## Gurth's Puzzles

Everything about Sudoku that doesn't fit in one of the other sections
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`. 5 .|. . 1|. 7 .8 . .|. 3 .|. . .. . 1|5 . .|4 . .-----+-----+-----2 . .|. 6 .|5 . .. 4 .|7 . .|. 2 .. . 5|. 8 3|. . 4-----+-----+-----. . 2|. . 5|3 . .. . .|. 1 .|. . 6. 9 .|3 . .|. 5 . ER=10.5, gsfr=99838, it starts with 7 singles`

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`. 5 .|. 4 .|. 9 .6 . .|. . 2|. . .. . 3|5 . .|2 . .-----+-----+-----. 3 .|. 9 .|5 . .1 . .|. . 8|. . 3. . 5|2 7 .|. 1 .-----+-----+-----. . 4|. . 5|1 . .. . .|4 . .|. . 8. 7 .|. 2 .|. 5 . ER=10.5, gsfr=99839, it starts with 7 singles`

Are they isomorph?
Thes two jade starts with 7 singles and then a SE step of rating 10.5!, but after the 7 singles we have a 90 degrees symmetry, and we *could* apply the symmetry technique to get an easier step, suggestions?

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`First non trivial SE step is a Dynamic Contradiction Forcing Chains (+ Multiple Forcing Chains) with 34 views6 . .|. . .|. . 9. 2 .|. 4 .|. 1 .. . 3|5 . .|2 . .-----+-----+-----. . .|. 9 .|5 . .. 1 .|. . 8|. 3 .. . 5|2 7 3|. . .-----+-----+-----. . 4|. . .|1 . .. 3 .|. 2 .|. 4 57 5 .|4 . .|. . 8 ER=11.0, it starts with 10 singles, then a SE step of rating 11.0`

After singles this jade has a 90 degree rotational symmetry, suggestions for an easier step?
Mauricio

Posts: 1174
Joined: 22 March 2006

### ST

You guys are proceeding by leaps and bounds - every time I come to post some congratulations, I see a whole lot of new stuff I must take away and digest - great work, guys!

RW is always hot when it comes to inventing new techniques, so is udosuk, and now Mauricio as composer you have produced these staggering 10.6 ultras that are "defenseless" as you so rightly say. Congratulations!!! Your 90-degree symmetry sounds like an exciting development, I must still look at it, and udosuk's non-morphing technique too.
____________________________________________________
gurth

Posts: 358
Joined: 11 February 2006
Location: Cape Town, South Africa

Mauricio wrote:
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`First non trivial SE step is a Dynamic Contradiction Forcing Chains (+ Multiple Forcing Chains) with 34 views6 . .|. . .|. . 9. 2 .|. 4 .|. 1 .. . 3|5 . .|2 . .-----+-----+-----. . .|. 9 .|5 . .. 1 .|. . 8|. 3 .. . 5|2 7 3|. . .-----+-----+-----. . 4|. . .|1 . .. 3 .|. 2 .|. 4 57 5 .|4 . .|. . 8 ER=11.0, it starts with 10 singles, then a SE step of rating 11.0`

After singles this jade has a 90 degree rotational symmetry, suggestions for an easier step?
Though fully symmetrical, the puzzle is still hard. My program has 14 points without using symmetry and 7 steps, when using.
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` *----------------------------------------------------------* | 6     478   178   | 378   138  2     | 3478  5     9     | | 5     2     789   | 3789  4    679   | 3678  1     367   | | 1489  4789  3     | 5     168  1679  | 2     678   467   | |-------------------+------------------+-------------------| | 3     678   2678  | 1     9    4     | 5     2678  267   | | 249   1     279   | 6     5    8     | 479   3     247   | | 489   4689  5     | 2     7    3     | 4689  689   1     | |-------------------+------------------+-------------------| | 289   689   4     | 3789  368  5     | 1     2679  2367  | | 189   3     1689  | 789   2    1679  | 679   4     5     | | 7     5     1269  | 4     136  169   | 369   269   8     | *----------------------------------------------------------*`
From the starting grid you can show r1c2<>8 and r2c9<>7 with singles, LC and pair, but it does not help much.
ravel

