Greatest divergance possible in dual solution puzzle?

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

Postby JPF » Wed Nov 22, 2006 6:45 pm

Red Ed wrote:
JPF wrote:What about the parity of the number of solutions ?
Even, since no solution grid is equal to its *foo*, where *foo* is any symmetry op of order 2. (Ocean and I did this for *foo* = transpose previously. There are only 6337 symmetry ops that have any fixed grids, and none of those ops have order 2.

t,u sym.ops
t^2=e
u^3=e
P a puzzle such that : tP=P

Can we have at the same time : uP=P ?
It seems to be the case when P={empty grid}.

Number of solutions = 2Nt + 3Nu + ...

JPF
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Postby Red Ed » Wed Nov 22, 2006 7:39 pm

If by "puzzle" you mean any subset of a solution grid, even with multiple solutions, then as you say P = {empty grid} satisfies uP = P. But that's a rather boring answer! Is this line of enquiry leading somewhere? I don't think so ... but am happy to be proved wrong:)
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Postby JPF » Wed Nov 22, 2006 10:47 pm

I was just trying to fully understand your statement that if tP=P and t^2=e, then the number of solutions of P is even.
But apparently, I missed something.

Anyway… I'm largely off-topic.

As you say this question is going nowhere
and it's exactly where this 58 unavoidable set is coming from.

JPF
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Unavoidable set of size 60.

Postby Ocean » Thu Nov 23, 2006 1:33 am

JPF wrote:The two solutions differ in 58 cells.
[...]
I've the feeling that we can do better on the same theme.

Hope it's ok with a slight variaton - only one clue changed:
Code: Select all
#
# 21 clues pseudo-puzzle. 2 solutions: differ in 60 cells.
#
1.....2...3..4......5..6......3..7...4..8..5...6..2..12..7..8......5..9......1..6
#
# The two solutions:
#
168973245732145689495826173529314768341687952876592431253769814614258397987431526
174538269639247518825916347918365724742189653356472981261794835487653192593821476
#
# The unavoidable sets (size 60):
#
.68973.457.21.568949.82.173529.14.683.16.79.287.59.43..53.69.146142.83.798743.52.
.74538.696.92.751882.91.347918.65.247.21.96.335.47.98..61.94.354876.31.259382.47.
#
 *-----------*
 |1..|...|2..|
 |.3.|.4.|...|
 |..5|..6|...|
 |---+---+---|
 |...|3..|7..|
 |.4.|.8.|.5.|
 |..6|..2|..1|
 |---+---+---|
 |2..|7..|8..|
 |...|.5.|.9.|
 |...|..1|..6|
 *-----------*
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Postby JPF » Thu Nov 23, 2006 1:45 am

Well done, Ocean !

JPF
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Postby Ocean » Thu Nov 23, 2006 2:01 am

JPF wrote:Well done, Ocean !

JPF

Thanks. A small improvement based on your surprising findings.

Wonder what is the ultimate size limit - with unavoidable sets of size 60 we should be rather close now. Can it be 64? A size 65 unavoidable set would imply a grid with 65 17s, which is probably a bit too much...
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Postby JPF » Thu Nov 23, 2006 2:27 am

This shape is very fruitful.

Here's an other 60 :

Code: Select all
Clues given:                        21
Cells that are the same:             0
Cells that differ in two solutions: 60

 3 . . | . . . | 2 . .
 . 7 . | . 4 . | . . .
 . . 5 | . . 6 | . . .
-------+-------+-------
 . . . | 3 . . | 7 . .
 . 4 . | . 8 . | . 5 .
 . . 6 | . . 2 | . . 1
-------+-------+-------
 2 . . | 7 . . | 8 . .
 . . . | . 5 . | . 9 .
 . . . | . . 1 | . . 6


368975214972143685415826379529314768143687952786592431251769843634258197897431526
394517268671248539825936147918365724742189653536472981263794815187653492459821376

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Postby coloin » Sat Nov 10, 2007 2:01 am

Here is another 60

Edited to display symetry

:
Code: Select all
+---+---+---+
|..3|...|..2|
|.8.|9..|.7.|
|5..|...|1..|
+---+---+---+
|.6.|4.9|...|
|...|..8|...|
|...|67.|.4.|
+---+---+---+
|..1|...|3..|
|.7.|..4|.8.|
|2..|...|..5|
+---+---+---+   2 grid solutions

