13-clue Sudoku X

For fans of Killer Sudoku, Samurai Sudoku and other variants

Postby udosuk » Sat Jul 28, 2007 3:51 pm

Ruud wrote:Came across a 13-clue Sudoku-X which starts with 41 consecutive empty cells...

Wonderful!:D Half of the grid is totally empty! So what can you do next, 5 empty rows?

IMHO the reference about coordinate-system is totally BS (pardon for my bluntness). The original poster of that idea must be a non-mathematician, he/she can't even distinguish between discrete number system and real number system, let alone finity and infinity. Let's leave the coordinates to trigonometry and calculus and complex numbers and such...
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Postby Ruud » Sun Jul 29, 2007 9:09 pm

gsf wrote:each of these can be swapped independently (for 4 combinations including the identity)
Code: Select all
r(46)
c(46)


After implementing this change in my canonicalization routine, invalid puzzles appeared in the results.

You need to swap rows 4-6 and columns 4-6 simultaneously, otherwise you are exchanging 2 pairs of digits between the diagonals. The number of pattern permutations is now 96. My program is now rechecking the earlier results.

Ruud
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Postby gsf » Sun Jul 29, 2007 9:28 pm

Ruud wrote:
gsf wrote:each of these can be swapped independently (for 4 combinations including the identity)
Code: Select all
r(46)
c(46)


After implementing this change in my canonicalization routine, invalid puzzles appeared in the results.

You need to swap rows 4-6 and columns 4-6 simultaneously, otherwise you are exchanging 2 pairs of digits between the diagonals. The number of pattern permutations is now 96. My program is now rechecking the earlier results.

aha
invalid on the diagonals
good catch
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Postby Mauricio » Mon Jul 30, 2007 3:00 am

Ruud wrote:The number of pattern permutations is now 96.

According to my own research, the number of sudoku-X morphisms is twice the number of sudoku automorphisms of this "sudoku":
Code: Select all
+-------+-------+-------+
| 1 . . | . . . | . . 2 |
| . 1 . | . . . | . 2 . |
| . . 1 | . . . | 2 . . |
+-------+-------+-------+
| . . . | 1 . 2 | . . . |
| . . . | . . . | . . . |
| . . . | 2 . 1 | . . . |
+-------+-------+-------+
| . . 2 | . . . | 1 . . |
| . 2 . | . . . | . 1 . |
| 2 . . | . . . | . . 1 |
+-------+-------+-------+

There are 96 of them, so that makes it 192. It is twice because you can rotate 90 degress to swap diagonals, and those are too X-sudokus.
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Postby udosuk » Mon Jul 30, 2007 4:32 am

Ruud wrote:
Code: Select all
. . .|. . .|. . .
. . .|. . .|. . .
. . .|. . .|. . .
-----\-----/-----
. . .|. . .|. . .
. . .|. . 1|2 . .
3 . .|. 4 .|. 5 .
-----/-----\-----
. . 2|6 . 7|. . .
. . .|8 . .|. . 1
7 . .|. . .|3 4 .

Ruud, I tried to solve this puzzle, and arrived at this state:
Code: Select all
 *-----------------------------------------------------------------------------*
 | 14      56789   3789    | 1245    678     2359    | 56      379     48      |
 | 2589    47      3789    | 45      678     359     | 56      1       239     |
 | 2589    56789   1347    | 145     678     359     | 48      379     239     |
 |-------------------------\-------------------------/-------------------------|
 | 589     145789  14789   | 37      2       6       | 48      39      3489    |
 | 89      4789    4789    | 37      5       1       | 2       6       3489    |
 | 3       2       6       | 9       4       8       | 1       5       7       |
 |-------------------------/-------------------------\-------------------------|
 | 14      14      2       | 6       3       7       | 9       8       5       |
 | 6       3       5       | 8       9       4       | 7       2       1       |
 | 7       89      89      | 25      1       25      | 3       4       6       |
 *-----------------------------------------------------------------------------*