Posts: 998
Joined: 21 February 2006

### Threefold symmetry

The simmetry may be threefold too. I'll show several examples, without simmetry steps, so people can work out the steps (if there are any(steps)).
Horizontal:
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`Version 11 . .|. . 4|. 8 .. 9 .|1 . .|. . 5. . 6|. 7 .|1 . .-----+-----+-----. 3 .|. . 2|4 . .5 . .|. 3 .|. . 2. . 2|6 . .|. 3 .-----+-----+-----. 8 .|3 . .|. . 1. . 1|. 9 .|3 . .3 . .|. . 1|. 7 . ER=??(Not tested)Hint: Observe boxes 1-2-3,4-5-6,7-8-9 and digits 1-1,2-2,3-3,4-5-6,7-8-9.`

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`Version 21 . .|. . 8|. . .. . .|2 . .|. . 9. . 7|. . .|3 . .-----+-----+-----6 . .|. . 7|. 3 .. 1 .|4 . .|. . 8. . 9|. 2 .|5 . .-----+-----+-----. 8 .|. . 2|4 . .5 . .|. 9 .|. . 3. . 1|6 . .|. 7 . ER=10.4Hint:Observe boxes 1-2-3,4-5-6,7-8-9 and digits 1-2-3,4-5-6,7-8-9.`

An easier sudoku (from the simmetry viewpoint and ER rating), and a different 3-fold simmetry
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`Version 31 . .|2 . .|3 . .. 5 .|. 6 .|. 4 .. . 9|. . 7|. . 8-----+-----+-----. . 8|. . 9|. . 7. 6 .|. 4 .|. 5 .3 . .|1 . .|2 . .-----+-----+-----. 3 .|. 1 .|. 2 .8 . .|9 . .|7 . .. . 5|. . 6|. . 4 ER=3.0Hint: Observe boxes 1-2-3,4-5-6,7-8-9 and digits 1-2-3,4-5-6,7-8-9`

I did not find any hard sudoku with this kind of threefold horizontal simmetry, so looking for this kind of simmetry may be harder
than just solving the puzzle.

Diagonal:
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`Version 19 . .|2 . .|. . .. 6 .|. . 8|. 3 .. . 8|. 5 .|1 . .-----+-----+-----. 1 .|9 . .|. . 5. . .|. 7 .|. 2 .. . 3|. . 6|9 . .-----+-----+-----. 7 .|3 . .|6 . .5 . .|. . 1|. 7 .. . 2|. . .|. . 8 ER=9.6Hint:Observe boxes 1-5-9,2-6-7,3-4-8; and digits 1-1,2-2,3-3,4-4,5-5,6-6,7-8-9.`

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`Version 2. . 9|. 3 .|. 2 .. . .|4 . .|6 . .. . .|. . 1|. . 4-----+-----+-----5 . .|. . .|1 . .. . 2|7 . .|. . 3. 4 .|. . .|. 5 .-----+-----+-----. . 6|. . 5|. . .. 1 .|. 6 .|. . .3 . .|2 . .|. 8 . ER=10.3Hint: Observe boxes 1-5-9,2-6-7,3-4-8 and digits 1-1,2-2,3-3,4-5-6,7-8-9.`

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`Version 36 . .|. . 9|. 5 .. . .|. 3 .|. . 2. 2 .|8 . .|9 . .-----+-----+-----. 7 .|. . 3|. 9 .. . 6|. 4 .|7 . .3 . .|. . .|. . 1-----+-----+-----2 . .|. 1 .|. . .. . 7|. . 8|1 . .. 8 .|4 . .|. . 5 ER=10.6Hint: Observe boxes 1-5-9,2-6-7,3-4-8 and digits 1-2-3,4-5-6,7-8-9.`

