Ask for some patterns that they don't have puzzles.

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

Postby RW » Thu May 24, 2007 3:05 pm

Thanks for the info JPF. What I'm actually even more interested to know is "what is the minimum number of solutions of a puzzle where all instances of 5 digits are given, but the remaining 4 digits are missing?" Like this:
Code: Select all
 *-----------*
 |..7|.31|6.2|
 |21.|6..|.37|
 |36.|.27|1..|
 |---+---+---|
 |7..|263|..1|
 |.36|1..|27.|
 |.21|.7.|.63|
 |---+---+---|
 |..3|712|..6|
 |1..|3.6|72.|
 |672|...|31.|
 *-----------* 288 solutions

Theoretically 24 solutions would be the minimum, in this case adding any three different digits as clues would give an unique puzzle. In the terminology of the structures thread, this would be a fully entwined quad. I think the 241 0111XXXXX puzzles could be a good starting point to look for low solution grids like this. Another good source for potential low solutions would probably be the FE-grids. If there is a FE-quad, then the next question is: how close to 120 solutions can we get when completely removing 5 digits?

RW
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Postby JPF » Fri May 25, 2007 10:18 pm

RW wrote:What I'm actually even more interested to know is "what is the minimum number of solutions of a puzzle where all instances of 5 digits are given, but the remaining 4 digits are missing?"

Here is a puzzle with 24 solutions :
6,7,8,9 are missing.

Code: Select all
 1 . 2 | . 3 4 | . . 5
 3 . 4 | 5 . . | . 2 1
 . 5 . | 1 2 . | 3 4 .
-------+-------+-------
 . 3 . | . 1 2 | . 5 4
 4 . 1 | . . 5 | 2 . 3
 2 . 5 | 3 4 . | . 1 .
-------+-------+-------
 5 2 . | . . 1 | 4 3 .
 . 1 . | 4 . 3 | 5 . 2
 . 4 3 | 2 5 . | 1 . .


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Postby claudiarabia » Sat May 26, 2007 10:01 am

RW wrote:Thanks for the info JPF. What I'm actually even more interested to know is "what is the minimum number of solutions of a puzzle where all instances of 5 digits are given, but the remaining 4 digits are missing?" Like this:
Code: Select all
 *-----------*
 |..7|.31|6.2|
 |21.|6..|.37|
 |36.|.27|1..|
 |---+---+---|
 |7..|263|..1|
 |.36|1..|27.|
 |.21|.7.|.63|
 |---+---+---|
 |..3|712|..6|
 |1..|3.6|72.|
 |672|...|31.|
 *-----------* 288 solutions

Theoretically 24 solutions would be the minimum, in this case adding any three different digits as clues would give an unique puzzle. In the terminology of the structures thread, this would be a fully entwined quad. I think the 241 0111XXXXX puzzles could be a good starting point to look for low solution grids like this. Another good source for potential low solutions would probably be the FE-grids. If there is a FE-quad, then the next question is: how close to 120 solutions can we get when completely removing 5 digits?

RW


Joker ;-) ! It just reminds me of those discussion: http://forum.enjoysudoku.com/viewtopic.php?t=3717&highlight=
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Postby RW » Sat May 26, 2007 7:02 pm

JPF wrote:Here is a puzzle with 24 solutions :
6,7,8,9 are missing.

Very nice! I wasn't all convinced that it was possible, but you proved me wrong again!:) I think it's fascinating that adding any clue to the grid will automatically always fix the remaining cells for the same digit also. And I think it's even more fascinating how hard it can be to solve only four clues. Here's a variant puzzle for you, digits 1,2,3 and 4 appear once only in every row column and box:
Code: Select all
 *-----------*
 |X.X|.XX|..X|
 |X.X|X..|.XX|
 |.X.|XX.|XX.|
 |---+---+---|
 |.X.|.XX|.XX|
 |X.X|..X|X.X|
 |X.X|XX.|.X.|
 |---+---+---|
 |XX1|2.X|XX.|
 |.X.|X.X|X.X|
 |3XX|XX.|X..|
 *-----------*

ER 9.1!

