## Symmetrical Givens

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

### Re: Symetrical Givens

dobrichev wrote:To my understanding your puzzles are minimal, i.e. have no redundant clues.

I took the seventh puzzle and ... hit bugs in my software tools.
Code: Select all
`16.4..9.2..4..2..69.26..34..1.7.42.5.........2.58.6.3..81..75.97..5..8..5.9..8.73 #Leren's #7 with (wrong 7, likely 4) automorphisms168473952374952186952681347816734295437295618295816734681347529743529861529168473 #its solution with (wrong 14, likely 4) automorphisms1 6 . 4 . . 9 . 2. . 4 . . 2 . . 69 . 2 6 . . 3 4 .. 1 . 7 . 4 2 . 5. . . . . . . . .2 . 5 8 . 6 . 3 .. 8 1 . . 7 5 . 97 . . 5 . . 8 . .5 . 9 . . 8 . 7 3 #Morph with 4x90deg rotational symmetry (+ other wrong symmetries = 7, but likely 4) automorphisms168473952374952186952681347816734295437295618295816734681347529743529861529168473 #morph's solution with (wrong 12, likely 4) automorphisms123456789456789123789123456231567894597834261864291537312678945678945312945312678 #minlex morph of the solution with 4? automorphisms1..4..7....6.891.3..91.3.56.........5.78.4.6.8.4.9.5.7..2..8..56..94.3.29..3.267. #puzzle in minlex solution with 4? automorphisms`

Assuming there is single error in the puzzle/solution automorphisms counting, and taking into account that puzzle automorphisms must be <= solution automorphisms, then both the puzzle and its solution should have 4x90 deg rotational symmetry only.
Can somebody confirm?

I confirm that

1. All 10 Leren's puzzles are minimal.
2. All 10 Leren's puzzles have 4 automorphisms each.
3. Solution grids for puzzles 3-10 have 4 automorphisms each, solution grids for puzzles 1-2 have 12 automorphisms each.
4. All 10 Leren's puzzles have neither quarter turn symmetry, nor double diagonal symmetry.

Serg
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### Re: Symetrical Givens

Last edited by champagne on Tue May 12, 2015 10:27 am, edited 1 time in total.
champagne
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### Re: Symetrical Givens

MD, although I'm out of my depth on this subject, for your possible interest I used Anchor-5 with its 90 degree symmetrical base pattern to canonicalise Leren's #7 grid which confirms 4 automorphisms.

.1..4..7..783.5..26.2.78..5..6..3..9.8945..2.12..89.5...........615...342...34.61
Code: Select all
` *--------------*--------------*--------------* |  3  <1>  5   |  6  <4>  2   |  9  <7>  8   |  |  9  <7> <8>  | <3>  1  <5>  |  6   4  <2>  |  | <6>  4  <2>  |  9  <7> <8>  |  3   1  <5>  |  *--------------*--------------*--------------*  |  4   5  <6>  |  1   2  <3>  |  7   8  <9>  |  |  7  <8> <9>  | <4> <5>  6   |  1  <2>  3   |   <n> = Givens  | <1> <2>  3   |  7  <8> <9>  |  4  <5>  6   |  *--------------*--------------*--------------* |  5   3   4   |  2   6   1   |  8   9   7   |  |  8  <6> <1>  | <5>  9   7   |  2  <3> <4>  |  | <2>  9   7   |  8  <3> <4>  |  5  <6> <1>  |  *--------------*--------------*--------------*`

As my interests are primarily in solving I note that
rABCcXYZ always contains a full set of digits where
ABC = any three rows from tiers 1,2,& 3 respectively
XYZ = 'complementary' columns in stacks 1,2,& 3 respectively such as 159 or 249

These sets can therefore be used as extra houses/sectors which should make them easier to compose.
Sadly for a manual solver I think it becomes too difficult to find symmetry/symmetries in the clue set when the puzzle is scrambled.

Finally perhaps a dumb question, but what's the difference between double diagonal symmetry and 180 degree (or Pi) rotational symmetry?

DPB
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### Re: Symetrical Givens

David P Bird wrote:Finally perhaps a dumb question, but what's the difference between double diagonal symmetry and 180 degree (or Pi) rotational symmetry?

The combination of the two diagonal symmetries gives a 180° symmetry, but there are a lot grids with 180° symmetry, which have no diagonal symmetry.

