The New Sudoku Players' Forum

by **m_b_metcalf** » Wed Jan 12, 2022 2:07 pm

Mathimagics wrote:This Perfect Troika seems to be highly rated?

Code: Select all
..17....2379.4..8.8...1......7.2.9...9.6....341...9......3..8.....4...36....82..5 ED=8.3/1.2/1.2 6....8.9......65.1.5.9..6.7........4..8..415...653..2.7.4.6.....8...72..9..1..... ED=8.4/1.2/1.2 .4..5.3.....2.......2..3.4.53.8.1.6.2...7..........7.8.2...5.191.5.9.....63...47. ED=8.4/1.2/1.2

How about defng Ultra Perfect Troikas, where all ratings are the same too?

Mike

by **Mathimagics** » Wed Jan 12, 2022 4:42 pm

m_b_metcalf wrote:How about defining Ultra Perfect Troikas, where all ratings are the same too?

OMG, enough of these definitions already! :lol:

Anyway, I would prefer perhaps just a sliding scale of perfection ... 8-)

by **Hajime** » Wed Jan 12, 2022 8:57 pm

Almost Sublime ...

Code: Select all: 861325497429781365357946182286134579174659238593278614932417856615892743748563921 solution 321321321213213213132132132321321321213213213132132132321321321213213213132132132 3 x symmetrical 27 clues ..1..5..7.2..8..6.3..9..1....6..4..9.7..5..3.5..2..6....2..7..6.1..9..4.7..5..9.. SE=8.3 minimal .6..2..9.4..7..3....7..6..2.8..3..7.1..6..2....3..8..4.3..1..5.6..8..7....8..3..1 SE=7.2 minimal 8..3..4....9..1..5.5..4..8.2..1..5....4..9..8.9..7..1.9..4..8....5..2..3.4..6..2. multiple solutions

by **marek stefanik** » Thu Jan 13, 2022 5:49 am

A Perfect Troika consisting of three isomorphic puzzles:

Code: Select all: 1..45..8..5......3..91..4.....5....7.6..9.2....7..1.6...2.4.9.86....83.2.7.3..... 7.2/7.2/2.6 ..3...7..4...89.2..8..23..6..1..4.9.5.48....1....3.5...1.6...7...5.7....9....26.. 7.2/7.2/2.6 .2...6..9..67..1..7......5.23..6.8.......7.3.89.2....43....5....4.9...1...8.1..45 7.2/7.2/2.6 123456789456789123789123456231564897564897231897231564312645978645978312978312645 solution

Marek

by **Mathimagics** » Thu Jan 13, 2022 6:02 am

Nice one, Marek! 8-)

by **Mathimagics** » Thu Jan 13, 2022 12:50 pm

Marek's contribution indicates that we are running out of Troika flavours!

Mike suggested "Ultra Perfect" for perfect troikas with same rating, but Marek's isomorphic discovery is even more deserving of the term ...

My generating exercise has turned up this case of equal-ratings, albeit not an isomorphic one :

Code: Select all: ..7.1..94.6.5....1....3.....9....6837.....1..6.8.23..........3..73.492...45.....8 ED=7.2/1.2/1.2 3..........9..73..5.2...8.62..........4.98..2.5.1...4782..514.....8...6.9....2.1. ED=7.2/1.2/1.2 .8.2.65..4...8..2..1.9.4.7...1475....3.6...5.......9....67....91.......5...36.7.. ED=7.2/1.2/1.2

Triple-isomorphism cases might be called "Supreme"?

[I'll have a Supreme, please, but hold the pineapple! 8-)

]

by **m_b_metcalf** » Thu Jan 13, 2022 8:52 pm

Well, there have been some astonishing results posted above. A troika with three isomorphic puzzles is a real coup.

