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`*-----------*`

|98.|...|...|

|6..|...|.5.|

|...|.12|3..|

|---+---+---|

|...|9..|6..|

|..1|...|9..|

|..4|..1|...|

|---+---+---|

|..7|89.|...|

|.5.|...|..4|

|...|...|.21|

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

12 posts
• Page **1** of **1**

No takers on the Two Trick Pony but this one is an absolute doddle !

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

|98.|...|...|

|6..|...|.5.|

|...|.12|3..|

|---+---+---|

|...|9..|6..|

|..1|...|9..|

|..4|..1|...|

|---+---+---|

|..7|89.|...|

|.5.|...|..4|

|...|...|.21|

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

- Leren
**Posts:**3284**Joined:**03 June 2012

Curious puzzle, no single digit solutions. Large deadly pattern?

Two steps: 1. Gurth's symmetrical placement => r5c5 = 5, then (3=7)r1c5 - r1c8 = r6c8 - (7=3)r6c4 => -3 r2c4; stte

Phil

Two steps: 1. Gurth's symmetrical placement => r5c5 = 5, then (3=7)r1c5 - r1c8 = r6c8 - (7=3)r6c4 => -3 r2c4; stte

Phil

- pjb
- 2014 Supporter
**Posts:**1959**Joined:**11 September 2011**Location:**Sydney, Australia

Close. Actually the puzzle has double diagonal symmetry, which gives immediately r5c5 = 5, r1c9 = 7, r9c1 = 3; stte. Is that one move or two? Not quite sure, care even less.

With that hint, someone may be brave enough to tackle the Two Trick Pony. Leren

With that hint, someone may be brave enough to tackle the Two Trick Pony. Leren

- Leren
**Posts:**3284**Joined:**03 June 2012

Thanks Leren

Can you please give a link to a source that explains the extra eliminations?

Phil

Can you please give a link to a source that explains the extra eliminations?

Phil

- pjb
- 2014 Supporter
**Posts:**1959**Joined:**11 September 2011**Location:**Sydney, Australia

Hi Phil, about the best I can do at the moment is to explain the principle of Diagonal Symmetry, using the current puzzle as an example.

What it boils down to is that if a puzzle has clues or solved cells that are symmetrical about a diagonal, with an isomorphism involving 3 sets of paired digits and three unpaired digits, then the unpaired digits are the only ones that can occupy the relevant diagonal.

So the Main diagonal has three unpaired digits 1, 5 and 9, so you can eliminate all digits except these three from the main diagonal.

The Anti Diagonal has three unpaired digits 3, 5 and 7, so you can remove all digits except these three from the Anti Diagonal.

I did a quick search to see whether I could find a statement of the theorem, but I came up blank. Like a lot of the more advanced solving topics, I haven't looked at them for so long I've forgotten most of what I knew when I coded them into my solver.

There are two other types of symmetry. Rotational Symmetry, which enables you do place r5c5, and Sticks Symmetry, which is so incredible that I wouldn't have been able to understand it, except that I came across a post by Mauricio on this topic with a worked example which I think might be here. After days of staring at it, was able to code it up.

Maybe one of the resident gurus, such as blue, Champagne or Jason, may be able to point you to a statement of the Diagonal Symmetry theorem.

Also, the two trick pony is one from a collection by blue, which has double diagonal symmetry but is otherwise very hard, with a rating of 10.5/10.5/9.8. No wonder I've had no takers so far on that one.

Leren

What it boils down to is that if a puzzle has clues or solved cells that are symmetrical about a diagonal, with an isomorphism involving 3 sets of paired digits and three unpaired digits, then the unpaired digits are the only ones that can occupy the relevant diagonal.

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

| 9 8 23 | 4 357 357 | 1 67 267 |

| 6 1 23 | 37 8 9 | 4 5 27 |

| 47 47 5 | 6 1 2 | 3 8 9 |

|-----------+----------------+-----------|

| 2 37 8 | 9 4 37-5 | 6 1 357 |

| 357 6 1 | 2 5-37 8 | 9 4 357 |

| 357 9 4 | 37-5 6 1 | 2 37 8 |

|-----------+----------------+-----------|

| 1 2 7 | 8 9 4 | 5 36 36 |

| 38 5 6 | 1 2 37 | 78 9 4 |

| 348 34 9 | 357 357 6 | 78 2 1 |

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

Main Diagonal (TLBR) Symmetry [24] [37] [68] + [1] [5] [9] => - 5 r4c6, r6c4, - 37 r5c5

So the Main diagonal has three unpaired digits 1, 5 and 9, so you can eliminate all digits except these three from the main diagonal.