In the solution to Leren's second puzzle, which is equivalent to the 9th in blues' complete list of all minimal puzzles having digital double sticks symmetry, a mini-column symmetry can be easily seen in the solution grid (blues' version used here):

Code: Select all
` +-------+-------+-------+ | 1 3 2 | 9 7 8 | 4 6 5 | | 8 7 9 | 6 4 5 | 2 1 3 | | 4 6 5 | 3 1 2 | 7 9 8 | +-------+-------+-------+ | 6 5 4 | 1 2 3 | 9 8 7 | | 9 8 7 | 4 5 6 | 3 2 1 | | 3 2 1 | 7 8 9 | 6 5 4 | +-------+-------+-------+ | 2 1 3 | 8 9 7 | 5 4 6 | | 7 9 8 | 5 6 4 | 1 3 2 | | 5 4 6 | 2 3 1 | 8 7 9 | +-------+-------+-------+`

Just switch colums 4 and 5. Then with the number cycles (1,3,2),(4,6,5),(8,7,9) you can change the columns in each stack cyclically (1->2->3->1), and get the same grid again.

But i see nothing in the solution to the first puzzle (?)

Puzzle 7's solution has an "almost" mini-column symmetry (you would get one, if you switch 2-3 in stack 1, 5-6 in stack 2 and 8-9 in stack 3).
Code: Select all
` +-------+-------+-------+ | 1 3 2 | 9 7 8 | 4 6 5 | | 9 7 8 | 5 6 4 | 3 1 2 | | 4 6 5 | 3 1 2 | 7 9 8 | +-------+-------+-------+ | 6 5 4 | 1 2 3 | 9 8 7 | | 7 8 9 | 4 5 6 | 1 2 3 | | 3 2 1 | 7 8 9 | 6 5 4 | +-------+-------+-------+ | 2 1 3 | 8 9 7 | 5 4 6 | | 8 9 7 | 6 4 5 | 2 3 1 | | 5 4 6 | 2 3 1 | 8 7 9 | +-------+-------+-------+`
eleven

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### Re: Symetrical Givens

eleven wrote:
Puzzle 7's solution has an "almost" mini-column symmetry (you would get one, if you switch 2-3 in stack 1, 5-6 in stack 2 and 8-9 in stack 3).
Code: Select all
` +-------+-------+-------+ | 1 3 2 | 9 7 8 | 4 6 5 | | 9 7 8 | 5 6 4 | 3 1 2 | | 4 6 5 | 3 1 2 | 7 9 8 | +-------+-------+-------+ | 6 5 4 | 1 2 3 | 9 8 7 | | 7 8 9 | 4 5 6 | 1 2 3 | | 3 2 1 | 7 8 9 | 6 5 4 | +-------+-------+-------+ | 2 1 3 | 8 9 7 | 5 4 6 | | 8 9 7 | 6 4 5 | 2 3 1 | | 5 4 6 | 2 3 1 | 8 7 9 | +-------+-------+-------+`

Hi eleven,

I think I wrote in a discarded puzzle false comments on puzzle #7, but that puzzle starts with

16.

the solution can't be

13.
champagne
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### Re: Symetrical Givens

Sorry, i just wanted to add, that this is blues' version (#32), which has a normalized form for the sticks symmetries.
eleven

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### Re: Symetrical Givens

eleven wrote: In the solution to Leren's second puzzle, which is equivalent to the 9th in blues' complete list of all minimal puzzles having digital double sticks symmetry .....

Congratulations eleven, you are spot on ! In fact all of the puzzles are fully scrambled versions of a sample of blue's double sticks symmetry puzzles.

By fully scrambled I mean: 1. Digits are scrambled. 2. Bands and stacks are scrambled. 3. Rows and columns are scrambled within bands and stacks. 4. The puzzle is rotated through 0, 90, 180 or 270 deg. 5. The puzzle is transposed (or not).

The scrambling is controlled by a randomizing process, so even I don't know what will come out of the scrambling in any one go. AFAIK there are 4,875,992,432,640 ways to scramble each puzzle, so at, say, one scrambling per second, I would expect that a puzzle would be unchanged every 154,616.71 years (on average) !

Why did I do this ? Well, as a solver, I noted that all sample stick symmetry puzzles I found invariably have the elimination cells in r258c456 or the transpose of this. There had to be more to life than this - stick elimination cells in other parts of the puzzle. The process also enabled me to check any bugs or functional deficiencies in my Stick Symmetry detector (which I did find and corrected).

I'll leave it as a challenge for the experts to determine which other of blue's puzzles are in my list, or I'll post the original list if you want.

Leren
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### Re: Symetrical Givens

Leren wrote:
eleven wrote: In the solution to Leren's second puzzle, which is equivalent to the 9th in blues' complete list of all minimal puzzles having digital double sticks symmetry .....

Congratulations eleven, you are spot on ! In fact all of the puzzles are fully scrambled versions of a sample of blue's double sticks symmetry puzzles.

