Canonical Grid Has No 19

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

Canonical Grid Has No 19

Postby Moschopulus » Mon Aug 28, 2006 11:02 pm

The canonical grid

123456789
456789123
789123456
231564897
564897231
897231564
312645978
645978312
978312645

has no puzzle with 19 clues. Checker did the search in about 15 hours.

This question has some history. First dukuso here showed there is no 18. He gave a 19. Then coloin in this thread pointed out that the 19 dukuso gave is a mistake.

I always thought it was odd that such a symmetric grid has a 19, and I know others did too. If any grid shouldn't have a 19, it's this one. So now I feel better.
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Postby coloin » Tue Aug 29, 2006 6:10 pm

yes I checked it too - no 19. It was always a strange one that its near identical canonical grid didnt have a 19.

I almost thought it didnt have a 20 untill they started appearing eventually.

Code: Select all
868/46656 (2332581 puzzles gen'd; 10 uniques found); ETTF 20d 22h 08m


This gives a prediction of 537 20-clue puzzles. This is near enough the 648 morph puzzles for each puzzle that Red Ed has suggested are in this symmetrical grid.

These 10 puzzles will all be isomorphic in this case
Code: Select all
100400080050700003009000000030060000504000000000201060000000900000008010070300005
100400089050700000000003000000060007000090200800001060002000000600000010070300005
100400089050000000700003000000060000500007030000200004002000070040900002008000600
100400089006700000000020000030000000500000001007200060000005070000008300900010005
100400089006000000700020000030000007004800030090000500000005000600070002000300040
100400080000009000700020006030060000504000030000201000000000900600070002008000040
100400009006700020080000000000000800000090001007200060002005000640000000000310005
100400009000080000700003050000000800500007030090000004002005000640000002000310000
100400000056000000000023400030060800500090001007000000000000070000008300900010005
100400000056000100000023000030060800000007000090200004000000070040900002008000600

C
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Postby Red Ed » Thu Aug 31, 2006 12:34 pm

coloin wrote:These 10 puzzles will all be isomorphic in this case
Yes, they are.

I hope that when using checker, you were fixing (at least) the first clue to be ...well... anything you like. You can get away with this because the 648 automorphisms mean that any single clue is equivalent to any other. This gives a 6x speedup.
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Postby coloin » Thu Aug 31, 2006 1:50 pm

I' m relieved that is the case !
Red Ed wrote:I hope that when using checker, you were fixing (at least) the first clue to be ...well... anything you like. You can get away with this because the 648 automorphisms mean that any single clue is equivalent to any other. This gives a 6x speedup.


Yes I think that is right, though I didnt use it.......in fact I too think you can fix more clues....

At least 3
Code: Select all
+---+---+---+
|1..|...|...|
|...|...|...|
|...|...|...|
+---+---+---+
|...|5..|...|
|...|...|...|
|...|...|...|
+---+---+---+
|...|...|9..|
|...|...|...|
|...|...|...|
+---+---+---+

We might easily generate what multiple of 648 the 21 clue and 22 clue stats are !

On a [sort of] similar theme
..........................................It may just be a coincidence but......have you noticed that all puzzles from any grid have from 5 to 9 clues of the following inverse/disjointed template pattern equivalent ?
Code: Select all
+---+---+---+
|1..|...|...|
|...|2..|...|
|...|...|3..|
+---+---+---+
|.4.|...|...|
|...|.5.|...|
|...|...|.6.|
+---+---+---+
|..7|...|...|
|...|..8|...|
|...|...|..9|
+---+---+---+

Most commonly, especially in many of Gfroyles 17s, there are 7 clues of the 9.

e.g a 17
Code: Select all
+---+---+---+
|...|...|.12|
|3..|...|.6.|
|...|.4.|...|
+---+---+---+
|9..|...|5..|
|...|..1|.7.|
|.2.|...|...|
+---+---+---+
|...|35.|4..|
|..1|4..|8..|
|.6.|...|...|
+---+---+---+

and a 7 clue "disjointed template" [couldnt find an 8 clue]
Code: Select all
+---+---+---+
|...|...|..2|
|...|...|...|
|...|.4.|...|
+---+---+---+
|9..|...|...|
|...|..1|...|
|...|...|...|
+---+---+---+
|...|3..|...|
|...|...|8..|
|.6.|...|...|
+---+---+---+


Did Gordon normalize them to this pattern and add the remaining clues ?

