## Symmetrical Givens

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

### Symmetrical Givens

I did one pass over the 817681 hardest puzzles and extracted those with symmetric givens, respectively with symmetric solutions.

Mauricio's 37-given was widely discussed elsewhere. Below are the rest.

Code: Select all
`. . . . . 1 . . 2. 3 . . . . 4 . .. . 2 . 5 . . 6 .. . . 3 . . 6 . .. . 7 . 2 . . 8 .4 . . . . 9 . . .. 1 . 8 . . 9 . .. . 8 . 6 . . . 52 . . . . . . 7 . #19.....8...7..6......5..4..3.....35...4..9..2...61....42..5.........8..7...1..6..5;11.30;11.30;10.40;IG;cham;9131;22;;.....1..2.3....4....2.5..6....3..6....7.2..8.4....9....1.8..9....8.6...52......7. diagonal (main)`

Code: Select all
`. . . . . 1 . . .. . 1 3 . . . . 2. 4 . . 6 . . 5 .. 9 . . . . 5 . 66 . . . 9 . . . 44 . 8 . . . . 9 .. 8 . . 4 . . 6 .3 . . . . 2 7 . .. . . 7 . . . . . #298.7.....6...9.7....7.65...4....3.2...86..9..........3.1..2...4..95..6.........1.;11.10;11.10;10.00;GP;kz0;11948;23;;.....1.....13....2.4..6..5..9....5.66...9...44.8....9..8..4..6.3....27.....7..... pi rotational (180 degrees)`

Code: Select all
`. . . . . 1 . . .. . 1 3 . . . . 2. 4 . . 6 . . 5 .. 9 . . . . 5 . 44 . . . 9 . . . 66 . 8 . . . . 9 .. 8 . . 4 . . 6 .3 . . . . 2 7 . .. . . 7 . . . . . #3.2......9..6.8.1..7......4...58..6...4..61........5.9..7.3........61..2...1..83..;11.10;11.10;10.00;elev;3295;3684;23;;.....1.....13....2.4..6..5..9....5.44...9...66.8....9..8..4..6.3....27.....7..... pi rotational (180 degrees)`

Code: Select all
`2 . . 1 . . . . .. 4 . . . 3 . . .. . 7 . 6 . 5 . .3 . . . . . 6 . .. . 8 . 7 . . . 5. 1 . . . 2 . 8 .. . 9 8 . . 7 . 6. . . . . 6 . 4 9. . . . 9 . 8 5 . #498.7..6..7.5.6.....64..8...8......3..5..9.7....7..2..15...8.9.....1...2......3..4;11.10;11.10;9.70;GP;13_03;891157;25;*;2..1......4...3.....7.6.5..3.....6....8.7...5.1...2.8...98..7.6.....6.49....9.85. diagonal (main)`

Code: Select all
`. 1 . . 3 . . 4 .6 . . . . 4 . . .. . . 7 . 6 8 . .. 2 . . . . 4 . 3. . 1 . . . 2 . .5 . 4 . . . . 1 .. . 7 9 . 8 . . .. . . 4 . . . . 9. 4 . . 5 . . 2 . #598.7.....7...6......6.95...4.....32...96....8.....39...1.....9...79....5....2.41.;11.10;1.20;1.20;GP;12_11;113603;24;*;.1..3..4.6....4......7.68...2....4.3..1...2..5.4....1...79.8......4....9.4..5..2. pi rotational (180 degrees)`

Edit: the above #5 was .....7..6..4.2.5...8.6...1...1....23....1....45....1...1...9.7...2.5.3..9..8..... pi rotational (180 degrees), duplicate of #10

Code: Select all
`. . . . . 2 . . 1. . . . 4 . . 3 .. . 5 6 . 1 . . 2. 3 . . . . 2 . 8. . 7 . . . 3 . .4 . 2 . . . . 7 .2 . . 9 . 5 6 . .. 7 . . 8 . . . .9 . . 2 . . . . . #698.7..6..7..9.......5.4....63...8.7..7.....3...2.....1...2.7..4....1.75......41..;11.10;1.20;1.20;GP;12_11;179218;24;*;.....2..1....4..3...56.1..2.3....2.8..7...3..4.2....7.2..9.56...7..8....9..2..... pi rotational (180 degrees)`

Edit: the above #6 was 7....5..8...79..3..1.8..7....1....2.6...5...7.3....4....6..9.4..2..86...9..5....6 pi rotational (180 degrees), duplicate of #11

