The New Sudoku Players' Forum

[Ocean]

The Bicycle Collection.


This is meant to be a collection of puzzles that are naturally solved with one or more bidirectional cycles (also called continuous nice loops), but not needing other 'chains'. For interesting discussions about the subject, see here. The purpose of this thread is to provide a useful training set, or examples grouped by taxonomy.

The theme for this thread is puzzles that are solved with (one or more) Continuous Simple Nice Loops as the highest ranked (most difficult) technique - not quite precise because ranking of techniques is debatable. Some properties of such chains are:
1. they describe a full/continuous cycle
2. they can freely start anywhere within the cycle (no specific starting point)
3. they work equally in both directions.
4. candidates may be eliminated from cells sharing a unit with two linked cycle nodes, or from nodes in the cycle itself (see the theoretical descriptions of nice loops elsewhere; links to be provided later).

Bidirectional cycles can be subdivided in I. y-cycles (=xy-rings; pure bivalue cycles), II. x-cycles (=coloring; only one digit), III. pure bilocation cycles, and IV. xy-cycles (mixed type).

A sample set of puzzles containing one y-cycle (of various lengths) is prepeared as a start. Will try to include the other cycle types as well, hoping to get help from forum members. Submitted puzzles should have some symmetry or other nice clues pattern, and preferably a low number of clues. Puzzles with more than one cycle (same type, or different types) are also welcomed.

The taxonomy can quickly become complicated. For instance, there may be puzzles with two alternative cycles, or two cycles that both are 'needed'. When judging whether the listed bi-cycle is needed or not, it may be useful to specify a simplest alternative method or bypassing technique(s)? Good examples are welcomed.

If we exclude the naked quadruples, four-rings can have six different configurations:

Code: Select all

A..|...|..D    A..|...|..D    A..|...|D..    A..|...|D..    A.C|...|..D    A.C|...|..D
...|...|...    ...|...|...    ...|...|...    ...|...|...    ...|...|...    ...|...|...
...|...|...    B..|...|..C    B..|...|..C    ..B|...|..C    B..|...|...    .B.|...|...
---+---+---    ---+---+---    ---+---+---    ---+---+---    ---+---+---    ---+---+---
...|...|...    ...|...|...    ...|...|...    ...|...|...    ...|...|...    ...|...|...
...|...|...    ...|...|...    ...|...|...    ...|...|...    ...|...|...    ...|...|...
...|...|...    ...|...|...    ...|...|...    ...|...|...    ...|...|...    ...|...|...
---+---+---    ---+---+---    ---+---+---    ---+---+---    ---+---+---    ---+---+---
...|...|...    ...|...|...    ...|...|...    ...|...|...    ...|...|...    ...|...|...
...|...|...    ...|...|...    ...|...|...    ...|...|...    ...|...|...    ...|...|...
B..|...|..C    ...|...|...    ...|...|...    ...|...|...    ...|...|...    ...|...|...

Type I         Type II        Type III       Type IV        Type V         Type VI


As the number of nodes increases, the possible configurations become more complex.


In addition to the listed examples, Mike Barker has provided (here) relevant puzzles from the zoo.


Part I: xy-rings (Bidirectional Y-Cycles)

#
# 4-ring
#

Code: Select all

#
# I.4.1 (Ocean)
#
*-----------*
|...|.1.|2..|
|..3|4.5|...|
|...|...|67.|
|---+---+---|
|...|..1|.89|
|...|...|...|
|75.|8..|...|
|---+---+---|
|.85|...|...|
|...|7.6|4..|
|..9|.2.|...|
*-----------*
#
# I.4.2 (Ocean)
#
*-----------*
|...|..1|.2.|
|...|.3.|..4|
|5..|...|6..|
|---+---+---|
|...|7..|86.|
|..3|...|5..|
|.91|..3|...|
|---+---+---|
|..9|...|..1|
|8..|.4.|...|
|.6.|5..|...|
*-----------*
#
# I.4.3 (Ocean)
# [Type I]
#
*-----------*
|...|...|..1|
|...|.2.|.34|
|2.5|.6.|...|
|---+---+---|
|...|..4|..2|
|..6|...|7..|
|3..|2..|...|
|---+---+---|
|...|.1.|6.8|
|93.|.5.|...|
|4..|...|...|
*-----------*
#
# I.4.4 (Ocean)
# [Type II]
#
*-----------*
|...|...|..1|
|..2|..3|...|
|..4|..2|..5|
|---+---+---|
|.6.|7..|..8|
|..5|...|4..|
|7..|..8|.1.|
|---+---+---|
|8..|1..|2..|
|...|9..|3..|
|9..|...|...|
*-----------*
#
# I.4.5 (Ocean)
# [Type I]
#
*-----------*
|...|...|.1.|
|...|.23|4..|
|4.5|...|..6|
|---+---+---|
|...|..7|2..|
|.1.|...|.8.|
|..6|8..|...|
|---+---+---|
|2..|...|3.7|
|..9|64.|...|
|.8.|...|...|
*-----------*
#
# I.4.6 (claudiarabia)
# [Type I]
#
 *-----------*
 |.4.|...|.1.|
 |...|5.7|...|
 |8..|.9.|..7|
 |---+---+---|
 |.2.|.1.|.9.|
 |..5|4.6|8..|
 |.3.|.8.|.7.|
 |---+---+---|
 |4..|.5.|..6|
 |...|2.9|...|
 |.7.|...|.3.|
 *-----------*
#
# I.4.7
# [Type II]
#
 *-----------*
 |..1|...|2..|
 |.2.|...|.3.|
 |4..|.5.|..6|
 |---+---+---|
 |...|1.7|...|
 |.8.|.6.|.1.|
 |...|2.9|...|
 |---+---+---|
 |5..|...|..8|
 |.3.|.4.|.6.|
 |..2|...|7..|
 *-----------*