I was inspired by Hajime's near-miss Sublime Troika pattern to attemt to improve the one in my initial post, where there were 17 redundant clues all together. The result is not perfect, but a lot better, with just 4 redundant clues in total:

Code: Select all: 3 . . 1 . . 4 . . . 5 . . 9 . . 3 . . . 4 . . 2 . . 5 8 . . 2 . . 7 . . . 1 . . 6 . . 9 . . . 3 . . 8 . . 2 4 . . 8 . . 5 . . . 7 . . 1 . . 4 . . . 5 . . 9 . . 7 ED=8.3/1.2/1.2, 2 redundant clues . . 7 . . 5 . . 9 6 . . 7 . . 2 . . . 8 . . 3 . . 7 . . . 6 . . 1 . . 3 5 . . 3 . . 8 . . . 4 . . 5 . . 1 . . . 9 . . 6 . . 1 2 . . 5 . . 9 . . . 6 . . 2 . . 8 . ED=7.1/1.2/1.2, 1 redundant clue . 2 . . 8 . . 6 . . . 1 . . 4 . . 8 9 . . 6 . . 1 . . . 9 . . 4 . . 5 . . . 2 . . 7 . . 4 7 . . 9 . . 6 . . . 3 . . 7 . . 2 . . . 8 . . 3 . . 6 1 . . 4 . . 3 . . ED=7.1/1.2/1.2, 1 redundant clue 327185469651794238984632175896241753512367894743958612439876521278513946165429387

It's close enough that one can believe that a perfect one exists.

Regards,

Mike

by **Mathimagics** » Thu Jan 13, 2022 11:42 pm

m_b_metcalf wrote:A troika with three isomorphic puzzles is a real coup.

Yes, indeed, as noted above!

I have since found 3 more of these:

Code: Select all: ..3...7..4...8..2..8..23..62.1..4.9.5.48....1....3.5...1.6...7...5.7....9....26.. .2.4.6..9..67.91..7......5..3..6.8.......7.3.8..2....43....5....4.9...1...8.1..45 1...5..8..5......3..91..4.....5....7.6..9.2...97..1.6...2.4.9.86....83.2.7.3..... ..3...7..4...89.2..8..23..6..1..4.9.5..8....1....3.5...1.6..97...5.7....9....26.. .2...6..9..67..1..7......5.23..6.8.......7.3.8..2....43....5....4.9...12..8.1..45 1..45..8..5......3..91..4.....5....7.64.9.2...97..1.6...2.4...86....83...7.3..... ..3...7..4...8..2..8...3..6..1..4.9.5.48....1....3.5...1.6..97...5.7....9....264. .2...6..9..67.91..7......5.23..6.8.......7.3.89.2....43....5....4.9...1...8.1...5 1..45..8..5......3..912.4.....5....7.6..9.2....7..1.6...2.4...86....83.2.7.3.....

All are on the MC grid, as is Marek's original find.

I suspect that these are to be found only on the MC grid, which of course has the most automorphisms (648).

by **marek stefanik** » Fri Jan 14, 2022 7:21 am

Once again three isomorphic puzzles, this time forming a Sublime Troika:

Code: Select all: 12....5..4..31.......6..23..64....7..9..56.......8..459.8.....3..28.7........17.9 1.2/1.2/1.2 ..39.8........28.77.9.....15..12.......4..31.23....6...7..64.......9..56.45....8. 1.2/1.2/1.2 ....7..64.56....9..8..45........39.88.7.....2..17.9......5..12.31....4..6..23.... 1.2/1.2/1.2 123978564456312897789645231564123978897456312231789645978564123312897456645231789 solution

Marek

by **Hajime** » Fri Jan 14, 2022 8:50 am

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

I was busy with my 3 template forms:

Code: Select all: 3 2 1 3 2 1 3 2 1 2 1 3 2 1 3 2 1 3 1 3 2 1 3 2 1 3 2 3 2 1 3 2 1 3 2 1 2 1 3 2 1 3 2 1 3 1 3 2 1 3 2 1 3 2 3 2 1 3 2 1 3 2 1 2 1 3 2 1 3 2 1 3 1 3 2 1 3 2 1 3 2