Code: Select all
`. . . . 2 . . 1 .. 3 . 4 . . . . 2. . 5 . . 6 4 . .. 6 . 8 . . 7 . .9 . . . . . . . 3. . 4 . . 5 . 2 .. . 7 2 . . 8 . .6 . . . . 7 . 9 .. 1 . . 6 . . . . #7....3..1..5...7..3..92..7....79...3.4.......5.2...86....6..38..2..6...4..1..2....;10.70;10.70;10.60;IG;H7;8014;24;;....2..1..3.4....2..5..64...6.8..7..9.......3..4..5.2...72..8..6....7.9..1..6.... diagonal and antidiagonal`

Code: Select all
`. . . . . 7 . . 6. . 4 . 2 . 5 . .. 7 . 8 . . . 1 .. . 1 . . . . 2 3. . . . 1 . . . .4 5 . . . . 1 . .. 1 . . . 6 . 9 .. . 2 . 5 . 3 . .8 . . 9 . . . . . #8........1..1.23...45.1..........6.7...45....2.7..1.8....23....5.1..9..6.8.....9..;10.70;10.70;2.60;dob;12_12_03;260512;23;*;.....7..6..4.2.5...7.8...1...1....23....1....45....1...1...6.9...2.5.3..8..9..... pi rotational (180 degrees)`

Code: Select all
`. 2 . . 6 . . . 71 . . 4 . . 8 . .. . 7 . . 9 . 3 .. 1 . . . . 2 . .2 . . . 5 . . . 1. . 3 . . . . 4 .. 6 . 8 . . 7 . .. . 1 . . 6 . . 95 . . . 9 . . 6 . #9..4..6..2.1..8.4..5..4...8...5.....9.9..1..5.7.....3...5...3..6..9.4..2.2..8..7..;10.60;10.60;10.60;IG;?;9200;25;;.2..6...71..4..8....7..9.3..1....2..2...5...1..3....4..6.8..7....1..6..95...9..6. diagonal and antidiagonal`

Code: Select all
`. . . . . 7 . . 6. . 4 . 2 . 5 . .. 8 . 6 . . . 1 .. . 1 . . . . 2 3. . . . 1 . . . .4 5 . . . . 1 . .. 1 . . . 9 . 7 .. . 2 . 5 . 3 . .9 . . 8 . . . . . #10........1..1.23...45.1..........6.7...45....2.8..1.6....23....5.1..7..9.9.....8..;10.60;10.60;2.60;dob;12_12_03;260580;23;*;.....7..6..4.2.5...8.6...1...1....23....1....45....1...1...9.7...2.5.3..9..8..... pi rotational (180 degrees)`

Code: Select all
`7 . . . . 5 . . 8. . . 7 9 . . 3 .. 1 . 8 . . 7 . .. . 1 . . . . 2 .6 . . . 5 . . . 7. 3 . . . . 4 . .. . 6 . . 9 . 4 .. 2 . . 8 6 . . .9 . . 5 . . . . 6 #11.....1..2....3..4...56..7...5.7..8..7.9.....38...1..7..68.2.....9...6..45..9..6..;10.50;10.50;6.60;dob;12_12_03;225747;25;*;7....5..8...79..3..1.8..7....1....2.6...5...7.3....4....6..9.4..2..86...9..5....6 pi rotational (180 degrees)`

Code: Select all
`. 7 . . . 5 . 8 .8 . . 7 . . . . 2. . 4 8 6 . . . .. . 1 . . . 2 . .. 6 . . 5 . . 7 .. . 3 . . . 4 . .. . . . 7 9 1 . .3 . . . . 6 . . 9. 9 . 5 . . . 6 . #1298.7.....6...9.5....4..6.9.4....3.8..63.....2..8.1.3...4...8.6....2....5...1....7;10.50;10.50;2.60;GP;13_03;847912;25;*;.7...5.8.8..7....2..486......1...2...6..5..7...3...4......791..3....6..9.9.5...6. pi rotational (180 degrees)`

Code: Select all
`. . . . . 1 . . 2. . 1 3 . . 4 . .. 5 . . 6 . . 7 .. 8 . 4 . . . . 5. . 4 . 3 . 8 . .9 . . . . 8 . 4 .. 2 . . 1 . . 9 .. . 8 . . 3 6 . .7 . . 6 . . . . . #139..8..7...8..7..6...7..9..54..3...5..2......1..5.9.6..1..2....6..8.6.5.......4.3.;10.40;10.40;10.40;GP;12_11;101459;25;*;.....1..2..13..4...5..6..7..8.4....5..4.3.8..9....8.4..2..1..9...8..36..7..6..... diagonal and antidiagonal`