Code: Select all
493761852186925473527843169762419538914538627835672941651287394379154286248396715
713845962682913574594762138167429853429358716358671249841596327975234681236187495


Every 22 clue puzzle has an equivalent in the other grid [as above]

Found by "accident" whilst looking for hard puzzles.....
Hard puzzles can have these big unavoidable sets.....
Code: Select all
+---+---+---+           +---+---+---+                           
|..3|...|..2|           |..3|...|..2|                           
|.8.|9..|.7.|           |.8.|9..|.7.|                           
|5..|...|1..|           |5..|...|1..|                           
+---+---+---+           +---+---+---+                           
|.6.|4.9|...|           |.6.|4.9|...|                           
|...|..8|...|           |...|..8|...|                           
|...|67.|.4.|           |...|67.|.4.|                           
+---+---+---+           +---+---+---+                           
|..1|.9.|3..|           |..1|...|3..|                           
|.7.|..4|.8.|           |.7.|..4|.8.|                           
|2..|...|..5|           |2..|.9.|..5|                           
+---+---+---+           +---+---+---+   Two Isomorphs    SE 10.6



C
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61 + disjoint 6

Postby dobrichev » Tue May 17, 2011 1:14 pm

Code: Select all
6 . . 8 9 . . . .
. 9 . 3 . . . . .
. . 5 . . . . 2 .
8 7 . 9 . . 6 . .
9 . . . 8 . 1 . .
. . . . . 4 . . .
. . . 6 1 . 8 . .
. . 2 . . . . 4 .
. . . . . . . . .  #Kite A, 20 givens, 2 solutions, 0 solvable cells

621897453498325761735146928874951632956283174213764589347612895182579346569438217 #solution A1
 ||  ||||| | ||||||| |||| |  | || || ||| | ||||||| ||||||  | |||| |||| ||||||||||
643892715297351486185467329871923654924586137356174298439615872562738941718249563 #solution A2
6..89.....9.3.......5....2.87.9..6..9...8.1.......4......61.8....2....4.......... #pseudo-puzzle Kite A
   ||                      |  |     |   |
6..98.....9.3.......5....2.97.8..6..8...9.1.......4......61.8....2....4.......... #pseudo-puzzle Kite B
643982715297351486185467329971823654824596137356174298439615872562738941718249563 #solution B2
 ||  ||||| | ||||||| |||| |  | || || ||| | ||||||| ||||||  | |||| |||| ||||||||||
621987453498325761735146928974851632856293174213764589347612895182579346569438217 #solution B1

6 . . 9 8 . . . .
. 9 . 3 . . . . .
. . 5 . . . . 2 .
9 7 . 8 . . 6 . .
8 . . . 9 . 1 . .
. . . . . 4 . . .
. . . 6 1 . 8 . .
. . 2 . . . . 4 .
. . . . . . . . .  #Kite B, 20 givens, 2 solutions, 0 solvable cells


Enjoy Sudoku!

MD
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62

Postby dobrichev » Tue May 17, 2011 7:46 pm

Here are 2 19s generating 2 different UA62.
Code: Select all
...........1..2..3.4..5..6.........1..2..37...5..8.........75....31....7.9.....8.
...........1..2..3.4..5..6.........1.6..7..4.2....8......6......7..4..9.8....1..2

MD
Last edited by dobrichev on Wed May 18, 2011 4:18 am, edited 1 time in total.
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Re: Greatest divergance possible in dual solution puzzle?

Postby coloin » Tue May 17, 2011 10:08 pm

Incredible.
They can both be morphed to show the symmetry
Code: Select all
+---+---+---+
|...|...|...|
|.1.|..2|..3|
|..4|.5.|.6.|
+---+---+---+
|...|...|..1|
|..5|.8.|...|
|.2.|..3|7..|
+---+---+---+
|...|..7|5..|
|..9|...|.8.|
|.3.|1..|..7|
+---+---+---+

+---+---+---+
|...|...|...|
|.3.|..2|..1|
|..6|.5.|.4.|
+---+---+---+
|...|6..|...|
|..9|.4.|.7.|
|.2.|..1|8..|
+---+---+---+
|...|..8|2..|
|..4|.7.|.6.|
|.1.|...|...|
+---+---+---+

Edit: better symmetry provided by ronk

Wouldnt bet on the possibility of a 63 though

C
Last edited by coloin on Tue May 17, 2011 11:40 pm, edited 1 time in total.
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Re: Greatest divergance possible in dual solution puzzle?