The inverse template [8 out of 9] can be found in in both Oceans 2 hardest puzzles.

All puzzles can be classified as to which maximum of 5,6,7,8 or 9 clue inverse templates they have within !

Here is a grid, any minimal puzzle from which can only have a 5 !
Code: Select all
+---+---+---+
|...|...|782|
|...|...|164|
|...|...|935|
+---+---+---+
|732|984|...|
|684|512|...|
|159|376|...|
+---+---+---+
|965|123|...|
|347|698|...|
|218|745|...|
+---+---+---+


C
Last edited by coloin on Wed Sep 13, 2006 9:47 am, edited 1 time in total.
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Postby Moschopulus » Sun Sep 10, 2006 7:09 am

Some more 20s in the canonical grid:

020400700006080100000003000000060090000807000090000504300000000040000002008010600
023400700000080100000000006001000090500000030090200004000005000040900000008010600
023400700000080000000100006000000090060800001007000500010005000005070300900002000
023400700000009000000100050001060000060007200800030000000000008005900010070000600
020400700000000003009100050000060090060807000000000504300000000005900010070002000
023400700000009100000000050000060000004800000090001500010000008600000002008300040
020400080050009100000000006000060000004800000090001500300000900000000012078300000
020406080050000100009000000000000007000800030090001500300005000600070000008300040
020406080050000003700000000001500000004090000800001060000000900000008010070300005
020400080006000000700003050000060000500007030000200004300000900000900012078000000
020406080000000003700000050001000000500007030090200000000005900040900002000010600
020406080000000100009000050000500007060007200000030004300000000005900010070002000

Are they equivalent to coloins puzzles above?

Looks like the full search would take 5-7 days, assuming 11 is a clue, which we can do.
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Postby gsf » Sun Sep 10, 2006 12:07 pm

Moschopulus wrote:Some more 20s in the canonical grid:
Are they equivalent to coloins puzzles above?

yes
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Postby coloin » Tue Sep 12, 2006 10:31 pm

Well, I have taken a bit of a liberty in stating that there is 648 20s in the MC grid here.

But the chances are it has to be correct.