I had 1.5M tries to find 1000 unique-minimal puzzles for the template 1 in a night, using the PatternsGame concept.
There were only 4 of them with a unique counterpart in template 2 and only 1 was minimal, that was shown in my previous post.
The template 3 produced only multiple solutions.
In stead of trying an extra night for the next 1.5M tries, I started to tweak template 2 and 3 a bit by exchanging 1 or 2 cells, keeping up the symmetry of-course. And template 1 unchanged.
But no success up till now.
Was this a good idea? Does my original 3 templates have a Sublime Troika solution?

by **Mathimagics** » Fri Jan 14, 2022 8:58 am

.
I have completed the processing 12 random grids from the general population (ie not having automorphisms), looking for Perfect Troikas.

The source band is listed, along with the minimal 27 pool size used, the number of disjoint pairs in the pool (NDP), the number of pairs for which the implicit 3rd puzzle was valid (VP3), and finally the number that were minimal, ie perfect.

Code: Select all: Grid Band Pool27 NDP VP3 Perfect ----------------------------------------------------- A 44 2126289 2826229 5148 4 B 135 1730944 2108685 1919 4 C 96 2048220 2231452 3468 2 D 125 2011642 2389234 6710 13 E 17 2002224 2311007 6153 4 F 50 2052428 2676814 6357 7 G 47 2196087 3101542 5050 3 H 14 2035202 2560706 3562 3 I 44 2011307 2691965 5142 9 J 6 2037933 2352356 7122 9 K 249 2132868 2543875 7407 5 L 7 2104344 2753880 4981 9

It does seem safe to that, in all likelihood, that every grid has Perfect Troikas, and in very large quantities!

by **marek stefanik** » Fri Jan 14, 2022 10:28 am

Mathimagics wrote:I suspect that these [Sublime Troikas] are to be found only on the MC grid, which of course has the most automorphisms (648).

Well, actually...

I tried a different grid and it didn't even take 10 seconds to get a Supreme Sublime Troika with a rating 6 times higher than the one I have already posted. Enjoy. I will post more results when I rate them. (I rate them with skfr, so please tell me if the SERs are a bit different.)

Code: Select all: 12.6.....4.69..8...8......159...3..8.......12...7..34..7...4.......37.5...526...9 7.2/1.2/1.2 ....7...4.5.....37..9..526....12.6..8..4.69....1.8......859...3.12......34....7.. 7.2/1.2/1.2 ..3..859.....12...7..34......4....7..37.5....26...9..56.....12.9..8..4.6.....1.8. 7.2/1.2/1.2 123678594456912837789345261594123678837456912261789345678594123912837456345261789

Added: 8.8 pearl? (8.8/8.8/2.6 skfr)

Code: Select all: 1.3....9..5.....377.9...2......2.67....4.6..2....8.3.5..85.4...91.8......4..61... .....85.4...91.8......4..61.9.1.3....37.5....2..7.9...67.....2...2...4.63.5....8. .2.67....4.6..2....8.3.5...5.4.....88.....91..61....4.....9.1.3....37.5....2..7.9 123678594456912837789345261594123678837456912261789345678594123912837456345261789

Code: Select all: 1..6.8....569......8..4....59.....7...7....122....93.5.....4.23...83.4......617.. 9.1/1.2/1.2 .23.....44.....83.7......61...1..6.8....569......8..4..7.59.....12..7...3.52....9 9.1/1.2/1.2 ....7.59.....12..7..93.52....4.23...83.4......617.....6.8...1..9......56.4.....8. 9.1/1.2/1.2 123678594456912837789345261594123678837456912261789345678594123912837456345261789

Marek

by **Mathimagics** » Fri Jan 14, 2022 12:02 pm

Ok, my bad, 54 automorphisms is ok ...

So, now, how low can you go? 8-)

And that leads to another question ... how reliable are any ratings for automorphic grids. They tend to have repeating mini-rows and/or -cols

by **JPF** » Fri Jan 14, 2022 12:54 pm

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.