Postby Serg » Tue May 17, 2011 11:08 pm

Hi, dobrichev!
I confirm that your last posted UA sets are 2-valent strongly minimal UA61 set (1 set) and 2-valent strongly minimal UA62 sets (2 sets). Great! 62 is the largest UA sets size now.

Puzzles "Kite A" and "Kite B", producing the same UA61 set, are twins, i.e. they can be transformed to each other by simple UA6 transformation. (See example below.)
Code: Select all
+-----+-----+-----+     Puzzle "Kite A", producing 2-valent UA61 set
|6 . .|8 9 .|. . .|
|. 9 .|3 . .|. . .|
|. . 5|. . .|. 2 .|
+-----+-----+-----+
|8 7 .|9 . .|6 . .|
|9 . .|. 8 .|1 . .|
|. . .|. . 4|. . .|
+-----+-----+-----+
|. . .|6 1 .|8 . .|
|. . 2|. . .|. 4 .|
|. . .|. . .|. . .|
+-----+-----+-----+

+-----+-----+-----+     UA6, transforming "Kite A" to "Kite B" and vice versa
|. . .|8 9 .|. . .|
|. . .|. . .|. . .|
|. . .|. . .|. . .|
+-----+-----+-----+
|8 . .|9 . .|. . .|
|9 . .|. 8 .|. . .|
|. . .|. . .|. . .|
+-----+-----+-----+
|. . .|. . .|. . .|
|. . .|. . .|. . .|
|. . .|. . .|. . .|
+-----+-----+-----+

+-----+-----+-----+     Puzzle "Kite B", producing 2-valent UA61 set
|6 . .|9 8 .|. . .|
|. 9 .|3 . .|. . .|
|. . 5|. . .|. 2 .|
+-----+-----+-----+
|9 7 .|8 . .|6 . .|
|8 . .|. 9 .|1 . .|
|. . .|. . 4|. . .|
+-----+-----+-----+
|. . .|6 1 .|8 . .|
|. . 2|. . .|. 4 .|
|. . .|. . .|. . .|
+-----+-----+-----+

It is interesting that almost all diagonal symmetric clues of puzzles "Kite A" and "Kite B" have the same values (cells r2c4 and r4c2 are exceptions).

dobrichev wrote:Here are 2 19s generating 4 UA62.

I don't see 4 UA62, but 2 UA62 only. What do you mean?
Serg

[Edited: I deleted my silly question about diagonal symmetric valid puzzles having the same values in the symmetric clue cells. Sorry.]
[Edited: Really only one UA61 set was presented, not two sets - corrected]
Last edited by Serg on Wed May 18, 2011 12:21 pm, edited 3 times in total.
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Re: Greatest divergance possible in dual solution puzzle?

Postby dobrichev » Wed May 18, 2011 4:18 am

Serg wrote:
dobrichev wrote:Here are 2 19s generating 4 UA62.

I don't see 4 UA62, but 2 UA62 only. What do you mean?

My mistake. There are only 2 different UA62. Corrected in the post.
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Re: Greatest divergance possible in dual solution puzzle?

Postby Serg » Wed May 18, 2011 12:15 pm

"Twin" puzzles "Kite A" and "Kite B" really produce the same UA61 set (because both puzzles have the same completions). I'll edit my previous post.

Serg
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A 60+ collection

Postby dobrichev » Fri Mar 23, 2012 4:08 pm

Hi,

I spent some CPU cycles in expanding the high-clue UA search.
The resulting collection is available for downloading from https://sites.google.com/site/dobrichev/sudoku-puzzle-collections.

It contains 44 UA62, 7813 UA61, and 211908 UA60 generating sub-puzzles.

