The New Sudoku Players' Forum

[Mathimagics]

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I'm a Kakuro man, if a clue pattern is not diagonally symmetric then it's dead to me!

[tarek]

I know. When you become more comfortable with designing puzzles, you become interested in these things: symmetry, minimality and difficulty. I reached a point where I started even including my signature into the puzzle through isomorphic transformations

Tarek

[Mathimagics]

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I implemented a 10-clue search, there are 376992 patterns, and 516 ways to assign the clue-values:

• (9 values => one pair with same value) 36 ways

• (8 values => two pairs with same value) 360 ways

• (8 values => one triplet with same value) 120 ways

You can see why this method becomes impractical for 12 or more clues ... this enumeration took a few hours for each type (PW, PWX). The 9-clue cases took only a couple of minutes.

I have also just noticed that my first run failed to invoke the 3rd clue-value assignment option (1 triplet), so I need to rerun them both ...

The interim results (1st 2 options only): 171,384 puzzles for PW and 5,603,016 puzzles for PWX.

The clue-value assignment rules are universal, so don't depend on symmetry choice. Thus the same program could easily be changed to enumerate orthogonally-symmetric cases, but I don't propose to do that.

[Mathimagics]

you become interested in these things: symmetry, minimality and difficulty.

Well, our 8-clue symmetric puzzles have all of those, in spades!

[tarek]

Indeed

I was hopeful that we’d encounter an easy 8 clue pwx puzzle. I’m not sure if you tried this: If your fast solver recursive/BF routine was called 0 times (0 guesses) then your puzzle was solved with the basic techniques that you implemented to speed your solver (most programmers use singles +/- intersections)

That should be an easy filter to implement

Tarek

[Mathimagics]

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10-clue symmetric puzzle enumeration found additional 3,752 PW puzzles, and 247,024 PWX puzzles.

So the final counts are 175,136 puzzles for PW, and 5,940,040 for PWX.

I will look into the problem tarek mentioned of finding the "easiest" 8C puzzles ...

[Mathimagics]

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tarek, my solver is Mladen's fsss2 solver with PWX tables plgged in. As far as I can tell it uses only singles, no fancy tricks. I counted the number of times it had to make a guess.

Here are the 5 SudokuPWX 8-clue puzzles that have the lowest guess count (2 or 3) so far :

Code: Select all

.........45........8.3................1.7......2..........................9......
...4.......7..9............5........36..................2........1...............
......7.......9...6.......5.......183.....................4......................
.....6.8......................................32.........9.....5.........1......7
.........46.3......8..................5........2.................1.............9.


Good luck with these, I hope they are more "doable"!

[tarek]

Great. So in your case 0 guesses means solvable by singles!!! That is a good filter for solvable puzzles in this tangled up sudoku variant.

The problem with anything over 0 guesses that it can be of unpredictable difficulty. My recursive solver(s) also have singles only and use it regularly to filter out the easy from the non easy. Categorising the non easy is the tricky one as I use my other human style solvers but that can be very slow!!!

I’ll check your posted puzzles hopefully later!!!

Tarek

[Mathimagics]

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These are "no-guess" minimal puzzles. The first one is 10-clue, the rest are 11.

The underlying grid is the same in all cases and for this grid the smallest puzzle size is 9 clues.

The 10-clue puzzle is the smallest one on this grid, and the only 10-clue puzzle with guesses = 0. For 11-clues there are many such occurrences.

Code: Select all

..................3........96.5.3...1............8...4..................7...4....
..............................5.7.....................38....1.6.47....8....9.4...
........................4.....3...........57...3..2..4...7...9.................68
......................5...............16...............3.9......1.8....7......584
....................6...........2.7......8.....14...............5.9.....76.....9.


[tarek]

The low guess 8 clue PWXs are still very difficult for manual solving

All the guess = 0 puzzles you posted solve with singles only. So IMO it is safe to post them labelled as "solve with singles only" or "Easy"

tarek

[Mathimagics]

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OK, thanks very much!

I'll filter any puzzles that I post (where I invite P&P solving) from now on!

[tarek]

Just a quick observation on relabelling clue values in symmetric patterns. You don’t need to relabel anything to have a value beyond 4. That would be in the 9 clue version: 1 + 8 + 7 + 6 + 5 = 27 puzzles per pattern not 36. This should work for rotational and orthogonal symmetrical puzzles in any 9x9 sudoku variant. It should work too for all symmetries if you have the main diagonals as constraints as in PWX.

I didn’t look into the 10 clue puzzles or beyond but there could be similar short cuts based on the same logic!!

Tarek

[Mathimagics]

You don’t need to relabel anything to have a value beyond 4. That would be in the 9 clue version: 1 + 8 + 7 + 6 + 5 = 27 puzzles per pattern not 36.

Sorry if I'm being obtuse, but I don't understand what you mean.

For any 9-clue puzzle (no matter how the pattern is selected) there are 36 distinct ways to assign 8 clue values - one for each pair of clues that must have the same value. Each of these is a different puzzle that needs to be tested.

Surely this is true?

[tarek]

Forgive me I wasn’t saying that you were wrong. I was saying that you don’t need to test the entire 36 ways if the pattern is symmetric

[Mathimagics]

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Don't worry about my sensibilities (I'm wrong on a regular basis!), just explain your reasoning - aim it at the idiot level to be on the safe side!

It's all moot really because 9-clue testing for any variant takes only a few minutes in any case ... but still, I'm curious