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 » 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
User avatar
m_b_metcalf
2017 Supporter
 
Posts: 13094
Joined: 15 May 2006
Location: Berlin

Re: Troika Sudokus. A challenge.

Postby 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-)
User avatar
Mathimagics
2017 Supporter
 
Posts: 1870
Joined: 27 May 2015
Location: Canberra

Re: Troika Sudokus. A challenge.

Postby 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
User avatar
Hajime
 
Posts: 1003
Joined: 20 April 2018
Location: Netherlands

Re: Troika Sudokus. A challenge.

Postby 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
marek stefanik
 
Posts: 250
Joined: 05 May 2021

Re: Troika Sudokus. A challenge.

Postby Mathimagics » Thu Jan 13, 2022 6:02 am

Nice one, Marek! 8-)
User avatar
Mathimagics
2017 Supporter
 
Posts: 1870
Joined: 27 May 2015
Location: Canberra

Re: Troika Sudokus. A challenge.

Postby 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-) ]
User avatar
Mathimagics
2017 Supporter
 
Posts: 1870
Joined: 27 May 2015
Location: Canberra

Re: Troika Sudokus. A challenge.

Postby 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
User avatar
m_b_metcalf
2017 Supporter
 
Posts: 13094
Joined: 15 May 2006
Location: Berlin

Re: Troika Sudokus. A challenge.

Postby 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).
User avatar
Mathimagics
2017 Supporter
 
Posts: 1870
Joined: 27 May 2015
Location: Canberra

Re: Troika Sudokus. A challenge.

Postby 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
marek stefanik
 
Posts: 250
Joined: 05 May 2021

Re: Troika Sudokus. A challenge.

Postby 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?
User avatar
Hajime
 
Posts: 1003
Joined: 20 April 2018
Location: Netherlands

Re: Troika Sudokus. A challenge.

Postby 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!
User avatar
Mathimagics
2017 Supporter
 
Posts: 1870
Joined: 27 May 2015
Location: Canberra

Re: Troika Sudokus. A challenge.

Postby 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
Last edited by marek stefanik on Fri Jan 14, 2022 12:06 pm, edited 1 time in total.
marek stefanik
 
Posts: 250
Joined: 05 May 2021

Re: Troika Sudokus. A challenge.

Postby 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
User avatar
Mathimagics
2017 Supporter
 
Posts: 1870
Joined: 27 May 2015
Location: Canberra

Re: Troika Sudokus. A challenge.

Postby 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
JPF
2017 Supporter
 
Posts: 5897
Joined: 06 December 2005
Location: Paris, France

Re: Troika Sudokus. A challenge.

Postby 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.
coloin
 
Posts: 2176
Joined: 05 May 2005
Location: Tenerife

PreviousNext

Return to General