Significance/example of symetry?

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Significance/example of symetry?

Postby Elroy » Sun Mar 12, 2006 7:53 am

In the vernacular of the gurus and in angusj's overview of his Simple SuDoku program, the word "symetry" seems to command an certain respect in SuDoku puzzles --
-- Of the admittedly-few I've done, even the fiendish" ones in Gould's book, very few hinted of any symetry at all, neither in the initial nor the final number patterns:?:
- What is it's significance? How does one recognize it?
-- Any simple example of symetry?
I did search but I apologize if I've missed something dumbly obvious.:(
(neat site)

Elroy

BTW, I meant to include that I also saw no particular symetry in the number of digits (0-9) represented (ie: some puzzles start without some of the digits) nor in the final candidate-pairs immediately prior to chaining-out the final solution.
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Postby TKiel » Sun Mar 12, 2006 2:24 pm

Elroy,

Symmetry refers to the initial clue set and there are many different kinds, though it is possible that only certain kinds are used in setting Sudoku puzzles. It's generally prettier to look at and easier to spot a mistake if entering clues manually, so it does serve a purpose. It's not strictly necessary, but some puzzle generators only make symmetric Sudokus. See this link for more info on the various kinds of symmetry www.sudoku.com/boards/viewtopic.php?t=3070&highlight=

Tracy
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Postby udosuk » Sun Mar 12, 2006 5:29 pm

The following thread is also a very nice one about this topic, contributed by one of our senior members (senpai as in Japanese) - tso:

http://forum.enjoysudoku.com/viewtopic.php?t=1332
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Postby Smythe Dakota » Sun Mar 12, 2006 8:52 pm

How about a different type of symmetry, one implemented with an additional rule: Whenever the digit N appears (either as a clue or as a final), the digit 10-N appears in the diametrically opposite location.

This would imply, of course, that the center square must contain the digit 5.

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Postby Moschopulus » Sun Mar 12, 2006 9:57 pm

Code: Select all
9 7 8 | 3 1 2 | 6 4 5
3 1 2 | 6 4 5 | 9 7 8
6 4 5 | 9 7 8 | 3 1 2
- - - - - - - - - - -
7 8 9 | 1 2 3 | 4 5 6
1 2 3 | 4 5 6 | 7 8 9
4 5 6 | 7 8 9 | 1 2 3
- - - - - - - - - - -
8 9 7 | 2 3 1 | 5 6 4
2 3 1 | 5 6 4 | 8 9 7
5 6 4 | 8 9 7 | 2 3 1
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Postby Red Ed » Sun Mar 12, 2006 11:57 pm

I reckon there are 59709063168 ways of producing such grids.

The permutation that takes a grid to its diametric opposite is (1 81)(2 80)...(40 42) in the notation of Frazer's sudoku symmetries page. This is in conjugacy class 79, so has 9! x 155492352 essentially-invariant grids. Each induces a symbol perm of the form (p[x1] p[x2]) (p[x3] p[x4]) (p[x5] p[x6]) (p[x7] p[x8]) where p[1...9] are the values along the top row (say) and the x* are distinct. We want the symbol perm (1 9)(2 8)(3 7)(4 6) which, for any x*, happens for one in 9*7*5*3*1 possible p, giving us 59709063168 grids overall.
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Postby vidarino » Mon Mar 13, 2006 12:11 am

Smythe Dakota wrote:How about a different type of symmetry, one implemented with an additional rule: Whenever the digit N appears (either as a clue or as a final), the digit 10-N appears in the diametrically opposite location.


Fun idea.:)

In practice, it's only half a puzzle to solve, though, since whenever you are given a clue, or you fix or eliminate a number, you can fix or eliminate its counterpart in the opposite cell.