Code: Select all
`. . . . . 1 . . .. . . 3 . . . 1 2. . 4 . 6 . 5 . .. . 9 . . . . 5 88 . . . 9 . . . 44 6 . . . . 9 . .. . 6 . 5 . 8 . .3 7 . . . 2 . . .. . . 7 . . . . . #1498.7.....6..5.......4.6.9..4...8.3...98....2...3....1..4..3.6.....2....1.....5.7.;10.40;10.40;10.40;GP;12_11;111820;23;*;.....1......3...12..4.6.5....9....588...9...446....9....6.5.8..37...2......7..... pi rotational (180 degrees)`

Code: Select all
`. 1 . . . . . . .4 . . . 2 . . 3 .. . 3 4 . . 5 . .. . 2 . 1 . . 6 .. 3 . 6 . . 7 . .. . . . . 8 . . 9. . 1 . 6 . . 7 .. 5 . 7 . . 4 . .. . . . . 9 . . 8 #159..8..7...8..7..6......5...7....94...4.5...3...2.....14....83...3.6...5...1.....2;10.40;10.40;10.30;GP;12_11;100669;23;*;.1.......4...2..3...34..5....2.1..6..3.6..7.......8..9..1.6..7..5.7..4.......9..8 diagonal and antidiagonal`

Code: Select all
`2 1 . . . . . . .5 . 4 3 . . . . .. 3 6 . . . 4 . .. 4 . 7 . . 2 . .. . . . . 1 . 8 .. . . . 5 . . . 9. . 3 2 . . 6 . .. . . . 9 . . . 1. . . . . 8 . 5 2 #1698.7.....7.6...8...54......8..9..4......64.3......5..2.7.4..1......3..5......2..6;10.30;10.30;10.30;GP;12_11;115583;23;*;21.......5.43......36...4...4.7..2.......1.8.....5...9..32..6......9...1.....8.52 diagonal (main)`

Taking the first puzzle
Code: Select all
`. . . . . 1 . . 2. 3 . . . . 4 . .. . 2 . 5 . . 6 .. . . 3 . . 6 . .. . 7 . 2 . . 8 .4 . . . . 9 . . .. 1 . 8 . . 9 . .. . 8 . 6 . . . 52 . . . . . . 7 . #1`

the paired givens are (3,3), (2,2), (1,4), (5,7), (6,8), (9,9) and applying Gurth's Symmetrical Placement main diagonal is directly resolved by
r1c1=9 (the only self-paired candidate is 9. 5,6,7,8 can't live on the main diagonal)
r9c9=3 (for the same reason)
r8c8=2 (there is already 3 in the box, 2 remains)

Enjoy!
Last edited by dobrichev on Sun Apr 26, 2015 3:56 pm, edited 1 time in total.
dobrichev
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### Re: The hardest sudokus (new thread)

Hi dobrichev and good work. I always liked the 180 rotational symmetrical clues to be designed so that they add up to 10. Therefore r5c5 is always 5. Any self rotating clue should be 5.

Tarek

tarek

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### Re: The hardest sudokus (new thread)

As far as I can see, puzzle 5 is identical to 10, and 6 to 11, although the ratings are wildly different

Regards,

Mike

m_b_metcalf
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### Re: The hardest sudokus (new thread)

Thank you. It was copy/paste error for #5 and #6. I did the corrections in the original post.
dobrichev
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### Re: symmetrical givens

dobrichev wrote:I did one pass over the 817681 hardest puzzles and extracted those with symmetric givens, respectively with symmetric solutions.
...
Code: Select all
`. . . . 2 . . 1 .. 3 . 4 . . . . 2. . 5 . . 6 4 . .. 6 . 8 . . 7 . .9 . . . . . . . 3. . 4 . . 5 . 2 .. . 7 2 . . 8 . .6 . . . . 7 . 9 .. 1 . . 6 . . . . #7....3..1..5...7..3..92..7....79...3.4.......5.2...86....6..38..2..6...4..1..2....;10.70;10.70;10.60;IG;H7;8014;24;;....2..1..3.4....2..5..64...6.8..7..9.......3..4..5.2...72..8..6....7.9..1..6.... diagonal and antidiagonal`

Maybe I missed something... But this puzzle (#7), to my mind, has neither diagonal, nor antidiagonal symmetries among its givens. For example, if we consider 4 cells - r1c8, r2c9, r8c1, r9c2, we can see, that for both types of reflections 1 --> 2 and 1 --> 6. No relabelling can restore such transformations to original state. If you meant 2 successive transformations - first diagonal AND second antidiagonal, then such combination of transformations will be equivalent to 180 degree rotation.