#
# 5-ring
#

Code: Select all

#
# I.5.1
#
 *-----------*
 |...|...|..1|
 |...|2.3|...|
 |.45|...|6..|
 |---+---+---|
 |..7|..6|.3.|
 |.8.|...|.9.|
 |.1.|8..|7..|
 |---+---+---|
 |..3|...|95.|
 |...|4.1|...|
 |2..|...|...|
 *-----------*
#
# I.5.2
#
 *-----------*
 |...|...|..1|
 |.2.|...|3..|
 |.34|.5.|...|
 |---+---+---|
 |...|.67|..3|
 |1..|...|..8|
 |9..|54.|...|
 |---+---+---|
 |...|.3.|67.|
 |..5|...|.4.|
 |8..|...|...|
 *-----------*
#
# I.5.3
#
 *-----------*
 |...|...|.1.|
 |...|.12|3..|
 |4.5|..6|...|
 |---+---+---|
 |...|5..|..7|
 |2..|...|..4|
 |8..|..1|...|
 |---+---+---|
 |...|4..|8.6|
 |..8|97.|...|
 |.9.|...|...|
 *-----------*


#
# 8-ring
#

Code: Select all

#
# I.8.1
#
*-----------*
 |...|...|12.|
 |3..|...|4..|
 |15.|6..|...|
 |---+---+---|
 |...|7.2|3..|
 |...|...|...|
 |..2|4.8|...|
 |---+---+---|
 |...|..1|.69|
 |..8|...|..5|
 |.47|...|...|
 *-----------*

#
# 9-ring
#

Code: Select all

#
# I.9.1 (JPF)
#
 *-----------*
 |...|...|...|
 |..1|2.3|4..|
 |.5.|4.6|.7.|
 |---+---+---|
 |.68|...|25.|
 |...|...|...|
 |.35|...|19.|
 |---+---+---|
 |.4.|7.1|.6.|
 |..2|8.4|9..|
 |...|...|...|
 *-----------*

#
# 11-ring
#

Code: Select all

#
# I.11.1 (JPF)
#
 *-----------*
 |1..|...|..2|
 |.3.|4.5|.1.|
 |...|...|6..|
 |---+---+---|
 |..6|...|2..|
 |...|3.7|...|
 |..8|...|9..|
 |---+---+---|
 |..2|...|...|
 |.7.|1.9|.5.|
 |6..|...|..3|
 *-----------*

#
# xy-ring + one 'extreme' technique:
#

Code: Select all

#
# I.Extreme.1 (re'born)
#
 *-----------*
 |27.|6..|.8.|
 |86.|.2.|.1.|
 |..9|...|4..|
 |---+---+---|
 |...|7..|..3|
 |...|8.2|...|
 |7..|..6|...|
 |---+---+---|
 |..7|...|9..|
 |.5.|.6.|.41|
 |.1.|..9|.68|
 *-----------*
#
# I.Extreme.2 (JPF)
#
 *-----------*
 |1..|...|..2|
 |.3.|2.4|.5.|
 |...|...|6..|
 |---+---+---|
 |..6|...|1..|
 |...|7.3|...|
 |..8|...|9..|
 |---+---+---|
 |..1|...|...|
 |.7.|5.2|.3.|
 |8..|...|..9|
 *-----------*

Part II: x-cycles (Bidirectional X-Cycles)

Code: Select all

#
# II.multi.1 (JPF)
# Two Bidirectional X-Cycles.
# (First 8 nodes, second 6 nodes).
#
 *-----------*
 |...|...|...|
 |.12|34.|...|
 |.5.|.6.|7..|
 |---+---+---|
 |.8.|..7|...|
 |.39|...|57.|
 |...|4..|.6.|
 |---+---+---|
 |..4|.9.|.2.|
 |...|.78|13.|
 |...|...|...|
 *-----------*

Part III: Pure bilocation cycles (Bidirectional Cycles)
Room for examples...

