The Bicycle Collection.

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

Postby ravel » Mon Jun 04, 2007 4:34 pm

Code: Select all
 *---------------------------------------*
 | 3   8  4  | 19    17  79  | 5  2  6   |
 | 6   7  1  | 2     5   3   | 4  8  9   |
 | 2   5  9  | 4     8   6   | 3  7  1   |
 |-----------+---------------+-----------|
 |#19  6  8  |#139   4   79  | 2  5  37  |
 |#19  2  7  |#1359  13  58  | 6  4  38  |
 | 4   3  5  | 6     27  28  | 1  9  78  |
 |-----------+---------------+-----------|
 | 5   4  3  | 7     9   1   | 8  6  2   |
 | 7   1  2  | 8     6   4   | 9  3  5   |
 | 8   9  6  | 35    23  25  | 7  1  4   |
 *---------------------------------------*
Alternative to the cycle: UR 19 in r45c14 with strong link for 9 in row 5 => r4c4<>1, strong link for 1 in row 4 => r5c4<>9.
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Postby re'born » Mon Jun 04, 2007 6:02 pm

ravel wrote:
Code: Select all
 *---------------------------------------*
 | 3   8  4  | 19    17  79  | 5  2  6   |
 | 6   7  1  | 2     5   3   | 4  8  9   |
 | 2   5  9  | 4     8   6   | 3  7  1   |
 |-----------+---------------+-----------|
 |#19  6  8  |#139   4   79  | 2  5  37  |
 |#19  2  7  |#1359  13  58  | 6  4  38  |
 | 4   3  5  | 6     27  28  | 1  9  78  |
 |-----------+---------------+-----------|
 | 5   4  3  | 7     9   1   | 8  6  2   |
 | 7   1  2  | 8     6   4   | 9  3  5   |
 | 8   9  6  | 35    23  25  | 7  1  4   |
 *---------------------------------------*
Alternative to the cycle: UR 19 in r45c14 with strong link for 9 in row 5 => r4c4<>1, strong link for 1 in row 4 => r5c4<>9.


There is an interesting alternative using this uniqueness. To avoid a BUG grid:
Code: Select all
 
 *-----------------------------------------------------------*
 | 3     8     4     | 19    17    79    | 5     2     6     |
 | 6     7     1     | 2     5     3     | 4     8     9     |
 | 2     5     9     | 4     8     6     | 3     7     1     |
 |-------------------+-------------------+-------------------|
 | 19    6     8     | 13+9  4     79    | 2     5     37    |
 | 19    2     7     | 59+13 13    58    | 6     4     38    |
 | 4     3     5     | 6     27    28    | 1     9     78    |
 |-------------------+-------------------+-------------------|
 | 5     4     3     | 7     9     1     | 8     6     2     |
 | 7     1     2     | 8     6     4     | 9     3     5     |
 | 8     9     6     | 35    23    25    | 7     1     4     |
 *-----------------------------------------------------------*


we must have r4c4=9, r5c4=1 or r5c4=3. But, trivially, r5c4=9 would preclude all three of these things from happening, hence r5c4<>9.
Last edited by re'born on Mon Jun 11, 2007 7:44 am, edited 1 time in total.
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Postby JPF » Tue Jun 05, 2007 1:21 pm

an other one :

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

000000000012340000050060700080007000039000570000400060004090020000078130000000000
JPF
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Postby re'born » Tue Jun 05, 2007 3:19 pm

JPF wrote:an other one :

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

000000000012340000050060700080007000039000570000400060004090020000078130000000000
JPF



Wonderful puzzle JPF. SudoCue suggests 7 coloring deductions to solve this, but it can be done with just two bidirectional x-cycles (and they're even on the same digit). Very nicely done.