What might be a slightly more interesting variation would be to know that there exists a symmetry like the one described above, but without knowing which symbols are the opposite of eachother. Simply put, relabel the puzzle after generation.:) (E.g. if you solve a cell as a 6, and there is a 9 on the opposite side of the grid, you know that 6s and 9s are opposites all over, and you can put a 6 on the other side of all 9s and vice versa.)
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Postby Red Ed » Mon Mar 13, 2006 12:37 am

Maybe. But I would've thought a solution space of 59709063168 grids (if I got that right) would offer a fairly rich choice of puzzles without the need to make the solver deduce the symbol perm at the same time.
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Postby vidarino » Mon Mar 13, 2006 12:48 am

Red Ed wrote:Maybe. But I would've thought a solution space of 59709063168 grids (if I got that right) would offer a fairly rich choice of puzzles without the need to make the solver deduce the symbol perm at the same time.


True, it certainly does. However, as mentioned, in practice and quite literally, you're only solving half a puzzle; 40 cells. After copying the initial givens to your preferred half (e.g. top half to bottom half), you can solve the rest of the puzzle even if you cover the other half with a Post-it(tm), since both halves would carry exactly the same information.

That's why I suggested "concealing" the symbol symmetry, postponing the inevitable singles-avalance for later.:) The solution space is actually the same, since the only thing that needs to be done to create a puzzle is a random relabeling of the solution, to obfuscate which symbols mirror eachother.
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Postby Smythe Dakota » Mon Mar 13, 2006 2:20 am

vidarino wrote:.... In practice, it's only half a puzzle to solve ....

True, which brings up the question: what is the minimum number of clues necessary to solve such a puzzle? I'm betting it's LESS than half the minimum for the regular puzzles.

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Postby Elroy » Mon Mar 13, 2006 10:46 am

Thanks all. First for spelling lesson which indeed gained a few more search-hits (only 2000-fold) so I guess that qualifies "dumb" ok..
--
Reason I'd asked is I thought I'd glimpsed a remark somewhere suggesting that "SYMMETRICAL' puzzles are more likely to have a unique solution:
IMO multi-solutions erode the logic and fun of manual-solving published Sudoku puzzles, except MAYBE I wouldn't get as irritated/disenchanted if I had SOME inkling or clue beforehand..
.
You've suggested, I think, that symmetry in a published puzzle does NOT increase the odds or likelyhood of a unique solution. . . . . (:?: )

Elroy 8^|
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Postby tarek » Mon Mar 13, 2006 11:58 am

Elroy wrote:You've suggested, I think, that symmetry in a published puzzle does NOT increase the odds or likelyhood of a unique solution. . . . . (:?: )


Hi Elroy,

Symmetry has nothing to do with a unique solution........

a BIG PHYSICAL example is documented in the Sudoku of shame thread here

It has a 180 rotational symmetry & symmetry along both vertical & horizontal axes, however, It has >100 different solutions

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Postby ab » Mon Mar 13, 2006 2:48 pm

[quote="tarek]

Hi Elroy,

Symmetry has nothing to do with a unique solution........

a BIG PHYSICAL example is documented in the Sudoku of shame thread here

It has a 180 rotational symmetry & symmetry along both vertical & horizontal axes, however, It has >100 different solutions

Tarek[/quote]

I think you mean reflective symmetry in a horizontal and vertical axis. it does also have 180 rotational symmetry about a point:)

PS don't know why the quote thing didn't work!
Last edited by ab on Mon Mar 13, 2006 4:53 pm, edited 1 time in total.
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Postby tso » Mon Mar 13, 2006 5:40 pm

Actually, symmetry *can* have something to do with a puzzle having multiple solutions. Most (but not all) published puzzles in magazines, newspapers, etc have symmetricly placed clues. If a puzzle from these sources is *almost* symmetrical -- you have a higher than random chance of finding that it contains a typo -- a missing, misplaced or extra digit that will make the puzzle invalid -- either by giving it multiple solutions or none at all.
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Postby Smythe Dakota » Wed Mar 15, 2006 1:33 pm

ab wrote:.... PS don't know why the quote thing didn't work!

Because you omitted the closing quote in [quote="tarek"].

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