Part IV: xy-cycles (Bidirectional Cycles)

Code: Select all

#
# IV.multi.1 (claudiarabia)
# Two Bidirectional Cycles
# (First four nodes Type VI; Second four nodes Type II).
#
 *-----------*
 |..9|..7|...|
 |.4.|5..|.8.|
 |..3|...|1.5|
 |---+---+---|
 |1..|.3.|.4.|
 |...|6.5|...|
 |.8.|.9.|..2|
 |---+---+---|
 |4.5|...|6..|
 |.6.|..2|.3.|
 |...|8..|9..|
 *-----------*

Part V: Puzzles with several Bicycles

Code: Select all

#
# V.1 (claudiarabia)
# Two X-wings;
# One Bidirectional X-cycle (8 nodes/4 elements);
# Two xy-rings (8 and 9 nodes);
#
 *-----------*
 |4..|...|..2|
 |.8.|.9.|.7.|
 |...|3.6|...|
 |---+---+---|
 |..1|...|6..|
 |.6.|.4.|.9.|
 |..5|...|3..|
 |---+---+---|
 |...|5.2|...|
 |.7.|.6.|.8.|
 |9..|...|..4|
 *-----------*

Part XXX: Puzzles not yet categorized

Code: Select all

#
# JPF (Posted by JPF, June 15, 2007. to be categorized later...)
#
 *-----------*
 |...|...|...|
 |.1.|.2.|.3.|
 |...|456|...|
 |---+---+---|
 |..7|...|8..|
 |.35|.8.|91.|
 |..6|...|4..|
 |---+---+---|
 |...|695|...|
 |.7.|.1.|.8.|
 |...|...|...|
 *-----------*


#
# claudiarabia  (Posted June 16, 2007)
#
 *-----------*
 |.6.|.4.|2..|
 |...|..7|...|
 |9.8|5..|.1.|
 |---+---+---|
 |..7|1..|...|
 |..5|...|8..|
 |...|..6|3..|
 |---+---+---|
 |.8.|..5|9.3|
 |...|9..|...|
 |..1|.3.|.7.|
 *-----------*

#
# claudiarabia (Posted June 26, 2007)
#
 *-----------*
 |53.|.7.|...|
 |...|1.5|...|
 |.98|...|.6.|
 |---+---+---|
 |8..|.6.|...|
 |4..|8.3|..1|
 |...|.2.|...|
 |---+---+---|
 |.6.|...|28.|
 |...|4.9|..5|
 |...|...|.7.|
 *-----------*

#
# claudiarabia (Posted July 01, 2007)
#
 *-----------*
 |.5.|...|.7.|
 |9..|...|..3|
 |..8|1.2|6..|
 |---+---+---|
 |7..|...|..2|
 |.1.|3.7|.4.|
 |2..|.9.|..1|
 |---+---+---|
 |..2|6.8|5..|
 |4..|...|..9|
 |.8.|...|.2.|
 *-----------*

#
# claudiarabia (Posted July 01, 2007)
#
 *-----------*
 |61.|.4.|...|
 |.98|.17|...|
 |...|8..|...|
 |---+---+---|
 |8..|..9|.7.|
 |..7|2..|.45|
 |...|.3.|2..|
 |---+---+---|
 |..6|.9.|.1.|
 |1..|...|.69|
 |.7.|..3|..4|
 *-----------*

 
#
# claudiarabia (Posted July 03, 2007)
#
 *-----------*
 |2..|.9.|5..|
 |.7.|..3|...|
 |...|4..|.6.|
 |---+---+---|
 |..4|..9|..2|
 |9..|...|..8|
 |5..|6..|7..|
 |---+---+---|
 |.6.|..7|...|
 |...|5..|.4.|
 |..3|.8.|..1|
 *-----------*

#
# claudiarabia (Posted July 06, 2007)
#
 *-----------*
 |...|.7.|..2|
 |.5.|...|.9.|
 |6.3|..5|..7|
 |---+---+---|
 |8..|..9|4..|
 |.1.|...|...|
 |...|2..|.6.|
 |---+---+---|
 |.9.|..4|..1|
 |7..|6..|2..|
 |..5|...|..8|
 *-----------*