So, how are you doing on the 5 digits missing/120 solutions problem?:D

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Postby RW » Sat May 26, 2007 7:31 pm

Using that grid, it was no problem to construct an 8-clue odd-even:
Code: Select all
 *-----------*
 |,.,|.,5|..,|
 |,.,|,..|.,,|
 |.,.|,,.|1,.|
 |---+---+---|
 |.,.|.,,|.,,|
 |,.,|..,|,.,|
 |,.,|,,.|.,.|
 |---+---+---|
 |,,2|4.7|,,.|
 |.,.|,.,|,.3|
 |65,|,,.|,..|
 *-----------*
, = odd
. = even

Is this the minimum, or is the theoretical 7-clue puzzle possible? (If the 7-clue odd-even is possible, then the digit distribution 01111XXXX is possible in vanilla sudoku.)

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Postby JPF » Sat May 26, 2007 8:41 pm

RW wrote:So, how are you doing on the 5 digits missing/120 solutions problem?:D

Code: Select all
 1 . 2 | . 3 . | . . 4
 4 . . | . 2 1 | . . 3
 . 3 . | 4 . . | 1 2 .
-------+-------+-------
 2 . . | . . 4 | 3 1 .
 . . 1 | 3 . . | 4 . 2
 . 4 3 | . 1 2 | . . .
-------+-------+-------
 . 2 . | . 4 3 | . . 1
 . 1 4 | 2 . . | . 3 .
 3 . . | 1 . . | 2 4 .


145200 (= 1210 x 120) solutions !

Edit : a slight improvment :
Code: Select all
010200340340001020002043001130400002204100003000032410020310004003024100401000230
137160 solutions (1143 x 120)

btw, here is the max I got, up to now :
Code: Select all
120000340400230100300410200034102000000003421012004003201340000043000012000021034
3754560 solutions (31288 x 120)

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Postby udosuk » Mon May 28, 2007 2:52 am

Thanks RW for the nice puzzles!:)
RW wrote:Here's a variant puzzle for you, digits 1,2,3 and 4 appear once only in every row column and box:
Code: Select all
 *-----------*
 |X.X|.XX|..X|
 |X.X|X..|.XX|
 |.X.|XX.|XX.|
 |---+---+---|
 |.X.|.XX|.XX|
 |X.X|..X|X.X|
 |X.X|XX.|.X.|
 |---+---+---|
 |XX1|2.X|XX.|
 |.X.|X.X|X.X|
 |3XX|XX.|X..|
 *-----------*

ER 9.1!

This is equivalent to the following puzzle:
Code: Select all
 56789  1234   56789  | 134    56789  56789  | 1234   1234   56789 
 56789  1234   56789  | 56789  1234   1234   | 1234   56789  56789 
 124    56789  234    | 56789  56789  1234   | 56789  56789  1234   
----------------------+----------------------+----------------------
 124    56789  234    | 134    56789  56789  | 1234   56789  56789 
 56789  1234   56789  | 134    1234   56789  | 56789  1234   56789 
 56789  1234   56789  | 56789  56789  1234   | 1234   56789  1234   
----------------------+----------------------+----------------------
 56789  56789  1      | 2      34     56789  | 56789  56789  34     
 24     56789  24     | 56789  134    56789  | 56789  1234   56789 
 3      56789  56789  | 56789  56789  14     | 56789  124    124   

Where you arrive at the following state after a few basic moves:
Code: Select all
 56789  1234   56789  | 134    56789  56789  | 1234   1234   56789 
 56789  1234   56789  | 56789  124    1234   | 1234   56789  56789 
 124    56789  234    | 56789  56789  1234   | 56789  56789  1234   
----------------------+----------------------+----------------------
 124    56789  234    | 134    56789  56789  | 1234   56789  56789 
 56789  1234   56789  | 134    124    56789  | 56789  1234   56789 
 56789  1234   56789  | 56789  56789  1234   | 1234   56789  1234   
----------------------+----------------------+----------------------
 56789  56789  1      | 2      34     56789  | 56789  56789  34     
 24     56789  24     | 56789  13     56789  | 56789  13     56789 
 3      56789  56789  | 56789  56789  14     | 56789  124    124   

I'm still looking for an "elegant" way to solve this, but we can prove the uniqueness by the following moves:

r7c5=3 => contradiction
So r7c5=4.
Now r3c1=1 => contradiction
So r4c1=1.
And the puzzle is "solved" under SSTS.

Also, I found that r2c5=2 is a backdoor cell under SSTS.