To Ravel: I think that the puzzles Horizontal version 2 and Diagonal Version 3 are hard for your program.
Mauricio

Posts: 1174
Joined: 22 March 2006

Here's a sample of some diagonally and rotationally symmetric puzzles. So far I'm finding only about 1 symmetric puzzle in 20000 randomly selected grids. This may be in part due to my understanding of diagonally symmetric puzzles. As I understand them, when morphed to their symmetric forms the diagonal elements must map onto themselves. Because of this, there can only be three digits on the diagonal occurring 3 times each. For example:
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` 7 2 6 | 3 4 9 | 5 8 1 9 8 5 | 7 1 2 | 3 6 4 1 4 3 | 5 6 8 | 2 9 7 ------+-------+------ 3 7 4 | 8 9 6 | 1 5 2 5 6 1 | 2 3 4 | 8 7 9 2 9 8 | 1 5 7 | 6 4 3 ------+-------+------ 4 3 9 | 6 8 1 | 7 2 5 8 1 2 | 4 7 5 | 9 3 6 6 5 7 | 9 2 3 | 4 1 8`

This is different from a rotationally symmetric puzzle where only the r5c5 digit needs to map onto itself. For example:
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` 9 5 8 | 6 4 1 | 7 3 2 3 6 2 | 5 7 9 | 8 4 1 7 1 4 | 3 2 8 | 6 5 9 ------+-------+------ 6 8 3 | 4 9 2 | 5 1 7 4 9 1 | 7 6 5 | 2 8 3 5 2 7 | 1 8 3 | 4 9 6 ------+-------+------ 8 7 6 | 9 1 4 | 3 2 5 2 3 9 | 8 5 7 | 1 6 4 1 4 5 | 2 3 6 | 9 7 8`

Of course I could be easily missing symmetric puzzles in my search as well. Most of these should be pretty easy based on their suexrating. I haven't checked for isomorphisms (although that might be interesting to do).

Mauricio, what do you mean by threefold symmetry?