#
# Happy Fish (SE-7.0) (claudiarabia) (Posted Aug 04, 2007)
#
 *-----------*
 |...|...|...|
 |..8|3..|.5.|
 |.7.|.6.|2.4|
 |---+---+---|
 |9..|..8|..3|
 |.6.|...|.1.|
 |2..|..9|..6|
 |---+---+---|
 |.4.|.2.|9.7|
 |..6|1..|.8.|
 |...|...|...|
 *-----------*

#
# Planetary system (SE-6.8) (JPF in another thread; posted here by claudiarabia Aug 08, 2007)
#
 *-----------*
 |...|.12|3..|
 |...|4..|...|
 |..5|...|..6|
 |---+---+---|
 |.2.|...|..7|
 |3..|.8.|..9|
 |7..|...|.6.|
 |---+---+---|
 |4..|...|2..|
 |...|..9|...|
 |..9|67.|...|
 *-----------*

#
# claudiarabia (Posted Aug 17, 2007)
#
 *-----------*
 |..3|...|...|
 |.6.|...|431|
 |2..|...|.7.|
 |---+---+---|
 |.49|1.3|.6.|
 |7..|.5.|..2|
 |.2.|9.7|.1.|
 |---+---+---|
 |...|2..|6..|
 |...|31.|2.8|
 |...|..9|...|
 *-----------*

#
# claudiarabia (SE 7.2) (Posted Aug 18, 2007)
#
 *-----------*
 |...|.62|...|
 |...|4..|9..|
 |..2|...|.1.|
 |---+---+---|
 |1..|9..|...|
 |3..|.86|5..|
 |4..|.2.|.7.|
 |---+---+---|
 |.8.|5..|..6|
 |..6|...|.4.|
 |...|..1|3..|
 *-----------*

#
# claudiarabia  (Posted Sept 17, 2007)
#
 *-----------*
 |.7.|.3.|.8.|
 |9..|...|..4|
 |..1|6.5|9..|
 |---+---+---|
 |..4|...|7..|
 |3..|.5.|..9|
 |..6|...|5..|
 |---+---+---|
 |..7|4.3|1..|
 |5..|...|..2|
 |.8.|.2.|.6.|
 *-----------*

#
# Ocean (4-ring) (Posted Sept 17, 2007)
#
 *-----------*
 |...|...|..1|
 |...|.2.|.34|
 |5..|..6|...|
 |---+---+---|
 |...|..7|6..|
 |.8.|...|.2.|
 |..4|3..|...|
 |---+---+---|
 |...|4..|..9|
 |65.|.1.|...|
 |7..|...|...|
 *-----------*


[re'born]

Ocean,

Thanks for starting this thread. I think it should be a lot of fun and will help many of us increase our Sudoku Kung Fu.

I have a question about the taxonomy of your last puzzle.

Code: Select all

*-----------*
 |...|...|12.|
 |3..|...|4..|
 |15.|6..|...|
 |---+---+---|
 |...|7.2|3..|
 |...|...|...|
 |..2|4.8|...|
 |---+---+---|
 |...|..1|.69|
 |..8|...|..5|
 |.47|...|...|
 *-----------*


Can you also describe it as a 4-ring XY-cycle example?

Code: Select all

 *--------------------------------------------------*
 | 78   78   69   | 359* 59*  4    | 1    2    36   |
 | 3    2    69   | 1    8    79   | 4    5    67   |
 | 1    5    4    | 6    2    37   | 9    37   8    |
 |----------------+----------------+----------------|
 | 58   69   1    | 7    56   2    | 3    89   4    |
 | 4    78   3    | 59*  1    569  | 56   78   2    |
 | 57   69   2    | 4    3    8    | 56   179  17   |
 |----------------+----------------+----------------|
 | 2    3    5    | 8    4    1    | 7    6    9    |
 | 69   1    8    | 39-  7    369  | 2    4    5    |
 | 69   4    7    | 2    569* 569  | 8    13   13   |
 *--------------------------------------------------*


[r9c5]=9=[r1c5]=5=[r1c4]-5-[r5c4]-9-[r8c4]

and so r8c4<>9, solving the puzzle.

Given that it is extremely likely that I don't know what I'm talking about, perhaps we could add precise definitions of the three cycle types to the beginning of the post.

[Mike Barker]

Here are some examples from the zoo.