Out of interest - How many of these 21s from the MC [89 in total] are non-isomorphic
...4...8..5.7....37.9.......3..6....5.4.........2.1.6...2...9.......8.1..7.3....5
...4..78..5.7....3..9.......3..6....5.4.........2.1.6...2...9.......8.1..7.3....5
1......8....7....3..9.2.....3..6....5.4.........2.1.6......59....5..8.1..7.3....5
1......8....7....3..9.2.....3..6....5.4.........2.1.6......59...4...8.1..7.3....5
1......8....7....3..9.2.....3..6....5.4.........2.1.6......59..6....8.1..7.3....5
1......8...67....3..9.2.....3..6....5.4.........2.1.6......59.......8.1..7.3....5
1......8..5.7....3..9.......3..6....5.4.........2.1.6......59.......8.1..7.3.2..5
1......8..5.7....3..9.......3..6....5.4.........2.1.6.....459.......8.1..7.3....5
1......8..5.7....3..9.......3..6.8..5.4...........1.6......59.......8.1..7.3.2..5
1......8..5.7....3..9.......3..64...5.4.........2.1.6.....4.9.......8.1..7.3....5
1......8..5.7....3..9.......3.5.....564.........2.1.6.....4.9.......8.1..7.3....5
1......8..5.7....3..9.......3.56......4.........2.156.....4.9.......8.1..7.3....5
1......8..5.7....3..9.......3.56....5.4.........2.1.6.....4.9.......8.1..7.3....5
1......8..5.7....3..9.2.....3..6....5.4.........2.1.6......59.......8.1..7.3....5
1......8..5.7...23..9.......3..6.8..5.4...........1.6......59.......8.1..7.3....5
1......8.4..7....3..9.2.....3..6....5.4.........2.1.6......59.......8.1..7.3....5
1....6.8..5.7....3..........3..64...5.4......8..2.1.6.......9.......8.1..7.3....5
1....6.8..5.7....3..........3.56....5.4......8..2.1.6.......9.......8.1..7.3....5
1..4...8....7....3..9.2.....3..6....5.4.........2.1.6.......9....5..8.1..7.3....5
1..4...8....7....3..9.2.....3..6...75.4.........2.1.6..........6....8.1..7.3....5
1..4...8....7....3..9.2.....3..6.8..5.4...........1.6.......9....5..8.1..7.3....5
1..4...8...67....3..9.2.....3..6...75...........2.1.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9..........6.89.5.4...........1.6...2...9.......8.1..7.3....5
1..4...8..5.7....3..9.......3..6......48..........156.......9.....9.8.1..7.3....5
1..4...8..5.7....3..9.......3..6......48..........156.......9..6....8.1..7.3....5
1..4...8..5.7....3..9.......3..6......489.........156.......9.......8.1..7.3....5
1..4...8..5.7....3..9.......3..6....5.4......8....156.......9.......8.1..7.3....5
1..4...8..5.7....3..9.......3..6....5.4...2.......156.......9.......8.1..7.3....5
1..4...8..5.7....3..9.......3..6....5.4...2..8....1.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9.......3..6....5.48.....8....1.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9.......3..6...75.4.........2...6.......9.......8.1..7.3.2..5
1..4...8..5.7....3..9.......3..6...75.4.........2.1.6..........6....8.1..7.3....5
1..4...8..5.7....3..9.......3..6.8....4.........2.1.6........7.6....8.1..7.3....5
1..4...8..5.7....3..9.......3..6.8..5.4...........1.6.......9.......8.1..7.3...45
1..4...8..5.7....3..9.......3..6.8..5.4...........1.6.......9.......8.1..7.3.2..5
1..4...8..5.7....3..9.......3..6.8..5.4...........1.6.......9.......8.1.97.3....5
1..4...8..5.7....3..9.......3..6.8..5.4...........1.6.......9.......8.12.7.3....5
1..4...8..5.7....3..9.......3..6.8..5.4...........1.6.......9.......831..7.3....5
1..4...8..5.7....3..9.......3..6.8..5.4...........1.6.......9......78.1..7.3....5
1..4...8..5.7....3..9.......3..6.8..5.4...........1.6.......9.....9.8.1..7.3....5
1..4...8..5.7....3..9.......3..6.8..5.4...........1.6.......9...4...8.1..7.3....5
1..4...8..5.7....3..9.......3..6.8..5.4...........1.6.......97......8.1..7.3....5
1..4...8..5.7....3..9.......3..6.8..5.4...........1.6.3.....9.......8.1..7.3....5
