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Asymmetric vs. Symmetric Clues

 
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daj95376

Joined: 05 Feb 2006
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PostPosted: Wed Feb 15, 2006 5:10 pm    Post subject: Asymmetric vs. Symmetric Clues Reply with quote

I understand there's been considerable discussion on whether or not the clues to a puzzle should be symmetric. I'm not here to dig up those discussions. However, I have an observation and (hopefully) a new side-issue to discuss.

I've collected a number of puzzles offered in the forums and from web sites mentioned in the forums. So, I decided to test the symmetry of their clues.

I further tested the symmetric puzzles to see how many had naked singles immediately after loading.

In one group of 121 puzzles, 42 were asymmetric and 79 were symmetric. Of the latter, 36 had naked singles.

Another group was a collection of puzzles with 17-clues each, all 35,396 were asymmetric.

In the final group of 1,465 puzzles, 1,449 were asymmetric and 16 were symmetric. Of the latter, 9 had naked singles.

So, my observation is that most of the puzzles that I've collected are asymmetric. Of the symmetric puzzles, many of them have naked singles showing immediately after loading.

My topic for discussion, comes in two parts:

(1) If you have a generator, does it generate symmetric or asymmetric puzzles?

(2) Should initial naked singles count towards whether or not a puzzle is considered symmetric?

Thanks!
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Ruud
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PostPosted: Wed Feb 15, 2006 7:18 pm    Post subject: Reply with quote

Hi,

the collection of 17-clue Sudokus is completely asymmetrical because nobody has found a symmetrical 17 clue sudoku yet. The set is not representative for all sudokus.

The top1465 contains a lot of 17 clue sudokus from Gordon's collection, so you're counting a lot of them twice.

I have a mixed collection of more than 600,000 symmetrical and asymmetrical sudokus. My generator can produce either of them, and also various types of symmetry (vertical, horizontal, diagonal, rotational)

So far, I have not found any correlation between symmetry and complexity. The only difference is that a symmetrical sudoku cannot be fully minimalized without losing symmetry.

A naked single to start of with is not very rare, but a sudoku that requires naked subsets or any other advanced technique right from the start is very rare.

The collections you checked are made for a special purpose. The 17-clue collection is obvious, but top1465 is assembled from most difficult to solve sudokus by a backtracking solver. These sudokus tend to be a little more difficult than average, but there are also some very easy sudokus in the collection.

I'm not sure where you got the group of 121, so I cannot say anything about it.

Ruud.
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vidarino

Joined: 10 Feb 2006
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PostPosted: Wed Feb 15, 2006 7:28 pm    Post subject: Reply with quote

Ruud wrote:
The only difference is that a symmetrical sudoku cannot be fully minimalized without losing symmetry.


That's true in many cases, but not all. Here's a symmetric and minimal puzzle:
Code:
+-------+-------+-------+
| 9 3 8 | . 2 4 | . . . |
| . 1 . | . . . | . . . |
| . 5 6 | . . 7 | . . . |
+-------+-------+-------+
| . 2 . | 1 . 5 | 8 . . |
| . . . | . . . | . . . |
| . . 3 | 4 . 8 | . 1 . |
+-------+-------+-------+
| . . . | 7 . . | 2 4 . |
| . . . | . . . | . 3 . |
| . . . | 2 3 . | 1 6 5 |
+-------+-------+-------+


That was from a collection of 2682 symmetric puzzles. Only a dozen or two were minimal, so it's certainly not commonplace.

Vidar
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Ruud
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PostPosted: Wed Feb 15, 2006 8:04 pm    Post subject: Reply with quote

You are correct, Vidar.

It should have been: The only difference is that a symmetrical sudoku cannot always be fully minimalized without losing symmetry.

Ruud.
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fermat

Joined: 05 Feb 2006
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PostPosted: Thu Feb 16, 2006 12:07 am    Post subject: Reply with quote

Ruud wrote:
You are correct, Vidar.

It should have been: The only difference is that a symmetrical sudoku cannot always be fully minimalized without losing symmetry.

Ruud.


Cool. I expected that symmetry ALWAYS paid a price.
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Cristian2006

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PostPosted: Mon Feb 27, 2006 12:36 pm    Post subject: Reply with quote

It's true what you said about minimizing (it's of course very easy to minimize if it doesn't matter what does it look like) but at least IMO there is a correlation between

making more easily a hard puzzle with (for example) 24 givens if it's symmetric
and
making equally hard puzzle with the same amount of givens if it's asymmetric

So I think if you want to compare symmetric and asymmetric puzzles, they all should be with the same amount of givens. At least I have seen very hard puzzles with more than 30 givens, but having equally difficult asymmetric puzzles are more rare, I think.

I think someone could also examine, what might have been the most minimal puzzles for example to be solved only with hidden and naked singles and opposite to that what might have been the most maximal puzzles for example requiring X-Wing. And at the same time check how many there are asymmetric, how many symmetric. I think there has to be more correlations in sudoku that has examined so far.

Me myself call only symmetric puzzles sudoku. Asymmetric is "Number Place" to me. I really like symmetric.
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Richard

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PostPosted: Tue Feb 28, 2006 6:45 pm    Post subject: Reply with quote

Personally, I think it's good for a puzzle, however difficult, to have at least a couple of singles (naked or hidden) to start with. It helps the human solvers get into it. There's nothing more demoralising than looking at a new puzzle and taking five minutes or more just to work out your first number for one cell.

I always generate symmetric puzzles for my site because I like the symmetry and I think it's good to have some singles to get people started.

But talking about symmetry, one observation I have made is that it seems to be easier to make hard puzzles with diagonal symmetry than with horizontal or vertical symmetry. I haven't taken the time to figure out why but maybe someone else will find the observation interesting. Smile

Richard
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Cristian2006

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PostPosted: Fri Mar 03, 2006 12:12 am    Post subject: Reply with quote

Richard wrote:

But talking about symmetry, one observation I have made is that it seems to be easier to make hard puzzles with diagonal symmetry than with horizontal or vertical symmetry. I haven't taken the time to figure out why but maybe someone else will find the observation interesting. Smile

Richard


That's true and the explanation is easy, I think. You know, sudoku as a game isn't diagonal at all. If you make puzzles with horizontal or vertical symmetry, already when you are making cells interfere much with each other. And, there's also other explanation: relations in those simple sudoku technics are technically ALL horizontal and vertical. But advanced technics can affect much more cells that simple leave out, and those related cells could be diagonally placed...
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Richard

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PostPosted: Fri Mar 03, 2006 8:33 am    Post subject: Reply with quote

Ahh, of course. I hadn't thought about it that way. Thank's Cristian.
Richard
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Lummox JR

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PostPosted: Sun Mar 19, 2006 9:27 pm    Post subject: Reply with quote

As a related thought here, there are four different ways to consider diagonal symmetry. One is the traditional crossword style which is simply a 180° rotation, which I presume is what you meant. Another is to mirror the clues along the NW-SE axis, and another along the NE-SW axis. Finally the last form does both.

Surprisingly, I've found that mirroring along both diagonal axes does tend to produce harder puzzles than you might expect, about the same as crossword symmetry--which is interesting considering there are more cells per symmetrical group.

My own generator is presently capable of assymetry, crossword-style, quad (90° rotations), horizontal, vertical, horizontal and vertical, and the dual diagonal form.
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