## Complete Sudoku Collection - Vol. 1

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

### Re: 5 empty boxes impossible

Ocean wrote:For 4 empty boxes, there are five nonisomorphic cases:
Code: Select all
`...00X 0XX -> Probably none.XX0`

For 3 empty boxes, there are four nonisomorphic cases:
Code: Select all
`...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:

Code: Select all
`. . . 6 3 1 9 5 7. . . 8 7 9 2 1 4. . . 4 2 5 6 3 81 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 . . . . . .`
kjellfp

Posts: 140
Joined: 04 October 2005

dukoso wrote: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
Code: Select all
`. . . 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
Last edited by coloin on Tue Jan 03, 2006 10:09 am, edited 1 time in total.
coloin

Posts: 1638
Joined: 05 May 2005

### Re: 5 empty boxes impossible

I have now completed an exhaustive search on the form

Code: Select all
`00X0XXXXX`

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

Code: Select all
`00X0XXXXX00X0XXXX000X0X0XXX`

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

Posts: 140
Joined: 04 October 2005

### Re: 5 empty boxes impossible

kjellfp wrote: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.
Moschopulus

Posts: 256
Joined: 16 July 2005

### Re: 5 empty boxes impossible

kjellfp wrote:I have now completed an exhaustive search on the form

Code: Select all
`00X0XXXXX`

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

Code: Select all
`00X0XXXXX00X0XXXX000X0X0XXX`

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:

Code: Select all
`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...]

Code: Select all
`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
kjellfp wrote: 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.
Last edited by Ocean on Mon Jan 02, 2006 6:48 pm, edited 1 time in total.
Ocean

Posts: 442
Joined: 29 August 2005

### Re: 5 empty boxes impossible

Ocean wrote: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 32 3 7 1 5 4 8 9 64 8 6 3 9 7 5 2 19 1 2 . . . . . .3 4 5 . . . . . .6 7 8 . . . . . .. . . . . . 9 5 7. . . . . . 2 1 4. . . . . . 6 3 8`
kjellfp

Posts: 140
Joined: 04 October 2005

### Re: Complete Sudoku Collection - Vol. 1

Nick70 wrote: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
Code: Select all
`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!
Kiwy

Posts: 8
Joined: 25 October 2006

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.
ravel

Posts: 998
Joined: 21 February 2006

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