1..4...8..5.7....3..9.......3..6.8..5.4.......9...1.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9.......3..6.8..5.4......8....1.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9.......3..6.8..5.4....3......1.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9.......3..6.8..5.4.9.........1.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9.......3..6.89.5.4...........1.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9.......3.56......4.........2.156.......9.......8.1..7.3....5
1..4...8..5.7....3..9.......3.56......4......8....1.6.......9....5..8.1..7.3....5
1..4...8..5.7....3..9.......3.56......4......8....1.6.3.....9.......8.1..7.3....5
1..4...8..5.7....3..9.......3.56......4......8....156.......9.......8.1..7.3....5
1..4...8..5.7....3..9.......3.56......4......8.7..1.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9.......3.56......4...2.......1.6.......9.......831..7.3....5
1..4...8..5.7....3..9.......3.56......4...2.......1.6.......97......8.1..7.3....5
1..4...8..5.7....3..9.......3.56......4...2.......1.6.3.....9.......8.1..7.3....5
1..4...8..5.7....3..9.......3.56......4...2.......156.......9.......8.1..7.3....5
1..4...8..5.7....3..9.......3.56......4...2......31.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9.......3.56......4...23......1.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9.......3.56....5.4......8....1.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9.......3.56....5.4...2.......1.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9......23..6......4...........156.......9....5..8.1..7.3....5
1..4...8..5.7....3..9......23..6......4...........156.......9..6....8.1..7.3....5
1..4...8..5.7....3..9......23..6......4...........156..1....9.......8.1..7.3....5
1..4...8..5.7....3..9......23..6......4.....1.....156.......9.......8.1..7.3....5
1..4...8..5.7....3..9......23..6.....64...........156.......9.......8.1..7.3....5
1..4...8..5.7....3..9......23..6....5.4...........156.......9.......8.1..7.3....5
1..4...8..5.7....3..9......23..6.8..5.4...........1.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9......23.56......4...........156.......9.......8.1..7.3....5
1..4...8..5.7....3..9......23.56....5.4...........1.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9......231.6......4...........156.......9.......8.1..7.3....5
1..4...8..5.7....3..9....5..3.56......4...2.......1.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9...4...3..6...756..........2.1.........9.......8.1..7.3....5
1..4...8..5.7....3..9...4...3..6.8..5.4...........1.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9...4...3..6.8..564...........1.........9.......8.1..7.3....5
1..4...8..5.7....3..9..3....3.56......4......8....1.6.......9.......8.1..7.3....5
1..4...8..5.7....3..9.2.....3..6.8..5.4...........1.6.......9.......8.1..7.3....5
1..4...8..5.7....3.89.......3..6.8..5.4...........1.6.......9.......8.1..7.3....5
1..4...8..5.7....37.9.......3.56......4......8....1.6.......9.......8.1..7.3....5
1..4...8..5.7...23..9.......3..6.8..5.4...........1.6.......9.......8.1..7.3....5
1..4...8..5.7.9..3..9.......3..6.8..5.4...........1.6.......9.......8.1..7.3....5
1..4...8..5.78...3..9.......3..6.8..5.4...........1.6.......9.......8.1..7.3....5
1..4...8.45.7....3..9.......3..6.8..5.4...........1.6.......9.......8.1..7.3....5
1..4..78..5.7....3..9.......3.56......4......8....1.6.......9.......8.1..7.3....5
1..4.6.8..5.7....3..........3..6....5.4......8....156.......9.......8.1..7.3....5
1..4.6.8..5.7....3..........3..6....5.4......8..2.1.6.......9.......8.1..7.3....5
1..4.6.8..5.7....3..........3.56....5.4......8....1.6.......9.......8.1..7.3....5
1..4.6.8..5.7....3..9......23..6......4...........156.......9.......8.1..7.3....5
1.34...8..5.7....3..9.......3.56......4......8....1.6.......9.......8.1..7.3....5


