . . . . . . . . 38 givens, 180 degrees rotational symmetry

Pairing (4,6), (1,7), (2,5), (3,8) leaves only (9,9) which must occupy the central cell, ~stte.

Note that both puzzles are not only symmetric in pattern geometry, but also have automorphism.

I suppose horizontal/vertical mirror symmetry resolves the central column/row in a similar way that diagonal symmetry resolves the respective diagonal.

I am still curious whether rotational symmetries can do more, less obvious eliminations in the non-central cells. Thinking about the "symmetry of the symmetries" there should be either more eliminations or the central cell should be very special from the solver's perspective.

Cheers,

MD

congratulations for these examples of "symmetry" with many clues.

"symmetry" gives some equivalences that lead to much simpler chains when the puzzle in not immediately solved.

Many examples can be found in that forum,

- champagne
- 2017 Supporter
**Posts:**6609**Joined:**02 August 2007**Location:**France Brittany

Mladen, Symmetrical solution grids are rich in what I call 'lattice houses'. These are Swordfish cell cell sets with each row and column in a different band that contain a full set of digits 1-9.

In this puzzle there are three lattice sets that hold the diagonal cells; a = r147c147, b= r269c269, & c= r358c358

Clearly if the coordinates of these lattice houses can be predicted from the recognised symmetry they would provide extra openings for naked and hidden singles & doubles etc that could be used to solve them.

Using the symmetries listed in < Eleven's Thread > I find that 12 of the 26 'normalised' and completely symmetrical grids give a full set of 9 non-overlapping lattice houses. However, not having any knowledge of group theory, I then get lost as it seems that either different sub-patterns are able to follow different symmetry rules or the progression of the digits in their cyclic groups aren't aligned.

Perhaps you may find this an interesting area to explore.

David

In this puzzle there are three lattice sets that hold the diagonal cells; a = r147c147, b= r269c269, & c= r358c358

- Code: Select all
`*-----------*-----------*-----------*`

| 5a 3 2 | 7a 9 1 | 6a 8 4 |

| 6 8b 7 | 5 3 4b | 9 2 1b |

| 4 1 9c | 6 2c 8 | 5 3c 7 |

*-----------*-----------*-----------*

| 1a 5 3 | 9a 7 2 | 4a 6 8 |

| 9 6 4c | 1 8c 3 | 2 7c 5 |

| 7 2b 8 | 4 6 5b | 1 9 3b |

*-----------*-----------*-----------*

| 3a 9 5 | 2a 4 7 | 8a 1 6 |

| 8 4 6c | 3 1c 9 | 7 5c 2 |

| 2 7b 1 | 8 5 6b | 3 4 9b |

*-----------*-----------*-----------*

Clearly if the coordinates of these lattice houses can be predicted from the recognised symmetry they would provide extra openings for naked and hidden singles & doubles etc that could be used to solve them.

Using the symmetries listed in < Eleven's Thread > I find that 12 of the 26 'normalised' and completely symmetrical grids give a full set of 9 non-overlapping lattice houses. However, not having any knowledge of group theory, I then get lost as it seems that either different sub-patterns are able to follow different symmetry rules or the progression of the digits in their cyclic groups aren't aligned.

Perhaps you may find this an interesting area to explore.

David

- David P Bird
- 2010 Supporter
**Posts:**1040**Joined:**16 September 2008**Location:**Middle England

Nice solution, JC.

I have one comment.

The thing that's true, is that if the puzzle has a unique solution, then it must have the same symmetry.

For an illustrative counterexample (for anyone who's interested) ... the puzzle below is a slightly modified version of dobrichev's puzzle.

It has the same initial singles, and the same symmetry in the givens.

It has two solutions, neither of which has that symmetry.

(Applying the symmetry operation, transforms each solution into the other).

I have one comment.

JC Van Hay wrote:The solutions of the puzzle must have the same symmetries.

The thing that's true, is that if the puzzle has a unique solution, then it must have the same symmetry.

For an illustrative counterexample (for anyone who's interested) ... the puzzle below is a slightly modified version of dobrichev's puzzle.

It has the same initial singles, and the same symmetry in the givens.

It has two solutions, neither of which has that symmetry.

(Applying the symmetry operation, transforms each solution into the other).