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`*-----------------------------------------*`

| 9 8 23 | 4 357 357 | 1 6-7 7-26 |

| 6 1 23 | 37 8 9 | 4 5 2-7 |

| 47 47 5 | 6 1 2 | 3 8 9 |

|-------------+-------------+-------------|

| 2 37 8 | 9 4 37 | 6 1 357 |

| 357 6 1 | 2 5 8 | 9 4 357 |

| 357 9 4 | 37 6 1 | 2 37 8 |

|-------------+-------------+-------------|

| 1 2 7 | 8 9 4 | 5 36 36 |

| 8-3 5 6 | 1 2 37 | 78 9 4 |

| 3-48 4-3 9 | 357 357 6 | 78 2 1 |

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

Anti Diagonal (TRBL) Symmetry [19] [26] [48] + [3] [5] [7] => - 7 r1c8, r2c9, -26 r1c9, - 3 r8c1, r9c2, - 48 r9c1

The Anti Diagonal has three unpaired digits 3, 5 and 7, so you can remove all digits except these three from the Anti Diagonal.

I did a quick search to see whether I could find a statement of the theorem, but I came up blank. Like a lot of the more advanced solving topics, I haven't looked at them for so long I've forgotten most of what I knew when I coded them into my solver.

There are two other types of symmetry. Rotational Symmetry, which enables you do place r5c5, and Sticks Symmetry, which is so incredible that I wouldn't have been able to understand it, except that I came across a post by Mauricio on this topic with a worked example which I think might be here. After days of staring at it, was able to code it up.

Maybe one of the resident gurus, such as blue, Champagne or Jason, may be able to point you to a statement of the Diagonal Symmetry theorem.

Also, the two trick pony is one from a collection by blue, which has double diagonal symmetry but is otherwise very hard, with a rating of 10.5/10.5/9.8. No wonder I've had no takers so far on that one.

Leren

- Leren
**Posts:**3284**Joined:**03 June 2012

Not sure what you mean with "Diagonal Symmetry theorem".

Fact is, that if the givens are digit symmetric for a diagonal, i.e. for each given digit the given in the cell mirrored at the diagonal is the same (in the sample for the main diagonal 11,24,37,55,68,99), then also the solution is digit symmetric, if the puzzle is unique.

If it was not, you could mirror this solution and renumber the digits by changing them to the mirrored one, to get a valid (equivalent) grid, which is another solution for the givens.

Trivially the mirrored digits on the diagonal are the same and so those which mirror to themselves must occupy the solutions diagonal.

Fact is, that if the givens are digit symmetric for a diagonal, i.e. for each given digit the given in the cell mirrored at the diagonal is the same (in the sample for the main diagonal 11,24,37,55,68,99), then also the solution is digit symmetric, if the puzzle is unique.

If it was not, you could mirror this solution and renumber the digits by changing them to the mirrored one, to get a valid (equivalent) grid, which is another solution for the givens.

Trivially the mirrored digits on the diagonal are the same and so those which mirror to themselves must occupy the solutions diagonal.

- eleven
**Posts:**1798**Joined:**10 February 2008

Hi Leren,

No so many reactions to this thread and to his twin "Two trick Pony".

For me, not a lack of interest, quite the contrary: I have tried to understand and to deepen the topic of "automorphism"

First, thank you for your explanation of the diagonal symmetry. Thank you too for the link to Mauricio's post.

You mention having studied and coded Mauricio third sample puzzle. Could you give some hints on this one too, if I may ask so ? It looks so mysterious, at a first glance...

Reading Mauricio's post, I have caught that himself, ravel, Gurth, Red Ed, and others have discussed and eventually solved the issue of automorphic puzzles. This is rather old stuff, that can be found here and here

I need a lot of time to assimilate all this theory (new to me...)

Thanks for your teaching effort.

No so many reactions to this thread and to his twin "Two trick Pony".

For me, not a lack of interest, quite the contrary: I have tried to understand and to deepen the topic of "automorphism"

First, thank you for your explanation of the diagonal symmetry. Thank you too for the link to Mauricio's post.

You mention having studied and coded Mauricio third sample puzzle. Could you give some hints on this one too, if I may ask so ? It looks so mysterious, at a first glance...

Reading Mauricio's post, I have caught that himself, ravel, Gurth, Red Ed, and others have discussed and eventually solved the issue of automorphic puzzles. This is rather old stuff, that can be found here and here

I need a lot of time to assimilate all this theory (new to me...)