Then I couldn't find any elegant move to crack it.:( I could, using long contradiction chains, to prove r3c3=3 and r3c7=7 both lead to contradictions, and thus r3c3={14} and solve the puzzle. But it feels like trial and error to me.:(

Can you, or your program find any better way to solve it?:?: Thanks!
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Postby Ruud » Mon Jul 30, 2007 5:18 pm

udosuk:

The key moves are:

Dual ALS-xz r1c13|r2c23|r3c3 vs. r9c3 x=8/9 => r45c3<>89
XY-chain r2c2-r4c4-r5c4-r5c3 => r3c3|r5c2<>4
3D colors (4)r2c4=(4)r2c2=(4=1)r1c1=(1)r3c3=(1)r3c4) => r3c4<>4

After this last move, the puzzle falls after a few coloring moves on digit 3.

Note: The dual ALS-xz allows you to perform the ALS twice, with the x & z digits swapped.

Hard, but no guessing required. I found worse...

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Postby Para » Mon Jul 30, 2007 6:06 pm

udosuk wrote:Wonderful!:D Half of the grid is totally empty! So what can you do next, 5 empty rows?


There's always the puzzle on the right.
Don't know who made it. Came across it once.

Link to pic

greetings

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Postby udosuk » Wed Aug 01, 2007 8:21 am

Para wrote:Don't know who made it. Came across it once.

I happen to know who made it.:)

Link to pic

Apparently it's from a Japanese folk from the city of East Osaka. So there you go.:)

What I wanted to see is a Sudoku-X with 13 clues or fewer and 5 empty rows.:idea:

BTW:
Ruud wrote:After this last move, the puzzle falls after a few coloring moves on digit 3.

That's not true. After that you still need another colouring move on 7 to crack it. (Added later: it is actually a "pointing pair" move on the diagonal, which IMHO warrants at least a mentioning as normal non-X players wouldn't recognise it easily.)

I hated that "3D-colors". To me it smells and tastes like T&E. I think the Dual ALS-xz is very nice and the xy-chain is equivalent to a Y-Wing. I'll look for ways to work around that dreaded 3D-colors move.
Last edited by udosuk on Sat Aug 04, 2007 11:25 am, edited 1 time in total.
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Postby udosuk » Thu Aug 02, 2007 4:00 am

Alright here is a brief walkthrough to resolve the puzzle from the state I last posted:
Code: Select all
 *-----------------------------------------------------------------------------*
 | 14      56789   3789    | 1245    678     2359    | 56      379     48      |
 | 2589    47      3789    | 45      678     359     | 56      1       239     |
 | 2589    56789   1347    | 145     678     359     | 48      379     239     |
 |-------------------------\-------------------------/-------------------------|
 | 589     145789  14789   | 37      2       6       | 48      39      3489    |
 | 89      4789    4789    | 37      5       1       | 2       6       3489    |
 | 3       2       6       | 9       4       8       | 1       5       7       |
 |-------------------------/-------------------------\-------------------------|
 | 14      14      2       | 6       3       7       | 9       8       5       |
 | 6       3       5       | 8       9       4       | 7       2       1       |
 | 7       89      89      | 25      1       25      | 3       4       6       |
 *-----------------------------------------------------------------------------*

Dual ALS-xz: A=r1c1+r2c2+r123c3={134789}, B=r9c3={89}, x=8|9, z=9|8 => r45c3<>8|9
Y-Wing: r45c4={37}, r2c2,r5c3=4|7 => r3c3,r45c2<>4
d\: either r2c2=4 or r3c3=1 => r23c4 must have 5 => r19c4,r123c6<>5
r9c46=[25] => r1c6=2 (hidden single c6) => r1c14={14} => r1c9=8
Turbot fish: strong links of 3 in r1c38, r3c3+r4c4 => r4c8<>3 => r4c8=9
Box-line interaction: 9 on r1 locked in b1 => r23c123<>9
r5c1=9 (hidden single c1) => r5c2=8 (hidden single r5) => r9c2=9
r1c3=9 (hidden single r1,c3,b1) => r1c8=3 (hidden single r1) => r3c8=7
Pointing pair: 7 on d\ locked in r2c2+r4c4 => r4c2<>7
r4c12=[51] => r27c2=[74] => r4c4=3 => all naked singles

No chains, no colouring move. However, the 3rd line is a bit of elaborated logic, and one can argue it steps on the boundary of T&E.