Serg

[Edited. I corrected a typo.]
Last edited by Serg on Mon Apr 27, 2015 2:41 pm, edited 1 time in total.
Serg
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### Re: The hardest sudokus (new thread)

Hi Serg,
You got me.

Actually the puzzle has only 2 automorphisms, pi rotational (180 degrees). The fact that this symmetry can be split into 2 symmetries of the pattern (givens' positions, not their values) is a bonus here.
Following only the rotational symmetry the pairs are (1,1), (2,6), (3,9), (4,7), (5,8). 1 must occupy the central cell at r5c5. This reduces the first step to SE 9.1 and next is 9.2. Several SE 8+ steps are needed to the end.

The same should be applicable to puzzle #13, but its central cell is given.

Puzzle #9 has antidiagonal symmetry of givens, the main diagonal is a bonus.
Pairs are (1,6), (2,9), (3,8), (4,4), (5,5), (7,7). Populating the antidiagonal with self-paired values resolves cells at startup r3c7=4, r2c8=5, r6c4=7, r4c6=4, r7c3=4, r8c2=4, stte.

In puzzle #15 I don't see antidiagonal symmetry but gsf's famous tool says it has. Nevertheless it reports only one symmetry. The magic command I used is
Code: Select all
`gsfsolver -qFN -f'%#mg %(S)x %#C#Sc'`

In the topic 38 givens, symmetric in the Puzzles subforum some links to discussions on symmetry were kindly posted by the experts.
My plans are to export symmetrical puzzles from high-clue collections. There are no symmetric 39s, so far there are 11 symmetric 38s and about 100 37s. They are easy to identify but conversion to a geometrically symmetrical morph is unstable process as you can see.

Cheers,
dobrichev
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### Re: The hardest sudokus (new thread)

I am not planning to continue "symmetric" discussion in this thread, because it is off-topic. But nevertheless, I'd like to say some words concerning this theme.

You wrote additional pattern's symmetries (additional to clue values symmetry) are "bonus" for players. Do you know any way to use those "bonuses" in solution?

I turned my attention to puzzles having both diagonal and antidiagonal symmetries because it is the only possible "basic" symmetry not presented yet by published puzzles, having clue value symmetry. It is easy to prove that solution grids having vertical or horizontal symmetries, or both vertical AND horizontal symmetries, or full dihedral symmetry (4 symmetry axes) all are impossible. Puzzles (solution grids) having diagonal symmetry, 180 degree symmetry and 90-degree symmetry were published. So, the only possible symmetry type - double diagonal, having both diagonal and antidiagonal symmetry was not presented yet by puzzles (solution grids), having clue value symmetry. Do they exist?

Serg
Serg
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### Re: The hardest sudokus (new thread)

Serg wrote:Do you know any way to use those "bonuses" in solution?

No.

Nevertheless I wouldn't bet it is pure coincidence.
dobrichev
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### Re: The hardest sudokus (new thread)

Serg wrote: type - double diagonal, having both diagonal and antidiagonal symmetry was not presented yet by puzzles (solution grids), having clue value symmetry. Do they exist?

Serg

is this what you expect ?

1...6...5
.9...2.7.
...8..3..
..21...4.
4.......8
.8...96..
..7..4...
.3.6...1.
5...2...9
champagne
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### Re: The hardest sudokus (new thread)

first of all thanks for the puzzles. They seem to be quite challenging also with the symmetry property (since diagonal symmetry normally gives easier solutions, i tried #4 first, but needed a handful of chains, until it was solved).
It seems that under the hardest the digit symmetrical puzzles are relatively common (17 in 817681 seems to be much for me, having in mind that only one of 10000 solution grids has this property).

Since the gsf program options are not understandable from his manual, i don't know, if it is a bug, that "da" is reported, or your options are not the right ones.
Puzzle 15 can be presented this way to show the diagonal symmetry:
Code: Select all
` +-------+-------+-------+ | . . 5 | . . . | . . . | | . 8 . | . . 7 | . . 9 | | 7 . . | . 6 . | . 8 . | +-------+-------+-------+ | . . . | 1 . . | 2 . . | | . . 9 | . . 4 | . . 7 | | . 5 . | . 3 . | . 4 . | +-------+-------+-------+ | . . . | 2 . . | 1 . . | | . . 8 | . . 3 | . . 4 | | . 6 . | . 5 . | . 3 . | +-------+-------+-------+`
btw this one solves with singles after eliminating non-diagonal digits.
eleven

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### Re: The hardest sudokus (new thread)

Hi, champagne!
champagne wrote:
Serg wrote: type - double diagonal, having both diagonal and antidiagonal symmetry was not presented yet by puzzles (solution grids), having clue value symmetry. Do they exist?