Alternatively,

Code: Select all
 *-----------------------------------------------------------*
 | 3689  4     368   | 7     158   159   | 2     159   1359  |
 | 7     1     2     | 3     4     59    | 689   589   569   |
 | 389   5     38    | 189   6     2     | 7     4     139   |
 |-------------------+-------------------+-------------------|
 | 156   8     16    | 1259  3     7     | 4     19    129   |
 | 4     3     9     | 128A  128A  6     | 5     7     128   |
 | 15B   2     7     | 4     158B  159-  | 3     6     189   |
 |-------------------+-------------------+-------------------|
 | 18    67    4     | 15    9     3     | 68    2     567   |
 | 2     9     5     | 6     7     8     | 1     3     4     |
 | 138   67    138   | 125   125   4     | 689   589   5679  |
 *-----------------------------------------------------------*


A = {1,2,8} on r5c45, B = {1,5,8} on r6c15, x = 8, z = 1.

The ALS xz-rule implies r6c6<>1, solving the puzzle.
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Postby Ocean » Wed Jun 06, 2007 4:44 pm

Thanks Claudia and JPF for the nice puzzles!

Claudia's puzzle has two 4-node bidirectional cycles.
[r1c4]=3=[r7c4]=9=[r3c4]-9-[r2c6]-3-[r1c4]
[r3c5]=4=[r9c5]=6=[r9c6]-6-[r3c6]-4-[r3c5]

JPF's first puzzle has a 9-node xy-ring, and the second puzzle has 2 x-rings (8 and 6 nodes, respectively).

The three puzzles are included among the examples in the first post in this thread. In my opinion good and instructive examples!


Also, thanks to re'born and ravel for showing alternative solution paths.
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Postby re'born » Wed Jun 06, 2007 9:23 pm

Here is a doozy.

Code: Select all
 *-----------*
 |27.|6..|.8.|
 |86.|.2.|.1.|
 |..9|...|4..|
 |---+---+---|
 |...|7..|..3|
 |...|8.2|...|
 |7..|..6|...|
 |---+---+---|
 |..7|...|9..|
 |.5.|.6.|.41|
 |.1.|..9|.68|
 *-----------*


I used a finned x-wing, some (multi-)coloring, a UR single, an x-wing, an xy-wing and...a 6-ring bidirectional y-cycle. Now it is somebody else's turn to come in and say, "Well, yes, but if you look over here you'll see that the ALS xz-rule...":)
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Postby JPF » Wed Jun 06, 2007 10:12 pm

Here, re'born wrote:
JPF wrote:
Code: Select all
 
 1 . . | . . . | . . 2
 . 3 . | 2 . 4 | . 5 .
 . . . | . . . | 6 . .
-------+-------+-------
 . . 6 | . . . | 1 . .
 . . . | 7 . 3 | . . .
 . . 8 | . . . | 9 . .
-------+-------+-------
 . . 1 | . . . | . . .
 . 7 . | 5 . 2 | . 3 .
 8 . . | . . . | . . 9   ER=7.6
JPF,
You should submit this puzzle to the Bicycle Collection. It is a real beauty. Just write in the pencilmarks and there is a 4-ring bidirectional y-cycle staring you right in the face. After that an xy-wing and an empty rectangle will reduce it to a BUG+1. Fantastic.

Done. Thanks re'born.

JPF
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Location: Paris, France

Postby JPF » Wed Jun 06, 2007 10:40 pm

Still with the same pattern :

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


JPF
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Postby Ocean » Mon Jun 11, 2007 11:40 am

Thanks re'born and JPF for new good examples. Now we also have an 11-node xy-ring, plus two puzzles which were categorized as xy-ring + one 'extreme' technique (the 'extremes' being finned x-wing and empty rectangle).
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Postby JPF » Fri Jun 15, 2007 9:22 pm

Could it be qualified ?

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


JPF
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Postby re'born » Sat Jun 16, 2007 11:05 am

JPF wrote:Could it be qualified ?