I suspected that it was more than mere coincidence, that they were listed in the same relative order as thier isomorphs that list

eleven wrote:But i see nothing in the solution to the first puzzle (?)

If you take the solution to the isomorph from my list (the 3rd puzzle), transpose it, and swap columns 4 and 6, you get a grid with "jumping rows" symmetry.

Code: Select all
`1 8 5 | 3 7 6 | 2 9 44 3 9 | 5 2 8 | 6 1 72 7 6 | 1 9 4 | 3 8 5------+-------+------8 6 2 | 7 4 1 | 9 5 39 4 3 | 8 5 2 | 7 6 17 5 1 | 9 6 3 | 8 4 2------+-------+------5 2 7 | 6 1 9 | 4 3 83 9 4 | 2 8 5 | 1 7 66 1 8 | 4 3 7 | 5 2 9`

Cycling the stacks to the right (1->2->3->1), and relabeling (1->3->2->1), (8->7->9->8), (5->6->4->5), leaves the grid unchanged.
blue

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### Re: Symetrical Givens

eleven wrote:
David P Bird wrote:Finally perhaps a dumb question, but what's the difference between double diagonal symmetry and 180 degree (or Pi) rotational symmetry?

The combination of the two diagonal symmetries gives a 180° symmetry, but there are a lot grids with 180° symmetry, which have no diagonal symmetry.

Aha! yes, thanks for the enlightenment.

DPB
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### Re: Symetrical Givens

blue wrote:If you take the solution to the isomorph from my list (the 3rd puzzle), transpose it, and swap columns 4 and 6, you get a grid with "jumping rows" symmetry.

Ah thanks, i knew that sticks symmetry can come with mini-/jumping-/and gliding rows, but i did not spot that in the columns.
eleven

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### Re: Symetrical Givens

I make slow progress in crosschecking blues results for the DD catalog.

I think I can now go quicker.

I have tested 20 and 21 clues field.

considering minimal puzzles, I match blues results.

I have a list of 12 more puzzles (partly already published, as such or as a morph)

Hidden Text: Show
98.......6......5....21.3....41.......1...9.......96....7.98....5......4.......21
1..2..4...5.4...3.........842.1.........5.........9.862.........7...6.5...6..8..9
1...4.......2...7....3..5...479.....2.......8.....136...5..7....3...8.......6...9
98......36..........521......41.......1...9.......96......985..........47......21
5..1......9..........24.3..1.4..3.....2...8.....7..6.9..7.68..........1......9..5
98......36......5....21......41.......1...9.......96......98....5......47......21
98.......6......5....21......41.3.....1...9.....7.96......98....5......4.......21
98.......6......3....21......41.7.....1...9.....3.96......98....7......4.......21
96.1.....81.....3....2..7..1.4.....................6.9..3..8....7.....92.....9.41
5..26.1.........7....3....94.7......8...5...2......3.61....7....3.........9.48..5
93....4.57..2..........1..8.4....9......5......1....6.2..9..........8..35.6....71
92....4.54..3..........1..8.7....9......5......1....3.2..9..........7..65.6....81

These puzzles are "minimal" if we consider the symmetry of given, but they are not with the standard understanding of a "minimal puzzle"

I did not check for the final minimal form, but all of them have " n-1 clues" valid puzzles. Usually 2, but some have six "n-1 clues".

Here is the first one.

Code: Select all
`98.......6......5....21.3....41.......1...9.......96....7.98....5......4.......21`

the missing clues can be the '3' in row 3 or the '7' in row 7.
I did not check the rating of that puzzle, but this will come with a bigger sample.

IMO, for such a puzzle "nearly symmetry of given", a guess that it is a symmetry of given has to be considered (if it is a very hard puzzle).
Last edited by champagne on Wed May 13, 2015 9:53 am, edited 1 time in total.
champagne
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### Re: Symetrical Givens

I have a dumb question on terminology. I misunderstood the meaning of "minimal" further back, what I meant was that the location of the sticks elimination cells was non-standard.

My question relates to the term "morph". My understanding is that this term refers to number scrambling only but I could be wrong. That's why I took the safe route and described my puzzles as fully scrambled.

So, is a fully scrambled puzzle just a morph, or a morph plus other stuff ?

Leren
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### Re: Symetrical Givens

In my post I used morph as a synonym of (validity preserving) transformation.
dobrichev
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### Re: Symetrical Givens

dobrichev wrote:In my post I used morph as a synonym of (validity preserving) transformation.

I think that means a fully scrambled puzzle is a morph, so I can drop the term fully scrambled and just use morph.

Leren
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### Re: Symetrical Givens

I am not a fan of introducing new terminology. "Scrambled" is OK to me, except that I can't distinguish "full" from other types of scrambling
dobrichev
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