JPF

by **coloin** » Fri Jan 14, 2022 1:08 pm

Nice exercise to find these puzzles !!!
But as ever it does come down to the shear number of puzzles [at the 27 clue level] [per grid solution] as demonstrated clearly by Mathemagics

The isomorphic Troika puzzles with an equivalent pattern probably do have to come from automorphic grids ..

There are only 15 ED patterns of "diagonal" clues and the #1 does allow the one way [in each box] to insert the 2nd and 3rd converse isomorphic patterns as shown by marek

Code: Select all: # 1 ..1..2..3.2..1..4.5..6..1....2..4..1.5..6..7.7..2..3....7..5..8.8..7..9.9..3..2.. # C27/S4.da/M1.16.3 Patterns Game 16 ED = 10.6/10.6/9.8 # 2 2...4...6..4..2.3..7.8..1....24...1.1...3...2.3...54....5..9.7..1.6..2..6...7...9 # C27.M/S4.da/M1.31.2 Patterns Game 150 ED = 10.7/10.7/8.9 # 3 ..2..3.1..3..1.2..5..6....4..5..7.2..9..4.8..6..9....7.8..9..5.2..5..4....3..8..9 # C27/S2.d/M1.33.1 # 4 ..1.2.3...2...1..43..5...6...42...8.9...3...2.3...67...4...3..75..8...4...2.6.1.. # C27.M/S2.p/M1.13.4 # 5 1....2..3.9..4..1...73..5....21....9.6..2..5.9....56....3..41...7..8..2.2..6....8 # C27.M/S4.da/M1.11.6 # 6 2...3.1....14...2..5...6..3.6.2..4..3...5..7...7..3..65..8..7...9..1..4...6..2..9 # C27/S2.d # 7 .2..7...84..6..9....5..4.7..1.2..3..2...1...4..3..5.6..9.5..6....1..7..37...3..5. # C27.M/S4.da/M1.16.1 # 8 1..3..2...2..4..1...3..6..52...8..7..7.5..3....9..1..67...9.5...4.1....8..8..2.6. # C27/S2.d # 9 ..2.1.3...1...2.4.5..3....2..5..7.8.6...3...4.4.6..9..7....4..6.3.2...5...4.9.7.. # C27.M/S4.da #10 ..1..2..3.2..1..4.3..5..6....51..2...4..7...12....8.3...96...7..6..4.3..8....7..4 # C27 [asymmetric] #11 ..1..3..2.2..4..3.4..6..5....23...1..4...57..5...9...8..6.8..7..9.1..6..7....2..1 # C27/S2.d #12 2..1....3..1..2.4..5..3.6..5..7...9...6.8.2...8...4..1..7.6..8..6.4..1..3....7..5 # C27/S4.da #13 2...3.1....14...2..5...6..3.6.2..4..3....4.7...7.8...65..8..7...9..1..4...6..2..9 # C27/S2.d #14 6....2.7...8.1...5.9.5..3....2..1..3.1..2..4.5..3..2....4..8.9.1...9.8...3.4....7 # C27.M/S4.da #15 ..1.2...3.2...34..5..1...6...4..1.7.2...9...6.6.4..8...9...2..8..69...1.3...5.7.. # C27.M/S4.da

Of course they dont have to have the diagonal pattern as marek also found !!

Code: Select all: +---+---+---+ |12.|...|5..| |4..|31.|...| |...|6..|23.| +---+---+---+ |.64|...|.7.| |.9.|.56|...| |...|.8.|.45| +---+---+---+ |9.8|...|..3| |..2|8.7|...| |...|..1|7.9| +---+---+---+

nor do they have to have 3 clues in every box !!

Code: Select all: +---+---+---+ |12.|6..|...| |4.6|9..|8..| |.8.|...|..1| +---+---+---+ |59.|..3|..8| |...|...|.12| |...|7..|34.| +---+---+---+ |.7.|..4|...| |...|.37|.5.| |..5|26.|..9| +---+---+---+

but in both grid solutions the diagonal pattern is in the boxes eg B159 are identical.

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