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`#Diagonally Symmetric7..5.8....8...6.1....2...6537.1.....56..9....2....4...4..75.1.......3.7.....8...9 0056004....9....2.....45.71..8......19...6..85..3...5.6.3..7......92..6..7....3.......8 012627...6.........8.3..4.5.312..3..1.2.6..675....8.....4...........795..2.....8.4.6.3. 0071441....5....6.1..8..54.8......27....3....3..4..6....9..8..4.......8..2....2.95.8.76 005600#Rotationally Symmetric68.........7.....4.91..5...958.6.2..3.....14.............19.5.....5....1...3268.9 0063144...2.8........12717............84.....76....894.5.....6..423.8..3.......2.....9. 00570082.1...9.......24.6.9..4.....6.8....1..3......72569.......9.8..7....3.....5....64 010546..9....86.3.1.95..72...6..93..2.4..8..........4.....2..8....6......57....7..132.4 0085511.....3.8..87..192....5......49.1.................2734.21.....698..2....5....4... 005700.1.7...9535.8..21.4.....3.......86.4.9..6..7.5....7.8...35.........12....7......2 0055007.....45..62.8..........1.......6..94.3..76.......2.7...8.19.....14...92...3..... 005800.3..9....29.4.36......1....8...6.5.9..6.....84....2...6.......3..5..7486.....49.. 008372.513..2......7..1.....8.73.68.......3.2...4...49.......3.2..5..2..15......5...8.4 005600..1.....3.....2.5.89....1....9...2.5.4.......238.67......74...6..4....9..1..5..84 0179376..8......2..3.8...89.........97.4...7..25.96........85.7....6...3...15....1..9.. 0000003..5.2...5.6.147..9.....5...2..3..18.1372..............6.3...5......9.87.......2. 0078187.8...12............5...3...19.7....8...3..1.5..2..4.7...6.........95..3.43..7.9. 010144.24.3.6......1...3.....4.....6..5...4....3.753.....4.......6.51...5....9.9...7.2. 0058002...8.....7.6..4.5...49...7....5......7.....3.2..4.........57.6..58..9..39.....42 006603.1...3.2........6.649.1..8....8...97....52...3.1..7...13....6..........8.27.6..19 005500.........4.8.7.....13..2.8...6..14.....4.8.7...2..6.3.15..3.2...3.............69. 0058001.....4....7......4..23.96...6..9.3.9.....51.2....4.............723.6..5..39.7... 005700.4.9....6..5.6..7.7....3..82...7..3.6....84.......2........9....8...514..5.....87 011351....7.4..97...1.2...3..69....1....5.6....5..7....3.2...86.1.......4......47..839. 016317......93..82..91.....7...........82.5.1.......3.....4..2.4.6....47..8.6..6..1.5.. 019879..1..5......1..658...29....4.3.....91....3..28...6..4........83....349...97...... 007082.39.....5...8.9...846.3....3.....8...7.9...........16.........1.5.7..4....8.65.9. 0090248...2.....79..5...3...6..9.......8.....473...53........9.8...63..1.5..4.4.....2.. 011306.4..8...7..........35.94.....2.51.9......237..9.....8.9....5.6....3..9.....46.2.. 0090871...4......873.....3...1.74.513.........952........86.4.......2...6...8387....... 0057005...2..7..138.6.4.....15....8.........7....96..6...3...5.....376......82....79.1. 0056002...6..4......5.26.....2.177.3............93...15..4...15..7......6.......4293... 00570062....5....1....39...37....9....3....32..........6.2...7.8.5..6..47.........1.8.4 0091993.271...........5.....93..4...2...6.98....73.2.7......7.5.24.......3...7...6..5.9 005600..5.29.6.4..3.......8.7....8...5...2...1...4.2.6..39..3.....7...5....1.9.2..8.... 0093769.2..4.....15....8.....6.7.7.8...3......5............7.6.38....5...2.93.4..7..2.. 005800.9.8.6....5.1......2..9..535..4..3.1938.......7...5.6........47......6.....61...2 00560025.....3.....7.....3...8.....3..9.5....8..3....56...7..9.5.62.4...3..1..62....... 0058001..5.6.7...231...4.......5..............2...739..68....6....1...45......9..4..3.5 005800.2.64.1...84..5.7......9.26.15.6...8.......4.8.7.1.6.33.6............9......8.... 0056007.3..5..6.9.8.......5....4.6...1....14..9.87..3.2.......6.231..3.....6.........92 006920`
Mike Barker

Posts: 458
Joined: 22 January 2006

Mauricio,

again interesting puzzles.
I had a look at the first one. By changing the digits following your hints, when moving the stacks and rows (inside the bands) it maps to itself. (e.g. r1c123 -> r3c789 -> r2c456). So each assignment immediately leads to 2 more, e.g. a 6 in r5c2 would imply a 4 in r6c5 and a 5 in r4c8 (and a contradiction after a hard chain). But it is still too hard for me to solve it manually.

The ratings of my program are 12, 12, 0, 4, 13, 11.
ravel

Posts: 998
Joined: 21 February 2006

[edit: withdrawn]
Last edited by ronk on Sat Jan 13, 2007 1:08 pm, edited 1 time in total.
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

Mike Barker wrote:Mauricio, what do you mean by threefold symmetry?

ravel wrote:I had a look at the first one. By changing the digits following your hints, when moving the stacks and rows (inside the bands) it maps to itself. (e.g. r1c123 -> r3c789 -> r2c456). So each assignment immediately leads to 2 more, e.g. a 6 in r5c2 would imply a 4 in r6c5 and a 5 in r4c8 (and a contradiction after a hard chain). But it is still too hard for me to solve it manually.

ronk wrote:Comparing the standard and diagonal symmetric canonicals suggests that every puzzle can be morphed into a diagonal symmetric.