Thanks for your teaching effort.

Cenoman

- Cenoman
**Posts:**678**Joined:**21 November 2016**Location:**Paris, France

Hi Cenoman, about the best I can do at the moment is to present an example from my solver of a Sticks Symmetry solution. It's been so long since I looked at this I've forgotten exactly how it works.

3..9.57..17......2..8..14......1.8.9.3.6.7..............34.8..6.61...2..9..1....5

You can see from the PM that the Sticks Symmetry property requires r456c258 to contain 123 only, which solves the puzzle. I must admit that I've forgotten exactly how to work out the isomorphism or the location of the Sticks.

I'd have to revisit code in my solver that I haven't looked at for about 4 years. Sticks symmetry puzzles seem to be very rare but I do have a collection of them. If you want them I can post them in a hidden box.

Leren

3..9.57..17......2..8..14......1.8.9.3.6.7..............34.8..6.61...2..9..1....5

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

| 3 24 246 | 9 2468 5 | 7 168 18 |

| 1 7 456 | 38 3468 346 | 569 5689 2 |

| 256 9 8 | 27 267 1 | 4 56 3 |

|--------------------------+--------------------------+--------------------------|

| 24567 2-45 24567 | 235 1 234 | 8 23-4567 9 |

| 2458 3 2459 | 6 2-4589 7 | 15 12-45 14 |

| 245678 1 245679 | 2358 23-4589 2349 | 56 23-4567 47 |

|--------------------------+--------------------------+--------------------------|

| 257 25 3 | 4 2579 8 | 19 179 6 |

| 457 6 1 | 357 3579 39 | 2 4789 478 |

| 9 8 247 | 1 267 26 | 3 47 5 |

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

Column Sticks Symmetry : [45] [67] [89] [1] [2] [3] . Elimination Rows 4-6 Columns 258.

You can see from the PM that the Sticks Symmetry property requires r456c258 to contain 123 only, which solves the puzzle. I must admit that I've forgotten exactly how to work out the isomorphism or the location of the Sticks.

I'd have to revisit code in my solver that I haven't looked at for about 4 years. Sticks symmetry puzzles seem to be very rare but I do have a collection of them. If you want them I can post them in a hidden box.

Leren

- Leren
**Posts:**3284**Joined:**03 June 2012

Leren wrote:Hi Cenoman, about the best I can do at the moment is to present an example from my solver of a Sticks Symmetry solution. It's been so long since I looked at this I've forgotten exactly how it works.

3..9.57..17......2..8..14......1.8.9.3.6.7..............34.8..6.61...2..9..1....5

[...]

I'd have to revisit code in my solver that I haven't looked at for about 4 years. Sticks symmetry puzzles seem to be very rare but I do have a collection of them. If you want them I can post them in a hidden box.

Thank you, Leren for the example and the proposal.

As I said in my previous post, I have to assimilate all this. I need time to understand the theoretical aspects and their application to the example. I will ask for further help if I feel to be in a dead end.

Regards.

Cenoman

- Cenoman
**Posts:**678**Joined:**21 November 2016**Location:**Paris, France

See also my thread here, where all symmetries are listed with examples and solving techniques.

Btw. sticks symmetry is the most common symmetry, but of course also rare in random puzzles. I think, only 1 in 10000 grids has a symmetry (more exact 9770.1, and of course the percentage of puzzles for them, where all givens are symmetric, is small too) - and it would be well hidden, if it is not in "normal form".

Btw. sticks symmetry is the most common symmetry, but of course also rare in random puzzles. I think, only 1 in 10000 grids has a symmetry (more exact 9770.1, and of course the percentage of puzzles for them, where all givens are symmetric, is small too) - and it would be well hidden, if it is not in "normal form".

- eleven
**Posts:**1798**Joined:**10 February 2008

Hi eleven, the Mauricio, ER 9.2 puzzle example looks like the one I used to get the idea of sticks symmetry, but I thought i saw it in a separate post by Mauricio himself. Maybe that's just my memory playing tricks on me. Leren

- Leren
**Posts:**3284**Joined:**03 June 2012

As far as i know, it was found by Red Ed/ravel. In my thread all possible symmetries are covered (after extracting them from Red Ed's table).

Concerning stick symmetries i generated a lot of puzzles (in minutes), but almost all of them were directly solved by applying the numbers in the stick cells.

Concerning stick symmetries i generated a lot of puzzles (in minutes), but almost all of them were directly solved by applying the numbers in the stick cells.

- eleven
**Posts:**1798**Joined:**10 February 2008

12 posts
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