Everything about Sudoku that doesn't fit in one of the other sections
Ocean wrote:Locked Candidates plus one Bidirectional Cycle (5-ring) is enough to solve this puzzle:

[r1c1]=4=[r9c1]=1=[r9c9]-1-[r7c8]=8=[r1c8]-4-[r1c1] => r1c7<>4, r7c7<>1.

Thanks Ocean, the first diagonal puzzle with an empty box on this thread!

Is there anywhere a thread that explains the Bidirectional Cycle? I'd like to link it to the main post like I've done with the other techniques.

RW
RW
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RW wrote:Is there anywhere a thread that explains the Bidirectional Cycle? I'd like to link it to the main post like I've done with the other techniques.

RW

The Bidirectional Cycle, also called Continuous Nice Loop, is described by Jeff here. Eliminations can be done according to Theorem 1 (eliminations from cells that are part of the cycle) and Theorem 2 (eliminations from cells not part of the cycle). The current example is a Mixed Inference Loop.
Ocean

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ocean wrote:The Bidirectional Cycle, also called Continuous Nice Loop, is described by Jeff here.

Thanks Ocean, I moved the puzzle to nice loops. I must confess I never really read those threads, so I don't know exactly what is covered there. Most techniques are quite easy to use, but it's a lot harder to keep up with all the different names for similar techniques...

RW
RW
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What is the maximum number of eliminations with one technique? Here is an xy-ring (4-ring) which eliminates twelve candidates. Since it also seems to qualify as 'Effortless Extreme', it's posted in this thread.
Code: Select all
`  *-----------* |..1|..2|...| |.3.|.4.|.5.| |2..|6..|7..| |---+---+---| |..7|...|..8| |.1.|...|.3.| |6..|...|5..| |---+---+---| |..6|..8|..2| |.8.|.9.|.1.| |...|5..|9..| *-----------*# *---------------------------------------------------------------------* | 45-79  6      1      |#79     #57     2      | 8      49     3      | | 789    3      89     | 1-78-9  4      1-79   | 2      5      6      | | 2      459    4589   | 6       3-58   359    | 7      49     1      | |----------------------+-----------------------+----------------------| | -3459  2      7      |#39     #35     -3459  | 1      6      8      | | 589    1      589    | 278-9   26     5679   | 4      3      79     | | 6      49     3489   | 1-378-9 -378   1-3479 | 5      2      79     | |----------------------+-----------------------+----------------------| | 59     59     6      | 4       1      8      | 3      7      2      | | 34     8      234    | 237     9      37     | 6      1      5      | | 1      7      23     | 5       26     36     | 9      8      4      | *---------------------------------------------------------------------* [r1c4]-9-[r4c4]-3-[r4c5]-5-[r1c5]-7-[r1c4] => 12 eliminations.`
(Actually, if the xy-ring is applied one step earlier - instead of a hidden double - it leads to 13 eliminations, and makes the double unneccessary).
Ocean

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Ocean suggested that this puzzle may be valid here.

Jellyfish, or two.

Code: Select all
`. . 2 | 5 . 7 | 6 . . . 4 . | . . . | . 5 . . . 5 | 9 . 4 | 2 . . --------------------- 8 . 9 | . . . | 1 . 2 . 5 . | . . . | . 7 . 3 . 4 | . . . | 5 . 6 --------------------- . . 7 | 4 . 9 | 8 . . . 6 . | . . . | . 2 . . . 1 | 2 . 6 | 7 . . `
wapati
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wapati wrote:Ocean suggested that this puzzle may be valid here.

Jellyfish, or two.

Singles + one Jellyfish is enough and it is not solved by SSTS + URs alone, so yes it qualifies. That's the first Jellyfish here.

Ocean wrote:What is the maximum number of eliminations with one technique? Here is an xy-ring (4-ring) which eliminates twelve candidates. Since it also seems to qualify as 'Effortless Extreme', it's posted in this thread.

Very nice puzzle Ocean! Sorry for the slow response, I saw it briefly when it came and then forgot it. I'll add it to the list.

RW
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Ocean wrote:What is the maximum number of eliminations with one technique?

My "personal record" is 18. It would be interesting to see other examples of more than, let's say, 10 eliminations.

Carcul
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Would 13 eliminations for one digit be acceptable? Look here and ignore any reference to any digit other than <2>.
daj95376
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Carcul wrote:
Ocean wrote:What is the maximum number of eliminations with one technique?

My "personal record" is 18. It would be interesting to see other examples of more than, let's say, 10 eliminations.