RW wrote:Using that grid, it was no problem to construct an 8-clue odd-even:
Code: Select all
 *-----------*
 |,.,|.,5|..,|
 |,.,|,..|.,,|
 |.,.|,,.|1,.|
 |---+---+---|
 |.,.|.,,|.,,|
 |,.,|..,|,.,|
 |,.,|,,.|.,.|
 |---+---+---|
 |,,2|4.7|,,.|
 |.,.|,.,|,.3|
 |65,|,,.|,..|
 *-----------*
, = odd
. = even

It is equivalent to:
Code: Select all
 1379   2468   1379   | 268    1379   5      | 2468   2468   79     
 13579  2468   13579  | 1379   2468   2468   | 2468   3579   579   
 248    379    468    | 379    379    2468   | 1      3579   2468   
----------------------+----------------------+----------------------
 248    1379   468    | 268    13579  139    | 2468   13579  1579   
 13579  2468   13579  | 268    2468   139    | 3579   2468   1579   
 13579  2468   13579  | 13579  13579  2468   | 2468   13579  2468   
----------------------+----------------------+----------------------
 139    139    2      | 4      68     7      | 59     159    68     
 48     179    48     | 159    268    19     | 579    2468   3     
 6      5      1379   | 139    139    28     | 79     248    248   

After some "basic" moves (including a swordfish):
Code: Select all
 1379  2468  1379  | 268   1379  5     | 2468  2468  79   
 1579  2468  1579  | 179   248   2468  | 2468  3     79   
 248   379   468   | 379   379   2468  | 1     5     2468 
-------------------+-------------------+-------------------
 248  A19    468   | 268   579   3     | 2468  79   -15   
B579   2468 B579   | 268   248   19    | 3     2468 B15   
 1359  2468  1359  | 179   1579  2468  | 2468  79    2468 
-------------------+-------------------+-------------------
 39    39    2     | 4     68    7     | 5     1     68   
 48    17    48    | 5     26    19    | 79    26    3     
 6     5     17    | 139   139   28    | 79    248   248   

ALS-xz rule:
ALS A: r4c2={19}
ALS B: r5c139={1579}
restricted common: x=9
common: z=1

Therefore r4c9<>1, and singles will then solve all the odd cells:
Code: Select all
 7     2468  3     | 268   1     5     | 2468  2468  9     
 1     2468  5     | 9     248   2468  | 2468  3     7     
 248   9     468   | 7     3     2468  | 1     5     2468 
-------------------+-------------------+-------------------
 248   1     468   | 268   7     3     | 2468  9     5     
 5     2468  7     | 268   248   9     | 3     2468  1     
 3     2468  9     | 1     5     2468  | 2468  7     2468 
-------------------+-------------------+-------------------
 9     3     2     | 4     68    7     | 5     1     68   
 48    7     48    | 5     26    1     | 9     26    3     
 6     5     1     | 3     9     28    | 7     248   248   

And now we're back to the same "1234" puzzle above.:idea:
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Postby RW » Mon May 28, 2007 6:54 pm

JPF wrote:137160 solutions (1143 x 120)

I had a feeling that this would be a lot harder... But I'm surprised that the max amount of solutions you found is less than 4M!

udosuk wrote:Thanks RW for the nice puzzles!:)

Thank JPF, he made them possible! It could get even more interesting like this:
Code: Select all
 *-----------*
 |,.,|.,5|..,|
 |,.,|,..|.,,|
 |.,.|,,.|1,.|
 |---+---+---|
 |.,.|.,,|.,,|
 |,.,|..,|,.,|
 |,.,|,,.|.,.|
 |---+---+---|
 |,,.|..7|,,.|
 |.,.|,.,|,.3|
 |.5,|,,.|,..|
 *-----------*
, = odd
. = even

None of the even are given, but we know that there is only one possible way to fill in four different symbols so that each of them appear only once in each row, column and box. The solver may start by filling in digits 2, 4, 6 and 8 in the four '.'-cells of any unit and continue solving from there. A bit of luck is involved, as you will get very different puzzles depending on what unit you start from!:D

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Postby udosuk » Tue May 29, 2007 8:37 am

RW, do you have an "elegant walkthrough" to resolve this position::?:
Code: Select all
 7     2468  3     | 268   1     5     | 2468  2468  9     
 1     2468  5     | 9     248   2468  | 2468  3     7     
 248   9     468   | 7     3     2468  | 1     5     2468 
-------------------+-------------------+-------------------
 248   1     468   | 268   7     3     | 2468  9     5     
 5     2468  7     | 268   248   9     | 3     2468  1     
 3     2468  9     | 1     5     2468  | 2468  7     2468 
-------------------+-------------------+-------------------
 9     3     2     | 4     68    7     | 5     1     68   
 48    7     48    | 5     26    1     | 9     26    3     
 6     5     1     | 3     9     28    | 7     248   248   