I don't think that is true, but I may be wrong.
I understood what you said it is only valid for the SOLUTION grid, not for the puzzle; to apply the technique we have to have a symmetric puzzle, not just only a symmetric solution.
Mauricio

Posts: 1174
Joined: 22 March 2006

Interesting question, dont know yet. Probably there are different (equivalent) symmetrical solution grids and we would have to check for each one, if the givens all have a counterpart.

But i like the diagonal-symmetric canonical form. I did not suspect, that all grids would have one, but is seems to be very probable now.

What i wanted to add to the threefold symmetry puzzles:
Unlike to the perfectly and diagonal symmetrical puzzles i cannot see here a special technique, that is possible for them like we had it with the symmetrical placement and Emerald techniques. It is more like a third of a 3x3 sudoku, you can solve it by solving (e.g.) one stack. Maybe this is the reason, why they can be that hard: There seems to be no [added: specific] outside information from the other stacks [added:] , only a general constraint (from the symmetry), which is not very restricitive for each cell.
ravel

Posts: 998
Joined: 21 February 2006

### GC51 : A 3rd CHALLENGE

GC51 : A 3rd CHALLENGE

This new challenge once again features a major new technique.

First you get the opportunity to solve it with no help. That would give you maximum return.

As before, the challenge is to solve it without all the chains etc characteristic of an SE 8.5 puzzle.

Code: Select all
`#GC51#SE 8.5  7 2 . 5 . . . . .   3 . . . 1 . 9 . .  . . 8 . . 7 . 5 .  9 . . 7 . . 3 . .  . 7 . 1 8 . . . .  . . . . 6 . . . .  . 4 . . . 6 2 . 3  . . 7 . . . . 4 .  . . . . 3 . . 7 6`

______________________________________________________________________
gurth

Posts: 358
Joined: 11 February 2006
Location: Cape Town, South Africa

[edit: withdrawn]
Last edited by ronk on Sat Jan 13, 2007 1:09 pm, edited 1 time in total.
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

### Re: GC51 : A 3rd CHALLENGE

gurth wrote:This new challenge once again features a major new technique.
Hm, must be very new. My understanding of symmetrical techniques is generalised: Find a transformation, that maps a puzzle to itself. Doing this for each candidate you can eliminate those, which dont map to itself.
[Edit:]To say it more correct: Each transformation includes a mapping for each number. When cell A is transformed to cell B, cell B be can only contain the mapped numbers of cell A. If A = B, it only can have candidates, that map to themselves. (Are there more such transformations, where also cells are mapped to themselves, than with 1 or 9 of them?)
But in this puzzle the givens in the boxes are distributed in a way, that such a transformation seems to be impossible.
Can you give a hint ?
Last edited by ravel on Sat Jan 13, 2007 6:09 pm, edited 1 time in total.
ravel

Posts: 998
Joined: 21 February 2006

ronk wrote:Comparing the standard and diagonal symmetric canonicals suggests that every puzzle can be morphed into a diagonal symmetric.

That can't be true. Every diagonal symmetric grid is automorphic, morph it to get the same grid, but that cannot be done with all the grids. I believe quite few grids actually have this property.

RW
RW
2010 Supporter

Posts: 1000
Joined: 16 March 2006

RW wrote:
ronk wrote:Comparing the standard and diagonal symmetric canonicals suggests that every puzzle can be morphed into a diagonal symmetric.

That can't be true. Every diagonal symmetric grid is automorphic, morph it to get the same grid, but that cannot be done with all the grids.

I've apparently misused the term "diagonal symmetric" ... so have withdrawn my prior posts.
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

If I can expand the definition of an Emerald to be a puzzle in which there is a one-to-one pairing between digits under a given symmetry operation. Then for a rotational emerald we have Gurth's symmetric placement rule which paraphased states that a digit occupies r5c5 if and only if it maps onto itself. There is, of course, one and only one such digit. Based on the above observations there is a corresponding symmetric placement rule for a diagonal emerald. In this case a digit occupies the diagonal if and only if it maps onto itself. There are three and only three such digits.
Mike Barker

Posts: 458
Joined: 22 January 2006

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