Below are the 44 19-clue pseudo-puzzles which generate 45 UA of size 62.
Hidden Text: Show
Code: Select all
................12..3..4........53...1..6....2..7.......5..34...7.2...8.6...1...9   19   2
...............123..1..4.....4..56...7..2....8..3.......5..14...2..7....3..8.....   19   2
...............123..1..45....4...6...7..2....8..3.......5..14...2.....8.3.......7   19   2
...............123..4..5.......1...6...3..7....2..4.8...9..2.4..1..7....3..6.....   19   2
...............123..4..5.......1...6...3..7....8..2.5...2..8.4..1.6.....3...7....   19   2
...............123..4..5.....1..64...7..2....8..3.......6..15...2.8.....3...7....   19   2
...............123..4..5.....1..67...2..8....3..9.......6..75...8.2.....9...3....   19   2
...............123..4..5.....1..67...8..2....9..3.......6..75...2.8.....3...9....   19   2
...............123..4..5..6.....4..5.1..7....2..8.........1.8.....2...7...5..6..4   19   2
...............123..4..5..6.....4..5.1..7....2..8.........2.8.....1...7...5..6..9   19   2
...............123..4..5..6.....6..4.1..7....2..8.........2.8.....1...7...6..4..5   19   2
..............1..2234.....5....3..6....4..7..1....2..8..6...3...7.....4.5....8...   19   2
..............1234..1..5..6.....4..1.2.....7.3.....8....4.....5.8..3....7..2.....   19   2
..............1234..1..5..6.....6..1.2..7....3..8.........3.8.....2...7...6.....5   19   2
..............1234..5..6..1.....4..5.2..7....3..8.........3.8.....2...7...4.....6   19   2
..............1234..5..6..1.....5..6.2..7....3..8.........3.8.....2...7...6.....9   19   2
..............1234..5..6..7.....4..1.2..8....3..9.........3.8.....2...9...7.....6   19   2
..............1234..5..6..7.....7..1.2..8....3..9.........3.8.....2...9...7.....6   19   2
..............1234..5..6..7....2..8....3..9....4.....6..6.....1.2.8.....3...9....   19   2
..............1234..5..6..7....2..8....3..9....7.....1..4.....6.2.8.....3...9....   19   2
...........1..2..3.4..5..6.........1..2..37...5..8.........75....31....7.9.....8.   19   2
...........1..2..3.4..5..6.........1.6..7..4.2....8......6......7..4..9.8....1..2   19   2
........1........2..3..4..5.....536..7..1....8..2.......6..34...2.7.....1...8....   19   2
........1........2..3..4..5..4..5.3..1..6....2..7.......8..35...6..1....7..2.....   19   2
........1........2..3..4..5..4..6.3..7..1....8..2.......6..59...2.7.....1...8....   19   2
........1.......2...3..4.......1...5...2...6...4..73...1..6.....2.5.....78...34..   19   2
........1.......2...3..4.......1...5...2...6...4..73...1..6.....2.5.....78...94..   19   2
........1.......2...3..4.......1...5...2...6.47....3..1...6....2..5.....8.7..34..   19   2
........1.......2...3..4.......1..5....2....6..7..84...1..5.....2.6.....49...78..   19   2
........1.......2...3..4.......1..5....2....6.7...84..1..5.....2...6....4.8..73..   19   2
........1.......2...3..4.......1..5....2....647....8..1..5.....2...6....8.9..73..   19   2
........1.......2...3..45......1..6....2....7.8...54..1..6.....2...7....4....93..   19   2
........1.......2...3.45........34...1......62......7...548.3...6..2....7...1....   19   2
........1.......2...3.45........35...1.....6.2.......7..58.94...6..1....7...2....   19   2
........1.......2...3.45........63...1.....7.2.......8..69.34...7..1....8...2....   19   2
........1.......2...3.45........64...1.....7.2.......8..43.95...7..1....8...2....   19   2
........1.......2...3.45.....4..35...1.....6.2.......7..58..3...7..2....6...1....   19   2
........1.......2...3.45.....45..3...6...1...7....2.....8.3.5...2.....6.1.......7   19   2
........1.......2...3.45.....46..5...1.....7.2.......8..64..3...7..1....8...2....   19   2
........1.......2...3.456.......63...1......72......8...63..5...8..1....7...2....   19   2
........1.......23..4..5.....5..67...2..8....3..9.......7..46...8.2.....9...3....   19   2
.......12.....34..567...........53....1.8.....2.9.........1...6...2...7.3.....5..   19   2
........1.......2...3..4..5.....34.6.7..1....8..2.......6...3.9.2.8.....1...7....   19   3
........1.....2.3...4.5..........4...1.6....73...8..5....2....9..5....8.7.2..1...   19   3