#29 4-node XY-ring
9...1........6...4..5....1...2.31.9....9....1...2.57..8....7.5.4.1...9.6...8..... #29.4 (WXYZ-wing)
.......2...8.7.4..7..6.4....3...9..........5...6....81.7...1...9....3..86..5.2.49 #29.1
2...3....69.1.......1.7..9.3....59....62..7.........648.....3...57...........148. #29.3~
..3.56.....7.....6.42..3.1....89.....9.4...3...5.....2.6..........32...1..1..9..5 #29.2>

#33 5-node XY-ring
.....87....2.....335..4...6.....5.19..1..92.7.......4...9..........37...1..9.4.8. #33.4~
.2.4.9.6...4....3.7..5.1..8.............87.9..89.6..143..6......7...53....1...... #33.3~(3-link Advanced Coloring)
64......2.3.6..9..9.8...7.14...2......95......8..7.1.......3.......4..9..21...87. #33.1>(6-node XY-chain)
.3..4...8....8764......5.2..18.....6..923............73....9.1..6....8.59..7..... #33.2>

#38 6-node XY-ring
.9....6.8.....2...7.....3..34..5.....5..1.9.........8.6...91..41...7...2..7....3. #38.1>(A=1 cell ALS xz-rule)
67..8.9....94...5.......8...........8.2.........3..4.7.3...97..1...5.....68..13.2 #38.2y(UVWXYZ-wing)

#54 7-node XY-ring
4...7.8........5..6....1.42...8.......5.1.4..3.29......7...93.1....46..5....2.... #54.1y(3-element Nice Loop)

#74 4-link Advanced Colouring (Continuous)
...1.8.2.47..2.8.9.2.4.............5....4...6..9....3..8...9...5..3...9..6...5..3 #74.6~
4..8.3..92.5.............3156..9.........27.8...7.......63...4.75...6.8.8..4..... #74.9~(SueDeCoq)

#75 4-element Nice Loop (Continuous)
..59....3.7.14....9.3......23...5.4...8...6...1.7...92......9.8....29.7.4....13.. #75.1>
...6...8.7....9....56.............499...786.1.....3...6.2.1...4..1..58.33......1. #75.6>

[Ocean]

I have a question about the taxonomy of your last puzzle.
...
Can you also describe it as a 4-ring XY-cycle example?

Code: Select all

 *--------------------------------------------------*
 | 78   78   69   | 359* 59*  4    | 1    2    36   |
 | 3    2    69   | 1    8    79   | 4    5    67   |
 | 1    5    4    | 6    2    37   | 9    37   8    |
 |----------------+----------------+----------------|
 | 58   69   1    | 7    56   2    | 3    89   4    |
 | 4    78   3    | 59*  1    569  | 56   78   2    |
 | 57   69   2    | 4    3    8    | 56   179  17   |
 |----------------+----------------+----------------|
 | 2    3    5    | 8    4    1    | 7    6    9    |
 | 69   1    8    | 39-  7    369  | 2    4    5    |
 | 69   4    7    | 2    569* 569  | 8    13   13   |
 *--------------------------------------------------*


[r9c5]=9=[r1c5]=5=[r1c4]-5-[r5c4]-9-[r8c4]

and so r8c4<>9, solving the puzzle.

Thanks for analyzing the puzzle, and showing this alternative solution! The chain you list is shorter than the 8-node xy-ring (which I called 8-ring). But it's a discontinuous "cycle", so I would prefer not to call it a 4-ring as you suggest. It qualifies as a "5-node discontinuous nice loop", as far as I recall Jeff's descriptions.

perhaps we could add precise definitions of the three cycle types to the beginning of the post.

Agree! I see that there are various definitions around, more or less precise, various names covering more or less the same thing, but maybe not exactly the same, and some definitions are well hidden within more general descriptions. So, in addition to pointers to the "authorative definition files", we could try to define exactly what is meant by the three cycle types that is planned for here (Bidirectional Y-Cycle, Bidirectional X-Cycle, and Bidirectional XY-Cycle).

As a general statement I would say this thread is meant to cover chains that
1. describe a a full/continuous cycle
2. can freely start anywhere within the cycle (no specific starting point)
3. work equally in both directions.

This is obviously only a subset of all possible chain types. When applying a chain satisfying the listed specifications, we should ask/check what other chains (or other methods) are available. Are such alternatives simpler or harder (to find, or to understand), are they leading more effectively to the solution or not, etc.?

[re'born]

I have a question about the taxonomy of your last puzzle.
...
Can you also describe it as a 4-ring XY-cycle example?

Code: Select all

 *--------------------------------------------------*
 | 78   78   69   | 359* 59*  4    | 1    2    36   |
 | 3    2    69   | 1    8    79   | 4    5    67   |
 | 1    5    4    | 6    2    37   | 9    37   8    |
 |----------------+----------------+----------------|
 | 58   69   1    | 7    56   2    | 3    89   4    |
 | 4    78   3    | 59*  1    569  | 56   78   2    |
 | 57   69   2    | 4    3    8    | 56   179  17   |
 |----------------+----------------+----------------|
 | 2    3    5    | 8    4    1    | 7    6    9    |
 | 69   1    8    | 39-  7    369  | 2    4    5    |
 | 69   4    7    | 2    569* 569  | 8    13   13   |
 *--------------------------------------------------*


[r9c5]=9=[r1c5]=5=[r1c4]-5-[r5c4]-9-[r8c4]

and so r8c4<>9, solving the puzzle.