Serg

is this what you expect ?

1...6...5
.9...2.7.
...8..3..
..21...4.
4.......8
.8...96..
..7..4...
.3.6...1.
5...2...9

Yes! It is brilliant example of a puzzle having both diagonal and antidiagonal symmetries among its clue values (and in its solution grid), so both symmetries can be efficiently used for puzzle solving. It's curious, but Andrew Stuart solver says "Run out of known strategies", i.e. it cannot find solution path for this puzzle.

You found this puzzle so quickly! I thought such puzzles are very rare (if they do exist).

Serg
Serg
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### Re: The hardest sudokus (new thread)

Serg wrote:Hi, champagne!

You found this puzzle so quickly! I thought such puzzles are very rare (if they do exist).

Serg

I just had to dig in old files and I had no link to the source.

I don't know how many of them can be found, but this can be explored on my side in the following way:

Take a pattern having both diagonal symmetries
Find all puzzles having a symmetry of given for one diagonal (this is something I do when I play in the interactive game, it's not too long, usually less than one hour)
Extract if any those having both symmetry of given.

EDIT : this is with existing code on my side. a tailored made code for such a pattern (symmetry on both diagonals) would go very fast, I have something similar for R90 symmetry, but not for 2 diagonals.
Last edited by champagne on Tue Apr 28, 2015 4:43 pm, edited 1 time in total.
champagne
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### Re: The hardest sudokus (new thread)

eleven wrote:It seems that under the hardest the digit symmetrical puzzles are relatively common (17 in 817681 seems to be much for me, having in mind that only one of 10000 solution grids has this property).

There is conjecture that puzzles with symmetrical pattern are usually harder. Even if that is not true, the whole collection could be biased if the searchers believed in the symmetry. Symmetrical givens require symmetrical pattern, so that hard puzzles are in the halfway
dobrichev
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### symmetrical givens, backdoor 3

Below are some symmetrical puzzles with backdoor 3 in singles. All of them are hard (not hardest). Especially the last one has much hard steps in SE. All of them have 180 deg rotational symmetry and the central cell given.
Code: Select all
`. . . . . 7 . . 6. . 3 . 4 . 2 . .7 . . 8 . . . 4 .. . 1 . . . . 2 3. . . . 1 . . . .4 5 . . . . 1 . .. 3 . . . 6 . . 9. . 5 . 3 . 4 . .8 . . 9 . . . . . #bd3 1, pi rotational (180 degrees)`

Code: Select all
`. . . . . 7 . 6 .. . 4 . 3 . 5 . .. 7 . 6 . . . . 3. . 1 . . . . 3 2. . . . 1 . . . .4 5 . . . . 1 . .5 . . . . 8 . 9 .. . 3 . 5 . 2 . .. 8 . 9 . . . . . #bd3 2, pi rotational (180 degrees)`

Code: Select all
`. . . . . 7 . 6 .. . 4 . 3 . 5 . .. 8 . 6 . . . . 3. . 1 . . . . 3 2. . . . 1 . . . .4 5 . . . . 1 . .5 . . . . 9 . 7 .. . 3 . 5 . 2 . .. 9 . 8 . . . . . #bd3 3, pi rotational (180 degrees)`

Code: Select all
`. . . . . 1 . . .. . 2 4 . . . . 3. 5 . . 7 . . 6 .. 9 . . . . 6 . 77 . . . 9 . . . 55 . 8 . . . . 9 .. 8 . . 5 . . 7 .4 . . . . 3 1 . .. . . 2 . . . . . #bd3 4, pi rotational (180 degrees)`

Code: Select all
`. . . . . 1 . . 2. . . 2 . . . 3 .. . 4 . 5 . 6 . .. . 9 . . . . 7 66 . . . 9 . . . 55 4 . . . . 9 . .. . 5 . 6 . 7 . .. 1 . . . 8 . . .8 . . 3 . . . . . #bd3 5, pi rotational (180 degrees)`

tarek wrote:I always liked the 180 rotational symmetrical clues to be designed so that they add up to 10. Therefore r5c5 is always 5. Any self rotating clue should be 5.

I fully agree with you.
Unfortunately I have no tools in hand to beautify them, and know that almost every manual intervention leads to disaster.
dobrichev
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### Re: The hardest sudokus (new thread)

[withdraw]
Last edited by m_b_metcalf on Thu Apr 30, 2015 9:00 am, edited 1 time in total.

m_b_metcalf
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