The New Sudoku Players' Forum

[kjellfp]

For 4 empty boxes, there are five nonisomorphic cases:

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...

00X
0XX -> Probably none.
XX0

For 3 empty boxes, there are four nonisomorphic cases:

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...

00X
0XX -> Probably none.
XXX

Actually, those two are the same. If there happens to be a Sudoku with the three empty boxes, then we trivially can introduce the fourth.

Btw, here is one with only two solutions:

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. . . 6 3 1 9 5 7
. . . 8 7 9 2 1 4
. . . 4 2 5 6 3 8
1 5 9 2 8 6 . . .
2 3 7 1 5 4 . . .
4 8 6 3 9 7 . . .
9 1 2 . . . . . .
3 4 5 . . . . . .
6 7 8 . . . . . .

[coloin]

there are essentially 2865 different ways to form a valid B12347-subsudoku.
This is a sudoku-grid where the 36 entries in blocks 5,6,8,9 are set to zero.

Solving the 2865 grids for all the possible B5 [and each B6,B8 &B9 ?] combinations [average 402 for each B5......] will confirm whether there is a solution to a grid like

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. . . 6 3 1 9 5 7
. . . 8 7 9 2 1 4
. . . 4 2 5 6 3 8
1 5 9 2 8 6 . . .
2 3 7 1 5 4 . . .
4 8 6 3 9 7 . . .
9 1 2 . . . . . .
3 4 5 . . . . . .
6 7 8 . . . . . .

Its beyond my capabilities ...... but I'm not hopeful however ! C

[kjellfp]

I have now completed an exhaustive search on the form

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00X
0XX
XXX

No Sudoku exists, but there are plenty with only two solutions.

This completes the list of which empty-box sets there might be in a Sudoku. Ocean's list is complete, his assumtions were right. I.e. there are no Sudokus on any of the forms

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00X
0XX
XXX

00X
0XX
XX0

00X
0X0
XXX

I don't know the minimal number of solutions for the last one.

[Moschopulus]

No Sudoku exists, but there are plenty with only two solutions.

That's interesting. Could you post a list of them?

The only other example I know of a pseudo-puzzle with two completions is the one found by Gordon with 16 clues.

[Ocean]

I have now completed an exhaustive search on the form

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00X
0XX
XXX

No Sudoku exists, but there are plenty with only two solutions.

This completes the list of which empty-box sets there might be in a Sudoku. Ocean's list is complete, his assumtions were right. I.e. there are no Sudokus on any of the forms

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00X
0XX
XXX

00X
0XX
XX0

00X
0X0
XXX

I don't know the minimal number of solutions for the last one.

Great!


To sum it up, what you have established is that all Sudokus with emtpy boxes belong to (or can be transformed to) one of these six (nonisomorphic) cases:

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0XX  00X  0XX  00X  0XX  00X
XXX  XXX  X0X  XX0  X0X  XX0
XXX  XXX  XXX  XXX  XX0  XX0

I had prepared a trivial post on something analogous to a 'complexity hiearchy': [or maybe there is a better word...]

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00X
0XX (3 empty boxes)
XXX

00X
0XX (4 empty boxes, but 'similar to' or 'same as' above)
XX0

00X
0X0 (4 empty boxes, and at least as 'hard' as those above)
XXX

00X
0X0 (5 empty boxes, and at least as 'hard' as all above)
XX0

00X
0X0 (6 empty boxes, and at least as 'hard' as all above)
X00

Since no Sudoku exists for the 3-case, it's now obvious that neither exists for the 4/5/6-cases in this hierarchy. Further, the minimal number of solutions cannot decrease down the hierarchy.

With reference to

B)
XX0
X00
00X

...
I have written a program to do B) in the most efficient way I could find, and the program completed in 14 minutes. The smallest positive number of solutions found was 72.

this means that the minimal number of solutions for this one:

00X
0X0
XXX

is between 2 and 72.

[kjellfp]

this means that the minimal number of solutions for this one:

00X
0X0
XXX

is between 2 and 72.

This one has 15 solutions:

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1 5 9 2 8 6 4 7 3
2 3 7 1 5 4 8 9 6
4 8 6 3 9 7 5 2 1
9 1 2 . . . . . .
3 4 5 . . . . . .
6 7 8 . . . . . .
. . . . . . 9 5 7
. . . . . . 2 1 4
. . . . . . 6 3 8

[Kiwy]

I have just completed a long search that required several weeks of calculations.
I have found all puzzles where the clues are placed in the following pattern of 19:

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*...*....
...*...*.
..*...*..
.*...*...
*...*...*
...*...*.
..*...*..
.*...*...
....*...*

The end result is that there are exactly 851 different puzzles following that pattern.

the Oddest Forum
http://oddest.nc.hcc.edu.tw/phpBB
user Tthsieh find 4 puzzles
http://oddest.nc.hcc.edu.tw/phpBB/viewtopic.php?t=55

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100050000000100030002000400050004000300060001000700020004000700080006000000030005
100050000000100030002000400050006000300070001000400020004000700080007000000030005
100050000000100030002000400050003000300060001000400020004000700080006000000030005
100050000000100060007000300050006000300040001000200070004000800060002000000030005


I using GSF program to check there are not Isomorphs!
the way I use is

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sudoku -f"%3n %#0c %#0v" puzzles.txt >check.txt


So is there someing I missed or the 4 puzzles are not Isomorphs!

[ravel]

The puzzles are equivalent to lines 4,26,350 and 528 in nick70's collection. I evaluated it with gsf's program, so i suppose, you made a mistake, when comparing the canonicalized forms.