Or should I post it in the "Help with particular puzzles" section to see if others can tackle it?
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Postby RW » Tue May 29, 2007 9:27 am

Go ahead and post it in the help-section, there's a few template eliminations there that some fishermen might be interested in (r6c6<>2, r5c8<>4, r1c4&r6c6<>6). After this there's a short loop:

[r2c5]-8-[r7c5]-6-[r8c5](-2-[r5c5])-2-[r9c6]-8-[r6c6]-4-[r5c5]-8-[r2c5]

after which a bunch of more template eliminations on digit 8 solve the puzzle. I'd also be very interested in seing a more elegant solution.

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Postby Eioru » Sun Jun 03, 2007 4:48 pm

How about these pattern

Code: Select all
**.......
**.......
.........
**.**....
**.**....
.........
**.**.**.
**.**.**.
.........


Code: Select all
.........
.****....
.*.**....
.****....
.***.***.
....****.
....**.*.
....****.
.........
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Postby JPF » Mon Jun 04, 2007 8:36 am

Eioru wrote:How about these pattern

Code: Select all
**.......
**.......
.........
**.**....
**.**....
.........
**.**.**.
**.**.**.
.........

I don't think this pattern has any valid puzzle.
Here's an alternate proposal with 7 empty units (6RC+1B) :
Code: Select all
 1 2 . | 3 . . | . . .
 4 5 . | 6 . . | . . .
 . . . | . . . | . . .
-------+-------+-------
 5 7 . | 2 4 . | 8 . .
 9 1 . | 7 6 . | 3 . .
 . . . | . . . | . . .
-------+-------+-------
 6 9 . | 5 8 . | 1 2 .
 2 8 . | 1 9 . | 7 4 .
 . . . | . . . | . . .

See here for more about empty units.

Eioru wrote:
Code: Select all
.........
.****....
.*.**....
.****....
.***.***.
....****.
....**.*.
....****.
.........

This one is easy.
Here's an example
Code: Select all
 . . . | . . . | . . .
 . 1 2 | 3 4 . | . . .
 . 4 . | . 5 . | . . .
-------+-------+-------
 . 6 . | 1 3 . | . . .
 . 3 4 | 7 . 5 | 2 8 .
 . . . | . 2 9 | . 5 .
-------+-------+-------
 . . . | . 7 . | . 4 .
 . . . | . 8 2 | 5 9 .
 . . . | . . . | . . .

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Postby m_b_metcalf » Mon Jun 04, 2007 4:06 pm

Eioru wrote:How about these pattern

Code: Select all
.........
.****....
.*.**....
.****....
.***.***.
....****.
....**.*.
....****.
.........

In addition to JPF's, here are two more, each with interesting features, including a naked quad.
Code: Select all
 . . . . . . . . .
 . 8 2 3 4 . . . .
 . 9 . 6 1 . . . .
 . 6 3 2 9 . . . .
 . 2 1 5 . 6 4 9 .
 . . . . 8 3 2 6 .
 . . . . 6 7 . 5 .
 . . . . 2 1 8 4 .
 . . . . . . . . .   SE 6.6
 
Code: Select all
 . . . . . . . . .
 . 8 2 3 4 . . . .
 . 9 . 6 1 . . . .
 . 6 5 2 9 . . . .
 . 2 1 7 . 3 4 9 .
 . . . . 8 5 2 6 .
 . . . . 5 7 . 1 .
 . . . . 2 1 8 4 .
 . . . . . . . . .   SE 7.2

Regards,

Mike Metcalf
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Mission impossible?

Postby claudiarabia » Tue Jun 05, 2007 8:53 am

Code: Select all
. . . x . x . . .
. x . . x . . x .
. . . x . x . . .
x . x . x . x . x
. x . x . x . x .
x . x . x . x . x
. . . x . x . . .
. x . . x . . x .
. . . x . x . . .


The corner boxes left with one number in the middle.

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Postby JPF » Tue Jun 05, 2007 10:24 am

Here's an easy example :

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

SE=7.2

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