- Code: Select all
`Puzzle:`

+-------+-------+-------+

| . . . | . . 1 | . . . |

| . 5 . | . 3 4 | . 2 1 |

| . . . | 6 2 . | 5 3 . |

+-------+-------+-------+

| . . 3 | . 7 2 | . 6 . |

| . 6 4 | 1 8 3 | . 7 . |

| 7 2 . | 4 6 . | 1 . . |

+-------+-------+-------+

| . . 5 | . . 7 | . 1 6 |

| . 4 6 | 3 1 . | 7 5 . |

| . 7 . | . . . | 3 . . |

+-------+-------+-------+

Solutions:

+-------+-------+-------+ +-------+-------+-------+

| 6 3 2 | 5 9 1 | 8 4 7 | | 3 9 2 | 7 5 1 | 6 4 8 |

| 9 5 8 | 7 3 4 | 6 2 1 | | 6 5 7 | 8 3 4 | 9 2 1 |

| 4 1 7 | 6 2 8 | 5 3 9 | | 4 8 1 | 6 2 9 | 5 3 7 |

+-------+-------+-------+ +-------+-------+-------+

| 1 8 3 | 9 7 2 | 4 6 5 | | 5 1 3 | 9 7 2 | 8 6 4 |

| 5 6 4 | 1 8 3 | 9 7 2 | | 9 6 4 | 1 8 3 | 2 7 5 |

| 7 2 9 | 4 6 5 | 1 8 3 | | 7 2 8 | 4 6 5 | 1 9 3 |

+-------+-------+-------+ +-------+-------+-------+

| 3 9 5 | 8 4 7 | 2 1 6 | | 8 3 5 | 2 9 7 | 4 1 6 |

| 2 4 6 | 3 1 9 | 7 5 8 | | 2 4 6 | 3 1 8 | 7 5 9 |

| 8 7 1 | 2 5 6 | 3 9 4 | | 1 7 9 | 5 4 6 | 3 8 2 |

+-------+-------+-------+ +-------+-------+-------+

- blue
**Posts:**688**Joined:**11 March 2013

blue wrote:Nice solution, JC.

I have one comment.JC Van Hay wrote:The solutions of the puzzle must have the same symmetries.

The thing that's true, is that if the puzzle has a unique solution, then it must have the same symmetry.

Hi blue

it's good sometimes to refresh basics.

The proof of the "symmetry of given" final symmetry uses the uniqueness of the solution.

So, as for simpler rules as URs, the puzzle must be a valid sudoku (one and only one solution)

- champagne
- 2017 Supporter
**Posts:**6609**Joined:**02 August 2007**Location:**France Brittany

David P Bird wrote:Mladen, Symmetrical solution grids are rich in what I call 'lattice houses'. These are Swordfish cell cell sets with each row and column in a different band that contain a full set of digits 1-9.

...

Using the symmetries listed in < Eleven's Thread > I find that 12 of the 26 'normalised' and completely symmetrical grids give a full set of 9 non-overlapping lattice houses. However, not having any knowledge of group theory, I then get lost as it seems that either different sub-patterns are able to follow different symmetry rules or the progression of the digits in their cyclic groups aren't aligned.

Perhaps you may find this an interesting area to explore.

David

Hi David,

Thank you for the link. It looks valuable. I hope after some reading I will understand what the latice houses are.

- dobrichev
- 2016 Supporter
**Posts:**1621**Joined:**24 May 2010

Is JC's solution not the approach previously named "Gurth's symmetrical placement" (http://sudopedia.enjoysudoku.com/Gurth% ... ement.html)?

Phil

Phil

- pjb
- 2014 Supporter
**Posts:**2025**Joined:**11 September 2011**Location:**Sydney, Australia

Although "Gurth's symmetrical placement" is focused on 180-degree rotational symmetry, the approach is generally the same. For the secondary puzzle above, exactly this approach was demonstrated.

Thanks for the link.

Thanks for the link.

- dobrichev
- 2016 Supporter
**Posts:**1621**Joined:**24 May 2010

pjb wrote:Is JC's solution not the approach previously named "Gurth's symmetrical placement" (http://sudopedia.enjoysudoku.com/Gurth% ... ement.html)?

Phil

we had in that forum a long thread dedicated to all forms of the "symmetry of given". I made a quick search without any success, so it could be that the corresponding thread disappeared in the former crash of the forum.

Basically, the "symmetric placement" is valid in any symmetry accepting self symmetry (not valid for example in vertical or horizontal symmetry)

The most efficient use is when only 3 pairs are needed (giving 3 digits with a "self symmetry") this is true for diagonal and stick symmetry

EDIT I found the link to the thread I had in mind

here

- champagne
- 2017 Supporter
**Posts:**6609**Joined:**02 August 2007**Location:**France Brittany

dobrichev wrote:Hi David,

Thank you for the link. It looks valuable. I hope after some reading I will understand what the latice houses are.

To clarify, what I'm calling a lattice house is a full set of 9 digits contained in the intersection cells of three rows, one in each tier, and three columns, one in each stack.

- David P Bird
- 2010 Supporter
**Posts:**1040**Joined:**16 September 2008**Location:**Middle England

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