## Greatest divergance possible in dual solution puzzle?

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

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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.

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:#168973245732145689495826173529314768341687952876592431253769814614258397987431526174538269639247518825916347918365724742189653356472981261794835487653192593821476## 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| *-----------*`
Ocean

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Well done, Ocean !

JPF
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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...
Ocean

Posts: 442
Joined: 29 August 2005

This shape is very fruitful.

Here's an other 60 :

Code: Select all
`Clues given:                        21Cells 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 | . . 6368975214972143685415826379529314768143687952786592431251769843634258197897431526394517268671248539825936147918365724742189653536472981263794815187653492459821376`

JPF
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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
`493761852186925473527843169762419538914538627835672941651287394379154286248396715713845962682913574594762138167429853429358716358671249841596327975234681236187495`

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
coloin

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### 61 + disjoint 6

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 cells621897453498325761735146928874951632956283174213764589347612895182579346569438217 #solution A1 ||  ||||| | ||||||| |||| |  | || || ||| | ||||||| ||||||  | |||| |||| ||||||||||643892715297351486185467329871923654924586137356174298439615872562738941718249563 #solution A26..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 B643982715297351486185467329971823654824596137356174298439615872562738941718249563 #solution B2 ||  ||||| | ||||||| |||| |  | || || ||| | ||||||| ||||||  | |||| |||| ||||||||||621987453498325761735146928974851632856293174213764589347612895182579346569438217 #solution B16 . . 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
dobrichev
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### 62

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.
dobrichev
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### Re: Greatest divergance possible in dual solution puzzle?

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.
coloin

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### Re: Greatest divergance possible in dual solution puzzle?

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?

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.
dobrichev
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### Re: Greatest divergance possible in dual solution puzzle?

"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

Hi,

I spent some CPU cycles in expanding the high-clue UA search.

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
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`................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,nSolutions542968731687351924913724865291583476375416298864297153756142389429835617138679542   1   1   #solution,sNumber,nUnavoidables5...6....6...5...................................................................   4   n642958731587361924913724865291583476375416298864297153756142389429835617138679542   2   1   #solution,sNumber,nUnavoidables6...5....5...6...................................................................   4   n798532641451697823263184795912753486674918532835246917586421379327869154149375268   3   1   #solution,sNumber,nUnavoidables79853264.4516978.326.18.79.91275..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,nSolutions879346521651792834234158976927513468518624397346987152483265719165479283792831645   1   1   #solution,sNumber,nUnavoidables............79..................................9.7................79............   6   n879346521651972834234158976927513468518624397346789152483265719165497283792831645   2   1   #solution,sNumber,nUnavoidables............97..................................7.9................97............   6   n253879641896412735174356298568927413419635827327184956631248579945763182782591364   3   1   #solution,sNumber,nUnavoidables25387964.89641.7.517.3.6298568927.134.9.3582..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
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`..............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).
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Cheers,
MD
dobrichev
2016 Supporter

Posts: 1827
Joined: 24 May 2010

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