Thanks for analyzing the puzzle, and showing this alternative solution! The chain you list is shorter than the 8-node xy-ring (which I called 8-ring). But it's a discontinuous "cycle", so I would prefer not to call it a 4-ring as you suggest.

Oh, I wasn't suggesting. More like...guessing.

It qualifies as a "5-node discontinuous nice loop", as far as I recall Jeff's descriptions.

Ahh, yes. I always screw that up. I keep forgetting what ronk told me once about these things: the only time strong links should be used is when moving within a bivalue cell.

[Ocean]

Here are some examples from the zoo.

Thanks Mike for bringing in these illustrative examples from the zoo!
The zoo is a great collection - you did a really good job with that!

The example types you show are: 4-, 5-, 6- and 7-node XY-ring, 4-link Advanced Colouring (Continuous), and 4-element Nice Loop (Continuous). All these fit perfectly within what I had in mind. I also like the way you specify alternative solution steps.

I also observed that your zoo does not contain any "5+ link Advanced Coloring (Continuous)", or any "5+ element Nice Loop (Continuous)". A bit curious if this is because those techniques are very rare, or because they are hard to find, or because simpler techniques most often make them unneccessary. Will see what people eventually come up with here. The one shown below was previously posted in Effortless Extremes.

Locked Candidates plus one Bidirectional Cycle (5-ring) is enough to solve this puzzle:

Code: Select all

 
001002000030040050500600700008000006010000090600000800006007004090010030000500200
#
 *--------------------------------------------------------------------*
 |#479    6      1      | 3789   5      2      |-349   #48     389    |
 | 279    3      279    | 1789   4      189    | 6      5      1289   |
 | 5      8      249    | 6      39     139    | 7      124    1239   |
 |----------------------+----------------------+----------------------|
 | 39     24     8      | 12349  2379   5      | 134    1247   6      |
 | 37     1      357    | 2348   6      348    | 34     9      235    |
 | 6      24     359    | 12349  2379   1349   | 8      1247   1235   |
 |----------------------+----------------------+----------------------|
 | 123    5      6      | 239    2389   7      |-19    #18     4      |
 | 8      9      24     | 24     1      6      | 5      3      7      |
 |#134    7      34     | 5      389    349    | 2      6     #189    |
 *--------------------------------------------------------------------*
#
# [r1c1]=4=[r9c1]=1=[r9c9]-1-[r7c8]-8-[r1c8]-4-[r1c1] => r1c7<>4, r7c7<>1, r9c1<>3.
#


Edit: Changed the spelling =8= (strong link) to -8- (weak), in order to conform more strictly to the nice loop notation.

[re'born]

Locked Candidates plus one Bidirectional Cycle (5-ring) is enough to solve this puzzle:

Code: Select all

 
001002000030040050500600700008000006010000090600000800006007004090010030000500200
#
 *--------------------------------------------------------------------*
 |#479    6      1      | 3789   5      2      |-349   #48     389    |
 | 279    3      279    | 1789   4      189    | 6      5      1289   |
 | 5      8      249    | 6      39     139    | 7      124    1239   |
 |----------------------+----------------------+----------------------|
 | 39     24     8      | 12349  2379   5      | 134    1247   6      |
 | 37     1      357    | 2348   6      348    | 34     9      235    |
 | 6      24     359    | 12349  2379   1349   | 8      1247   1235   |
 |----------------------+----------------------+----------------------|
 | 123    5      6      | 239    2389   7      |-19    #18     4      |
 | 8      9      24     | 24     1      6      | 5      3      7      |
 |#134    7      34     | 5      389    349    | 2      6     #189    |
 *--------------------------------------------------------------------*
#
# [r1c1]=4=[r9c1]=1=[r9c9]-1-[r7c8]=8=[r1c8]-4-[r1c1] => r1c7<>4, r7c7<>1, r9c1<>3.
#


I'm guessing that the chain should be written

[r1c1]=4=[r9c1]=1=[r9c9]-1-[r7c8]-8-[r1c8]-4-[r1c1]

[Ocean]

I'm guessing that the chain should be written

[r1c1]=4=[r9c1]=1=[r9c9]-1-[r7c8]-8-[r1c8]-4-[r1c1]

Thanks. I agree that the chain might be simpler to read when written as you say. I used =8= to describe the link, because it is actually a strong link. In the chain a weak link -8- is enough, but a strong link can always replace a weak link in a chain. Maybe a bit bewildering... so your way of writing the chain is probably better (cleaner).