I've actually proposed it as a new form of ALS in this thread.
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Postby kjellfp » Thu Aug 02, 2007 6:15 am

This web page claimes to have (currently) 5503 12-clue Sudoku-X puzzels, like
Code: Select all
\---+---+---/
|...|...|...|
|...|...|...|
|...|...|...|
+---\---/---+
|...|...|12.|
|...|3..|.4.|
|..5|.67|...|
+---/---\---+
|..8|...|..6|
|.1.|4..|...|
|...|...|..8|
/---+---+---\
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Postby Mauricio » Thu Aug 02, 2007 6:56 am

kjellfp wrote:This web page claimes to have (currently) 5503 12-clue Sudoku-X puzzels

Of course, it is Ruud'd website.
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Postby kjellfp » Thu Aug 02, 2007 7:33 am

Sorry, I got the impression that 12s where unknown, and apparently didn't read the thread good enough.
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Postby Ruud » Sat Aug 04, 2007 9:25 am

The collection has now grown to 7193, but no 11 yet. Growth is slowing down a bit.

I haven't found a Sudoku-X with 5 empty rows and minimum number of clues, but this one is pretty close:

Code: Select all
. . .|. . .|. . .
. . .|. . .|. . .
. . .|. . .|. . .
-----\-----/-----
. . .|. . .|. . .
. . .|. . .|. . 1
. . 2|3 4 5|. . .
-----/-----\-----
5 6 .|. . .|. . .
1 . .|. . .|. 7 .
. . .|8 . .|2 3 .


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Postby udosuk » Sat Aug 04, 2007 3:21 pm

The latest one is not nearly as evil as the first two, but still need a reasonably tricky step to resolve:
Code: Select all
 *--------------------------------------------------------------------*
 | 38     79     1      | 49     2      6      | 3479   489    5      |
 | 269    38     679    | 5      78     349    | 3479   1      23479  |
 | 29     5      4      | 1     @78     39     | 6     @89    -2379   |
 |----------------------\----------------------/----------------------|
 | 3689   348    69     | 2      1      7      | 3489   5      349    |
 | 37     34     5      | 6      9      8      | 347    2      1      |
 | 789    1      2      | 3      4      5      | 789    6     #79     |
 |----------------------/----------------------\----------------------|
 | 5      6      8      | 7      3      2      | 1     *49    #49     |
 | 1      2      3      | 49     6      49     | 5      7      8      |
 | 4      79     79     | 8      5      1      | 2      3      6      |
 *--------------------------------------------------------------------*

ALS-xy-wing:
ALS A (@): r3c58={789}
ALS B (#): r67c9={479}
ALS C (*): r7c8={49}
restricted common between B,C: x=4
restricted common between A,C: y=9
common between A,B: z=7

Therefore r3c9<>7, and singles solve the rest.

Unfortunately I can't find a simple ALS-xz to resolve this.:(

I'd like to call this move a "semi-Y-Wing" because if r3c8 was {79} instead of {89} it would be just a normal Y-Wing.

And now r3c58 combined is effectively performing the role as a single cell of {79} in r3c8.

People who are into chains will also find it as a very simple xy-chain (I'm not one of them).

Ruud, is there any 12-clue Sudoku-X which is solvable using singles only? There're plenty of singles-solvable 17-clue Sudoku puzzles.

PS:
My latest moderator priviledges allowed me to change Para's image to a a pic link (I hope he doesn't mind), thus restoring the normal width of this very page for my tiny 1024x768 laptop screen. Way cool!:D

It's funny when I edit others' posts the "Last edited by..." messages don't appear, but when I edit my own they do. I guess it's fair that I can't change the things I wrote in the past without others noticing.:D
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