The New Sudoku Players' Forum

by 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

by **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 . .

JPF

by **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=

by 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?

RW

by 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.)

RW

by **JPF** » Sat May 26, 2007 8:41 pm

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

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)

JPF

by **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:

by 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!

RW

by **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?

by 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.

RW

by **Eioru** » Sun Jun 03, 2007 4:48 pm

How about these pattern

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

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

by **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 . . . . | . . . | . . .

JPF

by **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

by **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.

Claudia

by **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

JPF

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