Tha last 2 pseudo-puzzles generate 3-valent UA62 where the large UA degenerates to UA4 and UA6 respectively. Details:
Hidden Text: Show
Code: Select all
........1.......2...3..4..5.....34.6.7..1....8..2.......6...3.9.2.8.....1...7....   19   3   #pattern,nClues,nSolutions

542968731687351924913724865291583476375416298864297153756142389429835617138679542   1   1   #solution,sNumber,nUnavoidables
5...6....6...5...................................................................   4   n

642958731587361924913724865291583476375416298864297153756142389429835617138679542   2   1   #solution,sNumber,nUnavoidables
6...5....5...6...................................................................   4   n

798532641451697823263184795912753486674918532835246917586421379327869154149375268   3   1   #solution,sNumber,nUnavoidables
79853264.4516978.326.18.79.91275..8.6.49.8532.35.4691758.421.7.3.7.69154.493.5268   62   y


........1.....2.3...4.5..........4...1.6....73...8..5....2....9..5....8.7.2..1...   19   3   #pattern,nClues,nSolutions

879346521651792834234158976927513468518624397346987152483265719165479283792831645   1   1   #solution,sNumber,nUnavoidables
............79..................................9.7................79............   6   n

879346521651972834234158976927513468518624397346789152483265719165497283792831645   2   1   #solution,sNumber,nUnavoidables
............97..................................7.9................97............   6   n

253879641896412735174356298568927413419635827327184956631248579945763182782591364   3   1   #solution,sNumber,nUnavoidables
25387964.89641.7.517.3.6298568927.134.9.3582..271.49.6631.4857.94.7631.2.8.59.364   62   y


There are several "twin" puzzles not included in the collection but they by definition generate the same UA sets with same permutations but in the context of a different solution grid. Example twins:
Hidden Text: Show
Code: Select all
..............2.3...1..3425..3...24..6..7....8..9.......4...1...7..6....9..8.....
........1.....2..3..1..3452.......15.6..7....8..9.......5...2.4.9..8....7..6.....
........1.....2..3..1..3452.......25.6..7....8..9.......2....34.9..8....7..6.....
........1.....2..3..1..3452.......25.6..7....8..9.......2...3.4.9..8....7..6.....
........1.....2..3..1..3452.......25.6..7....8..9.......3...1.4.9..8....7..6.....
........1.....2..3..1..3452.......34.6..7....8..9.......3....25.7..6....9..8.....
........1.....2.34..1..45.2.......43.6..7....8..9.......4...2.5.7.6.....9...8....


Here are the 45 UA62 obtained from the 44 19-clue generating pseudo-puzzles (not double-checked).
Hidden Text: Show
Code: Select all
....12345.12.36.78394758612.46.752311.569348.78.2415.9.71329.562581.479.63958.1.4
....12345.13645728542.7896..6452387.2397.14.685749..13.78.645924.1859.3769.2371.4
....12345.34.67.18125843679.46.5819335962.4.78714.952..82394.6149.17.8526.72.5934
....12345.34.67.18159843672.46.5813732567.4.98714.952..87194.6349.72.8516.23.5794
....12345.34.67.81152843697.46.9817332967.45.8714.59.2.87129.3629.75.8146.53.4729
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..1.2345624.756189695.187.3.27395.643.986421.86..713.54365.297.57.1396.891864..32
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.12.34.563.578.1.9679521384.346.2.952.74986.198..53472.6324951849831.2.752..67.43
.12.34.563.75.829.68.291473.5612..471.93.768.473685129.94.165322.89.371.531472.68


Cheers,
MD
dobrichev
2016 Supporter
 
Posts: 1316
Joined: 24 May 2010

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