Code: Select all
 *--------------------------------------------------------------------*
 | 2589   56     28     | 13     7      13     | 25     24569  24589  |
 | 567    1      4      | 89     2      89     | 567    3      57     |
 | 23789  289    238    | 4      5      6      | 127    279    12789  |
 |----------------------+----------------------+----------------------|
 | 1249   249    7      | 1259   6      1249   | 8      25     3      |
 | 24     3      5      | 27     8      247    | 9      1      6      |
 | 1289   289    6      | 1259   3      129    | 4      257    257    |
 |----------------------+----------------------+----------------------|
 | 2348   248    1238   | 6      9      5      | 1237   247    1247   |
 | 456    7      9      | 23     1      23     | 56     8      45     |
 | 235    56     123    | 78     4      78     | 1235   2569   1259   |
 *--------------------------------------------------------------------*


where a Y-wing style (though I would like it we changed the name of this technique to something that does not refer to xy-wings and which correlates to what it really looks like, namely remote naked pairs...perhaps semi-remote naked pairs) between r1c2,r8c7<56> implies r1c7<>5 (there is also an xyz-wing that will make this same exclusion). After cleaning up the grid a bit, we get

Code: Select all
 *-----------------------------------------------------------*
 | 59    56A   8     | 13    7     13    | 2     456a9 459   |
 | 56a7B 1     4     | 89    2     89    | 567   3     57    |
 | 237b9 29    23    | 4     5     6     | 1     7B9b  8     |
 |-------------------+-------------------+-------------------|
 | 1249  249   7     | 1259  6     1249  | 8     25    3     |
 | 24    3     5     | 27    8     247   | 9     1     6     |
 | 1289  289   6     | 1259  3     129   | 4     257   257   |
 |-------------------+-------------------+-------------------|
 | 2348  248   123   | 6     9     5     | 37    247   1247  |
 | 456   7     9     | 23    1     23    | 56    8     45    |
 | 235   56    123   | 78    4     78    | 35   256A9- 1259  |
 *-----------------------------------------------------------*


Now a little 3D-coloring implies r9c8<>9, which solves the puzzle.
[r9c8]-9-[r3c8]-7-[r3c1]=7=[r2c1]=6=[r1c2]-6-[r1c8]=6=[r9c8]

I can find a few bidirectional cycles, but none yet that really crack the puzzle open.
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A sudoku with a bi-directional X-cycle

Postby claudiarabia » Sat Jun 16, 2007 12:15 pm

Code: Select all
. 6 . . 4 . 2 . .
. . . . . 7 . . .
9 . 8 5 . . . 1 .
. . 7 1 . . . . .
. . 5 . . . 8 . .
. . . . . 6 3 . .
. 8 . . . 5 9 . 3
. . . 9 . . . . .
. . 1 . 3 . . 7 .                                                   SE 6-6
A multisolutional sudoku came before my eyes yesterday (not here:-)). I changed some clues and out came this: Beside the bi-directional X-cycle, there are many other techniques like X-wing, 2 xy-wing and three turbot-fishes. The sudoku-Assistenten solves it with one X-wing and a sky-scraper only. I often discovered more turbots in one sudoku than it was announced by the program previously. When you find the right Turbot-fish, you may circumwent other steps.

Happy solving
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Joined: 14 May 2006

Re: A sudoku with a bi-directional X-cycle

Postby re'born » Sun Jun 17, 2007 12:08 pm

claudiarabia wrote:
Code: Select all
. 6 . . 4 . 2 . .
. . . . . 7 . . .
9 . 8 5 . . . 1 .
. . 7 1 . . . . .
. . 5 . . . 8 . .
. . . . . 6 3 . .
. 8 . . . 5 9 . 3
. . . 9 . . . . .
. . 1 . 3 . . 7 .                                                   SE 6-6
A multisolutional sudoku came before my eyes yesterday (not here:-)). I changed some clues and out came this: Beside the bi-directional X-cycle, there are many other techniques like X-wing, 2 xy-wing and three turbot-fishes. The sudoku-Assistenten solves it with one X-wing and a sky-scraper only. I often discovered more turbots in one sudoku than it was announced by the program previously. When you find the right Turbot-fish, you may circumwent other steps.

