Troika Sudokus. A challenge.

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

Re: Troika Sudokus. A challenge.

Postby m_b_metcalf » Fri Jan 14, 2022 2:20 pm

Hajime wrote:Congratulations Marek, and thanks to mike for the idea of such a nice puzzle.

You're most welcome; I'm glad that it's generated such interest. In case you're wondering how I came up with the idea, it was because of my rather unsuccessful HAPPY NEW YEAR 25x25 sudoku. I failed to generate the 'positive' image that I wanted, but it was trivial to generate a 'negative' one. I then wondered about negative puzzles in general. For 9x9, a negative puzzle would, typically, have 50 or more clues, which isn't very interesting. Then, :idea:, if there were to be three puzzles...
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Re: Troika Sudokus. A challenge.

Postby Hajime » Fri Jan 14, 2022 3:20 pm

JPF wrote:Congratulations to all for having found these sublime Troika so quickly!
May I suggest another challenge : find 4 valid and minimal puzzles forming a grid partition.

Within 5 days the speedy guys have solved the challenge. Way too fast for me.
Let them go from Troika to Quadriga chariots with 4x20 and r5c5=empty and same Troika definitions,
so I can think further about on 3x27... :lol:
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Re: Troika Sudokus. A challenge.

Postby marek stefanik » Fri Jan 14, 2022 4:44 pm

Mathimagics wrote:Ok, my bad, 54 automorphisms is ok ... :(

So, now, how low can you go? 8-)
I'm currently trying a grid without ropes, don't know how big of an effect it has on the number of automorphisms, can you check?
This is the very first one that popped up (6.7/1.2/1.2 skfr):

Code: Select all
1238.....4........7..36.5..6.7....9...8.5.271........4..5.4.......93...6....12.89
.....5.4...6...93..89....12...1238.....4.....5..7..36..9.6.7...271..8.5...4......
....9.6.7.5.271..8.....4....4......593...6....12.89...8.....123......4..36.5..7..
123895647456271938789364512647123895938456271512789364895647123271938456364512789


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Re: Troika Sudokus. A challenge.

Postby coloin » Fri Jan 14, 2022 7:34 pm

Hajime wrote:
JPF wrote:Congratulations to all for having found these sublime Troika so quickly!
May I suggest another challenge : find 4 valid and minimal puzzles forming a grid partition.

Within 5 days the speedy guys have solved the challenge. Way too fast for me.
Let them go from Troika to Quadriga chariots with 4x20 and r5c5=empty and same Troika definitions,
so I can think further about on 3x27... :lol:

Ah now I understand JPF's challenge ! .... although the r5c5 is the easy one [it might come in handy]
4 disjoint 20C in the same grid solution ... maybe the mc grid 20C x4
Last edited by coloin on Fri Jan 14, 2022 7:37 pm, edited 1 time in total.
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Re: Troika Sudokus. A challenge.

Postby Mathimagics » Fri Jan 14, 2022 7:36 pm

Hi Marek,

Code: Select all
123895647456271938789364512647123895938456271512789364895647123271938456364512789


Your grid has 108 automorphisms, which makes it almost as rare as the MC grid. It is one of only 3 grids with NAUT = 108

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Re: Troika Sudokus. A challenge.

Postby marek stefanik » Sat Jan 15, 2022 7:59 am

Thanks Mathimagics,

Can you give me a grid with the least automorphisms while still preserving box symmetry?

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Re: Troika Sudokus. A challenge.

Postby Mathimagics » Sat Jan 15, 2022 10:55 am

Hmmm, what exactly do you mean by "box symmetry"?
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Re: Troika Sudokus. A challenge.

Postby marek stefanik » Sat Jan 15, 2022 12:14 pm

The boxes 1, 5 and 9 are the same, also boxes 267 and 348.
I like to think of it as of a translation automorphism, where the grid remains the same when you drag it along the main diagonal.

With that, every minimal puzzle with exactly one given in each such automorphic triple of cells has two morphs that form a Troika with it.
Since the main diagonal itself is also automorphic, it's easy to get Supreme Sublime Troikas with it. ;)
That's why I called it a cheese in the new thread. :)

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Re: Troika Sudokus. A challenge.

Postby eleven » Sat Jan 15, 2022 1:32 pm

See Red Ed's list:

3 682439517537681429419572683823794165145863792796125834278946351354218976961357248 1 22 <BS>
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Re: Troika Sudokus. A challenge.

Postby marek stefanik » Sat Jan 15, 2022 8:53 pm

Thanks eleven.

For this one my script only found six Troikas, the highest rated one being a 7.5:
Code: Select all
6.2....1..3....4..4..57..83..3.9.1.5..58...9...........7.9.....3.4.1...6..13...48 7.5/1.2/1.2
....3.5....76.1.2..19..26...2.7.4...1...6.....961..83.2.8..6.5..5...89........... 7.5/1.2/1.2
.8.4.9..75...8...9.........8......6..4...37.27...25..4....4.3.1...2...7.96..572.. 7.5/1.2/1.2
682439517537681429419572683823794165145863792796125834278946351354218976961357248


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Re: Troika Sudokus. A challenge.

Postby eleven » Sun Jan 16, 2022 12:10 am

Nice. If you would know about the solution's symmetry, the puzzles would be completely solved by it.
Why does it not work for 4-fold puzzles ?
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Re: Troika Sudokus. A challenge.

Postby marek stefanik » Sun Jan 16, 2022 6:46 am

It could work, leaving the middle cell empty or having one of the puzzles non-minimal.
I was thinking about shuffling the grid a bit to make the middle cell required in one of the puzzles, but I haven't looked into it yet.
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Re: Troika Sudokus. A challenge.

Postby eleven » Sun Jan 16, 2022 5:22 pm

Tried it now for 1/2 an hour each for 3 grids with double diagonal symmetry, but no success (there is a big difference between 27 and 20 clue puzzles).
[Added:]It cannot work at all with DD symmetry, because in the center box a corner would be the same in 2 of the 4 symmetries. I would have to use quarter symmetry instead,
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Re: Troika Sudokus. A challenge.

Postby mith » Tue Jan 25, 2022 4:50 pm

I went looking for Perfect Troikas based on the 11.8 27c, and finally found one:

Code: Select all
........1......23......4..5...26.....23.784..86.4.3....7.63....34.8.2...6.2..7.8.  ED=11.8/1.2/1.2
986.2.....54196....3.7...........8.31..9...56.......1.......12...9.1..67...54.3..  ED=4.2/1.2/1.2
...3.574.7.......82.1.8.69.495..1.7............7.5.9.25.8..9..4......5...1......9  ED=1.5/1.2/1.2


There is likely a 1.2/1.2/1.2 here somewhere, though I'm not sure how much more effort I will put into it. :P

Some stats: Generated 10000000 minimals from the 54c; of those 400315 were 27c; of those 584 had a corresponding valid puzzle; and of those 2 were minimal.
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