Search for such puzzles resulted in a (seemingly) interesting collection. Most of them do not belong to this thread though, so they should be posted elsewhere (after a bit more checking).

But here is an Effortless Extreme - a puzzle that can be solved with an xy-ring.
Code: Select all
` *-----------* |...|...|.12| |...|...|345| |...|..3|67.| |---+---+---| |...|.87|9..| |...|539|...| |..5|42.|...| |---+---+---| |.81|2..|...| |763|...|...| |49.|...|...| *-----------*## After basic techniques: *--------------------------------------------------------------------* | 36     3457   4679   | 679    4569   456    | 8      1      2      | | 168    17     6789   | 16789  169    2      | 3      4      5      | | 128    1245  A48     |A18     145    3      | 6      7      9      | |----------------------+----------------------+----------------------| | 1236   1234  A46     |A16     8      7      | 9      5      1346   | | 168    147    4678   | 5      3      9      | 1247   26     1467   | | 9      137    5      | 4      2     B16     |B17     368    13678  | |----------------------+----------------------+----------------------| | 5      8      1      | 2      4679  B46     |B47     369    3467   | | 7      6      3      | 19     1459   1458   | 1245   289    148    | | 4      9      2      | 3      1567   1568   | 157    68     1678   | *--------------------------------------------------------------------*`

Two xy-rings (4-rings) A and B are now visible, one gives rise to 7 eliminations, the other 8 eliminations, in total 7+8=15 possible eliminations. What is interesting is that 14 of these are 'useless' - or they do not lead to progress - even if all those 14 candidates are eliminated. While the last one of the 15 possibles leads to direct solution (rest is singles).
Ocean

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Joined: 29 August 2005

Here is a nice puzzle with vertical symmetry that solves with one application of xyz-transport (or, if you like, it can be thought of as an xyt-chain or an xy(z)-chain, in the sense of Denis Berthier)
Code: Select all
`. 7 .|. 2 .|. 6 .1 . .|7 . 3|. . 5. . 5|. . .|2 . .-----+-----+-----8 . .|. 3 .|. . 1. 4 .|. . .|. 2 .. . 9|. . .|8 . .-----+-----+-----7 . 4|. . .|5 . 22 . .|4 . 1|. . 7. . .|. 8 .|. . .`

Code: Select all
`.------------------.------------------.------------------.| 4     7     3    | 59    2     59   | 1     6     8    || 1     2     8    | 7     6     3    | 49    49A'  5    || 69    69    5    | 1     4     8    | 2     7     3    |:------------------+------------------+------------------:| 8     56    2    | 569   3     4    | 7     59-   1    || 356   4     7    | 8     1     59   | 369B  2     69C  || 356   1     9    | 56    7     2    | 8     345   46   |:------------------+------------------+------------------:| 7     8     4    | 3     9     6    | 5     1     2    || 2     39    6    | 4     5     1    | 39A   8     7    || 59    359   1    | 2     8     7    | 3469  349   469  |'------------------'------------------'------------------'`

The xyz-wing in r5c79, r8c7 eliminates 9 from cells seeing all of the xyz-wing, of which there are none. Label the cells in the xyz-wing by A,B and C as in the grid. We may transport A to r2c8 (A'), via
[r2c8 = 9 = r2c7 - 9 - r8c7] and hence we may eliminate 9 from r4c8 since it sees B,C and A'.

As an xyt-chain (or xy(z)-chain), it is
{n9 n4}r2c8 - {n4 n9}r2c7 - {n9 n3}r8c7 - {n3 n6}r5c7 - {n6 n9}r5c9, =>r4c8<>9.
re'born

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re'born, What about the UR in [r29c78]? Singles complete the puzzle.

Code: Select all
`[r9c7]=3|6 =>                                     [r9c7]<>49[r9c8]=3   => [r8c7]=9 => [r2c7]=4 => [r9c7]=6 => [r9c7]<>49*--------------------------------------------------------------------*|  4      7      3     |  59     2      59    |  1      6      8     ||  1      2      8     |  7      6      3     | *49    *49     5     ||  69     69     5     |  1      4      8     |  2      7      3     ||----------------------+----------------------+----------------------||  8      56     2     |  569    3      4     |  7      59     1     ||  356    4      7     |  8      1      59    |  369    2      69    ||  356    1      9     |  56     7      2     |  8      345    46    ||----------------------+----------------------+----------------------||  7      8      4     |  3      9      6     |  5      1      2     ||  2      39     6     |  4      5      1     |  39     8      7     ||  59     359    1     |  2      8      7     | *49+36 *49+3   469   |*--------------------------------------------------------------------*`
daj95376
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daj95376 wrote:re'born, What about the UR in [r29c78]? Singles complete the puzzle.