C
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Postby ronk » Wed Sep 13, 2006 12:20 am

coloin wrote:Out of interest - How many of these 21s from the MC [89 in total] are non-isomorphic

If you're using checker to generate uniques from a canonical grid ... I think all those puzzles will be non-equivalent. On second thought, I would expect checker to generate non-equivalents from any grid.
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Postby gsf » Wed Sep 13, 2006 1:52 am

coloin wrote:Out of interest - How many of these 21s from the MC [89 in total] are non-isomorphic

all different
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Postby coloin » Wed Sep 13, 2006 2:08 am

Thanks

I didnt use checker to generate those 21s....but i think checker does generate different puzzles although, in this MC case, it generates many isomorphic puzzles .

Total computation time was 0d 01h 54m.
Only a partial search was done -- searched the branch whose first clues are 11, 44
There are 23 20-puzzles in this branch (all puzzles were solved).
Code: Select all
100000700000000023089100000030500090060007200000000004000040000005900000070002600
100000700000700023089000000030500090004000000800001060000040000600008010000300005
100000080400000003009020400030500000060090201007000000000040900000008000070300005
100000080050709020000000006030500000004000000800001060000040900000008300070300005
100000080050700003009000000030500000000007000800200064000040900600008010000010005
100000080050700003009000050030500000004090200000001000000040900000000002078300600
100000080050700003009000000030500007000090000800001004000045000600000012000300600
100000080006000000700023400030500000004090200090001000000040900600008010000000005
100000080000000003009020400000504000060090000807000060002040900005008010000300000
100000089000700100000023000030500007004000000800000060000040000600008002070300005
103000080000709000080020000000500000004090200007001060002040900600000010000000005
100000080000000003009020400030500000004800000800001060000040900605070010000002000
100000009050000003009020400000500000004090000800001060302040070000008010000000600
100000000050000003009020400030500800004090200000001000000040070000908000070000605
100000000006700020080003000030500800000000001007200060000040070040908000000000605
100000000050700003009020000000500090004090200000001060002640008000000300070000005
100000000006700020080003000030504090000000200007000060000600008040008300000010005
100000000006780003009000050001500090000000200000030004300040008040900000070002000
120000000000780003009003000001500090060000000800200004000040008005900010000000600
100000000050700003009020006030500007000090200000001000000040908605000000070000040
100000000050080003009020400000560007004007000890000000002040900000000010000300005
103000000050000020000020406030500007004090001800000000000040900000008000070300005
120000000000780003009003000000500007004090200000000060002040900600000000070010005


12 "different" if you fix {11,44,77} - all isomorph puzzles it is presumed
Code: Select all
100000000000000000000000000000500000000000000000000000000000900000000000000000000


1 "different" if you fix {11,44,98}
Code: Select all
100000000000000000000000000000500000000000000000000000000000000000000000000000040


21 update ! [Edited]

I have done the count with 11 and 44 fixed - only 500

Unfortunately....it would appear that checker cannont be used effectivly here to count puzle stats

It appears to stop at 500
It produces non minimal 21s [20s] as well !

Why did I want to do this ? Well it seem the best grid that we can get near estimating the number of puzzles in a grid !!!!!![Some say 10^16]

Puzzle stats for 10^6 generated puzzles using suex9s.exe .......which must be flawed - either as a bug or skewed by a probability quirk.
Code: Select all
         
20 5   
21 56 
22 1797
23 21631
24 116439 
25 287167 
26 330392 
27 184541 
28 50751   
29 6735   
30 466   
31 20

If 5 per million 20s are found that would make the total number of puzzles a paltry 130,0000000
No duplicates are produced which puts the number of puzzles at more than 10^11 !
Plus we know there are at least 60,000 33s ! - none appeared by random generation !

C
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Postby ronk » Fri Sep 15, 2006 5:01 pm

coloin wrote:I have done the count with 11 and 44 fixed - only 500

If I'm using gsf's sudoku.exe properly, 152 of those are equivalent to one of the other 348 so, as you said earlier, checker.exe generates equivalent puzzles even from a canonical solution grid.
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Postby Moschopulus » Thu Sep 21, 2006 1:40 pm

Just to reply to some points above about checker:

- checker finds ALL puzzles with n clues in a given grid. All means all; no checking is done for minimality or equivalence.

- checker stops saving puzzles after finding 500 puzzles. The number 500 can be increased by modifying the constant MAX_PUZZLES near the top of the source file checker.cpp and then recompiling checker. At the time of writing we didn't think anyone would want more than 500.


Checker was really written to see if a grid has any n-clue puzzles, not to see how many. That's why things were done this way. If the usage is now changing, maybe the program should be changed to suit this?


All suggestions for changes to checker are very welcome, but I can't promise they will be done. It took many months to make the first improvements .... the priority level is not high I'm afraid. Still it is good to know what people would like, in the event that time becomes available. Also, the source code is all there, anyone can play around with it.
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Postby coloin » Sat Oct 07, 2006 12:51 pm

Thankyou Moschopulus

I fully appreciate what checker was originally intended to do - and how well it has been improved.
If we wanted to we could easily modify it to count up the number of puzzles in the mc grid [at least in the 21,22,23 clue levels].