[re'born]

...but a strong link can always replace a weak link in a chain.

Of course, any time you have a strong link you get a weak link (with the the definition of strong link being XOR and weak link being NAND), but I've always been told by the notation guru's that in the nice loop notation, you cannot replace a weak link with a strong link.

Hey ref, can I get a ruling on this?

[Mike Barker]

Discontinuous nice loops seem to be much more common than continuous ones.

The zoo had very strict requirements (in general only one technique beyond locked candidates) so it was very hard to find advanced technqiues that fit the bill. In the case of the example you show there is a swordfish, XY-wing, and smaller nice loops (albeit unnecessary). In addition, I would have counted it as a 4-element (2 stong links and 2 bivalues) nice loop or equivalently as a 5-link nice loop, but counting elements seems to convey more information.

The problem labelling the link as a strong link is that -1-[r7c8]=8= is a discontinuity. Jeffs construction rules require the weak link and strong link labels into and out of a continuous node to be the same. Just because something can be done doesn't mean it should be . I think this is especially true in constructing nice loops where it is often better to ignore the fact that a link could be strong and focus on just what it needs to be to follow the rules.

[ronk]

[r1c1]=4=[r9c1]=1=[r9c9]-1-[r7c8]-8-[r1c8]-4-[r1c1]

A nice loop is about the propagation of inferences (implications), so your expression is the correct one.

r7c8<>1 implying r7c8<>8 wouldn't make any sense.

[Ocean]

Of course, any time you have a strong link you get a weak link (with the the definition of strong link being XOR and weak link being NAND), but I've always been told by the notation guru's that in the nice loop notation, you cannot replace a weak link with a strong link.

Hey ref, can I get a ruling on this?

The problem labelling the link as a strong link is that -1-[r7c8]=8= is a discontinuity. Jeffs construction rules require the weak link and strong link labels into and out of a continuous node to be the same. Just because something can be done doesn't mean it should be . I think this is especially true in constructing nice loops where it is often better to ignore the fact that a link could be strong and focus on just what it needs to be to follow the rules.

r1c1]=4=[r9c1]=1=[r9c9]-1-[r7c8][blue]-8-[/blue][r1c8]-4-[r1c1]

A nice loop is about the propagation of inferences (implications), so your expression is the correct one.

r7c8<>1 implying r7c8<>8 wouldn't make any sense.

Ok, thanks rep'nA, Mike and Ron for clearing this up!

Discontinuous nice loops seem to be much more common than continuous ones.

The zoo had very strict requirements (in general only one technique beyond locked candidates) so it was very hard to find advanced technqiues that fit the bill.

I see.
Also, thanks for analyzing the example from Effortless Extremes, and tips about how to classify puzzles.

Tried to study those from the zoo in more detail. For instance the last example:

Code: Select all

 *-----------*
 |...|6..|.8.|
 |7..|..9|...|
 |.56|...|...|
 |---+---+---|
 |...|...|.49|
 |9..|.78|6.1|
 |...|..3|...|
 |---+---+---|
 |6.2|.1.|..4|
 |..1|..5|8.3|
 |3..|...|.1.|
 *-----------*
 
 *--------------------------------------------------------------------*
 | 12     1234   9      | 6      2345   124    | 1347   8      57     |
 | 7      1234   3-48   | 12458  23458  9      | 134    36     56     |
 | 18     5      6      | 7      348    14     | 1349   39     2      |
 |----------------------+----------------------+----------------------|
 | 1258   1236  #378    | 125    256    126    |#-23-57 4      9      |
 | 9      2-3-4 #34     | 245    7      8      | 6     #235    1      |
 | 125    12-46 #47     | 12459  24569  3      | 257    257    8      |
 |----------------------+----------------------+----------------------|
 | 6      8      2      | 3      1      7      | 59     59     4      |
 | 4      79     1      | 29     269    5      | 8      267    3      |
 | 3      79     5      | 2489   24689  246    | 27     1      67     |
 *--------------------------------------------------------------------*
 [r4c3]-7-[r6c3]-4-[r5c3]-3-[r5c8]=3=[r4c7]=7=[r4c3]


Sudoku Explainer finds this Bidirectional Cycle. Strangely enough it does not deduct all possible eliminations from the chain. The eliminations of 2 and 5 from [r4c7] is done in two separate Forcing Chains (contradiction chains), but the eliminations from r2c3, r5c2 and r6c2 are enough to solve the puzzle.

When it comes to precise definitions, I find the descriptions of X-cycles and Y-cycles here quite instructive. Bidirectional Cycles (as they are called in Sudoku Explainer) are mostly equivalent to Continuous Simple Nice Loops (Jeff's definition).
The theme for this thread is puzzles that are solved with (one or more) Continuous Simple Nice Loops as the highest ranked (most difficult) technique - not quite precise because ranking of techniques is debatable.