True dat My feeling is that UR patterns are some of the easiest to spot. In this case, though, when I just looked at the UR pattern, I didn't see immediately how to make an elimination. Whereas, it took me longer to spot the xyz-wing, but then it was quite clear how to make it into something that made eliminations. Difficulty comparison: Toss up!
re'born

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Joined: 31 May 2007

Nice puzzle, but unfortunately it is solved with an UR+2kx:
Code: Select all
` *-----------------------------------------------------------* | 4     7     3     | 59    2     59    | 1     6     8     | | 1     2     8     | 7     6     3     |*49   *49    5     | | 69    69    5     | 1     4     8     | 2     7     3     | |-------------------+-------------------+-------------------| | 8     56    2     | 569   3     4     | 7     59    1     | | 356   4     7     | 8     1     59    | 369   2     69    | | 356   1     9     | 56    7     2     | 8     345   46    | |-------------------+-------------------+-------------------| | 7     8     4     | 3     9     6     | 5     1     2     | | 2     39    6     | 4     5     1     |#39    8     7     | | 59    359   1     | 2     8     7     |369-4 *349   469   | *-----------------------------------------------------------*`

All you need is to eliminate 4 from r9c7. Of course, you can also eliminate candidate 9 from the same cell with the same pattern, or with the UR+2kx in r89c27+r9c1. AS for difficulty comparison, I still think URs like this are easier than other patterns, because the pm grid always tells you exactly where to look for them. When you see a naked pair in the same unit, you know there's a potential UR, XYZ-wings don't reveal their existance quite that easily.

RW
RW
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RW wrote:Nice puzzle, but unfortunately it is solved with an UR+2kx:

RW wrote:AS for difficulty comparison, I still think URs like this are easier than other patterns, because the pm grid always tells you exactly where to look for them. When you see a naked pair in the same unit, you know there's a potential UR, XYZ-wings don't reveal their existance quite that easily.

As I said above, in general I agree with you. I happen to think though that some of the UR deductions are a lot harder to spot than others (unless of course you're looking for them). And when you are looking for xyz-wings specifically, they are a breeze to find.

Perhaps this transported wxyz-wing will fare better. An assist goes to JPF for more or less finding this puzzle
Code: Select all
`. . .|9 . .|. . 8. 1 .|. 3 4|2 6 .. . .|8 . 2|3 . .-----+-----+-----9 . 7|. 2 .|. 3 6. . .|6 . 3|. . .6 2 .|. 9 .|5 . 1-----+-----+-----. . 5|2 . 7|. . .. 7 1|3 4 .|. 8 .3 . .|. . 6|. . .`

After the easy stuff one gets to
Code: Select all
`.---------------------.---------------------.---------------------.| 457    3      2     | 9      6      15    | 147    1457   8     || 578    1      89    | 57A    3      4     | 2      6      79    || 457    49     6     | 8      157    2     | 3      14579  479   |:---------------------+---------------------+---------------------:| 9      458-   7     | 145C   2      15D   | 48     3      6     || 1      458    48    | 6      57-    3     | 4789   2479   2479  || 6      2      3     | 47B    9      8     | 5      47     1     |:---------------------+---------------------+---------------------:| 48     6      5     | 2      18     7     | 149    149    3     || 2      7      1     | 3      4      9     | 6      8      5     || 3      489    489   | 15     158    6     | 147    1247   247   |'---------------------'---------------------'---------------------'`

where a wxyz-wing implies that A, C or D is a 5. We may transport A to D via
[r2c6 = 5 = r1c6 - 5 - r2c4]
and hence either C or D is a 5. Thus, r5c5<>5 (also r4c2<>5), solving the puzzle.

An alternative way to make this deduction is the (2-color) nice loop:
r5c5 - 5 - {r9c5 = 5 = r9c4 = 1 = r4c4 - 1 - r4c6} - 5 - r5c5, => r5c5 <> 5.
re'born

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Joined: 31 May 2007

re'born wrote:As an xyt-chain (or xy(z)-chain), it is
{n9 n4}r2c8 - {n4 n9}r2c7 - {n9 n3}r8c7 - {n3 n6}r5c7 - {n6 n9}r5c9, =>r4c8<>9.

Translating the chain to NL notation ...

r2c8 =9= r2c7 -9- r8c7 -3- {ALS:r5c7 =3|9= r5c79} => r4c8<>9
ronk
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