Happy solving


Claudia, well done turning someone else's mistake into something that kept me entertained for far too long.:D I love the puzzles where you end up with a nearly cleaned up board and then you can apply tons of different techniques to solve it. My best solution thus far uses the empty rectangle in box 9 to eliminate 4 from r9c4, solving the puzzle. Although the fish experts might find the finned jellyfish that eliminates 4 from r8c3 a rather satisfying solution.
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Posts: 551
Joined: 31 May 2007

I did it again ... with the wikipedia-sudoku above right

Postby claudiarabia » Tue Jun 26, 2007 11:57 am

re'born wrote:Claudia, well done turning someone else's mistake into something that kept me entertained for far too long.:D I love the puzzles where you end up with a nearly cleaned up board and then you can apply tons of different techniques to solve it. ...


The sudoku on the wikipedia-page on the top right is very easy and has sooo many initial clues. After removing some clues I got a fantastic sudoku with 22 clues only. It is to solve with three xy-wings and a bi-directional Y-cycle.


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


Happy solving

Claudia
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Posts: 288
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Another solution wields the same result.

Postby claudiarabia » Sun Jul 01, 2007 3:52 pm

Ocean"][quote="claudiarabia wrote:
Code: Select all
4 . . . . . . . 2
. 8 . . 9 . . 7 .
. . . 3 . 6 . . .
. . 1 . . . 6 . .
. 6 . . 4 . . 9 .
. . 5 . . . 3 . .
. . . 5 . 2 . . .
. 7 . . 6 . . 8 .
9 . . . . . . . 4

This sudoku has a bi-directional X-Cycle next to two bi-directional Y-cycles next to 2 X-Wings and many other fancy techniques.


The Sudoku-Explainer found out this Bi-directional Y-cycle with 8 cells to eliminate the 7 in r1c3 and r7c9.
Code: Select all
 *--------------------------------------------------------------------*
 | 4      135    69-7   | 178    157    1578   |#89     36     2      |
 | 156    8      23     | 124    9      145    | 15     7      36     |
 | 17     125    #79    | 3      12578  6      | 4      15    #89     |
 |----------------------+----------------------+----------------------|
 | 2378   49     1      | 2789   2357   35789  | 6      24     578    |
 | 237    6      #78    | 127    4      1357   | 1278   9      157    |
 | 278    49     5      | 6      127    1789   | 3      124    178    |
 |----------------------+----------------------+----------------------|
 |#68     13     4      | 5     #78     2      |#79     13     1369-7 |
 | 1235   7      23     | 49     6      49     | 25     8      135    |
 | 9      1235  #68     | 178    1378   1378   | 1257   12356  4      |
 *--------------------------------------------------------------------*



I found a "simple" bi-directional cycle (below) which leads to the same result and eliminates also all clues in r7c9 except of the cycle-members 6 and 9.
Code: Select all
 *--------------------------------------------------------------------*
 | 4      135    69-7   | 178    157    1578   | 89     36     2      |
 | 156    8      23     | 124    9      145    | 15     7      36     |
 | 17     125    #79    | 3      12578  6      | 4      15    8#9     |
 |----------------------+----------------------+----------------------|
 | 2378   49     1      | 2789   2357   35789  | 6      24     578    |
 | 237    6      #78    | 127    4      1357   | 1278   9      157    |
 | 278    49     5      | 6      127    1789   | 3      124    178    |
 |----------------------+----------------------+----------------------|
 | 68     13     4      | 5      78     2      |79     13     #69-137 |
 | 1235   7      23     | 49     6      49     | 25     8      135    |
 | 9      1235  #68     | 178    1378   1378   | 1257  1235#6  4      |
 *--------------------------------------------------------------------*

The cycle comprises 6 cells with 4 cycle-relevant numbers
Code: Select all
       -7 |            |
          |            |
       79 |            |       9
---------------------------------
          |            |
       78 |            |
          |            |
---------------------------------
          |            |      96-137
          |            |
       68 |            |   6

r3c3   r3c9  r7c9  r9c8  r9c3  r5c3
97     9     96    6     68    78


For the explainer will always take the most easy technique, he doesn't see this cycle. Also note that two of the xy-wings in the puzzle are not necessary because the 8 in r1c5 and r5c9 is eliminated by the big bi-directional X-cycle as well.

Claudia
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