Effectivly by introducing furthur constraints we can increase the mcn to 18 - as has been mentioned - this will occur in bands [1,30,310,413] according to the index416 program. [as described previously][may not even be necessary]

We can fix equivalent clues I think according to this pattern which reduces the total number of grids by up to 648. .......its took me a while........and I wracked my brain.......but the answer was there all along [in this post !]

The essential clues are a partial 7/9 Rook pattern [ 2 classes only ] [there is no 8/9 in the 20 puzzle]


Every puzzle in the MC grid [apart from puzzles with 3 empty boxes - and they have a minimum of 23 clues] has this 7/9 rook pattern somewhere in it - it is difficult ?impossible to avoid it.

I had initially thought that all the 7/9 rook patterns were the same ....but it appears not.

Here is a 20 clue puzzle from the mc grid
Code: Select all
+---+---+---+
|12.|...|...|
|..6|...|..3|
|...|..3|45.|
+---+---+---+
|..1|.6.|8..|
|5..|..7|2..|
|.9.|...|...|
+---+---+---+
|...|..5|.7.|
|...|9..|...|
|..8|.1.|6..|
+---+---+---+
and one of the 7/9 rook patterns
+---+---+---+
|1..|...|...|
|...|...|..3|
|...|...|...|
+---+---+---+
|...|.6.|...|
|...|...|2..|
|.9.|...|...|
+---+---+---+
|...|...|.7.|
|...|...|...|
|..8|...|...|
+---+---+---+


Checker only allows me to insert clues according to precalculated sets..

Running checker with fixed clues as allowed - positions 11 and 45 and we will see what the output for fixing 2 clues is. I will then manually seive the 7/9 rookery pattern above...........................................

Code: Select all
Finished.  Total computation time was 0d 02h 06m.
Only a partial search was done -- searched the branch whose first clues are 11, 45
There are 11 20-puzzles in this branch (all puzzles were solved).

Code: Select all
100000080050009100700000006000060000004800030000001500002005970040000000008300000
103000080000709000080020000200060007004000030000000500000005000600070002900300040
100000009006700020080000000000064000500000031000200500002600070000008000900010040
100000009006080100700000050030060807004000000090200000000005000040900002000010600
120000009000780000009003000000060000500007030800200004300005070040000002000000600
100000000050000003009020400030060807004090000000200000000005900000008010070300005
100000000006080000009003450001060800000007030000200000300005070040000002070000600
100000000006000020080003400000060800000090001007200060002005970040008000000300000
120000000006000003000003450001060800500007200090000000000005070000900000008010600 X
100000000050009000080020406200060007004000030007000500010005900000070002000300000
103000000050000020000020406000060007000800000090001500010005900600070300008000000




Using the five 7/9 rookerys from the above grid that I can find - this is essentially the same for all the isomorphic 20 puzzles
Code: Select all
1................3.............6..........2...9..............7............8......
1......................3.......6..........2..................7....9.......8......
1................3.............6..........2...9............5..............8......
.2.....................3.......6....5........................7....9.......8......
.2......................4......6....5........................7....9.......8......




Code: Select all
1................3.............6..........2...9..............7............8......  1 solution
1......................3.......6..........2..................7....9.......8......  1 solution
1................3.............6..........2...9............5..............8......  1 solution
.2.....................3.......6....5........................7....9.......8......  1 solution
120000000006000003000003450001060800500007200090000000000005070000900000008010600
.2......................4......6....5........................7....9.......8......  2 solutions
020000009006780003000000400001060000500000000090200004000005070000900010008010600
.x.........x.....x......x....x.x....x.........x............x.x....x.......x.x.x..


So four out of the five rookery patterns define the isomorph puzzle there
One of the five has two solutions.......the two isomorphs have 14 clues in common......how strange.

But not all 7/9 rookery patterns define a puzzle. That would figure...There must be more than I realize.....
Red Ed wrote:The MC grid has [9/9] Rookery Templates =2241

Indeed there are.

C
coloin
 
Posts: 1638
Joined: 05 May 2005


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