[ArkieTech]

Just when I think I am beginning to get a grip or understanding.

Code: Select all

 *-----------*
 |...|...|..1|
 |...|2.3|...|
 |.45|...|6..|
 |---+---+---|
 |..7|..6|.3.|
 |.8.|...|.9.|
 |.1.|8..|7..|
 |---+---+---|
 |..3|...|95.|
 |...|4.1|...|
 |2..|...|...|
 *-----------*


 *-----------*
 |.32|...|.71|
 |671|2.3|.89|
 |.45|...|623|
 |---+---+---|
 |.27|..6|138|
 |.86|...|295|
 |.19|8..|746|
 |---+---+---|
 |163|7..|954|
 |7.8|4.1|362|
 |2.4|...|817|
 *-----------*

 
 *-----------------------------------------------------------*
 | 89    3     2     | 569   68    4589  | 45    7     1     |
 | 6     7     1     | 2     45    3     | 45    8     9     |
 | 89    4     5     | 19    17    789   | 6     2     3     |
 |-------------------+-------------------+-------------------|
 | 45    2     7     | 59    459   6     | 1     3     8     |
 | 34    8     6     | 13    17    47    | 2     9     5     |
 | 35    1     9     | 8     23    25    | 7     4     6     |
 |-------------------+-------------------+-------------------|
 | 1     6     3     | 7     28    28    | 9     5     4     |
 | 7     59    8     | 4     59    1     | 3     6     2     |
 | 2     59    4     | 36    36    59    | 8     1     7     |
 *-----------------------------------------------------------*


r1c4-9-r4c4=9=r4c5-5-r8c5=9=r9c6-9-r1c6=5=r1c4

Help! since weak side r1c4, r4c4 is part of the ring I thought one or the other point had to be a 9 and therefore the 9 at r3c4 could be eliminated. I am wrong of course. Can anyone help me?

dan

[claudiarabia]

At first, Ocean, let me express my joy over your strong comeback in the forum after a long time of being seldom here.
I feel very honored that my incentive to discuss bi-directional cycles fell on such a fertile ground.

Code: Select all

. 4 . . . . . 1 .
. . . 5 . 7 . . .
8 . . . 9 . . . 7
. 2 . . 1 . . 9 .
. . 5 4 . 6 8 . .
. 3 . . 8 . . 7 .
4 . . . 5 . . . 6
. . . 2 . 9 . . .
. 7 . . . . . 3 .


This one needs one Sky-craper which eliminates two 1s, one bi-directionyl-y-cycle in r5c59 and r8c59 with the numbers 1347. It ends up in a BUG1 Situation.

Claudia

[re'born]

At first, Ocean, let me express my joy over your strong comeback in the forum after a long time of being seldom here.
I feel very honored that my incentive to discuss bi-directional cycles fell on such a fertile ground.

Code: Select all

. 4 . . . . . 1 .
. . . 5 . 7 . . .
8 . . . 9 . . . 7
. 2 . . 1 . . 9 .
. . 5 4 . 6 8 . .
. 3 . . 8 . . 7 .
4 . . . 5 . . . 6
. . . 2 . 9 . . .
. 7 . . . . . 3 .


This one needs one Sky-craper which eliminates two 1s, one bi-directionyl-y-cycle in r5c59 and r8c59 with the numbers 1347. It ends up in a BUG1 Situation.

Claudia

As long as you are ending up in a BUG+1 situation, you might as well solve it before the y-cycle, when you are in a BUG+2 situation.

Code: Select all

 *--------------------------------------------------*
 | 2    4    7    | 68   36   38   | 5    1    9    |
 | 19   6    19   | 5    2    7    | 3    4    8    |
 | 8    5    3    | 1    9    4    | 2    6    7    |
 |----------------+----------------+----------------|
 | 67   2    8    | 37   1    5    | 46   9    34   |
 | 79+1 19   5    | 4    37   6    | 8    2    13   |
 | 16   3    4    | 9    8    2    | 16   7    5    |
 |----------------+----------------+----------------|
 | 4    19   2    | 37   5    13   | 79   8    6    |
 | 3    8    6    | 2    47   9    | 17+4 5    14   |
 | 5    7    19   | 68   46   18   | 49   3    2    |
 *--------------------------------------------------*


[r8c9]-4-[r8c7]=4|1=[r5c1]-1-[r5c9]=1=[r8c9], implying r8c9<>4, solving the puzzle.

Of course, if you're not into the whole uniqueness thing, you could always try the xy-chain

4-[r9c7]-9-[r9c3]-1-[r7c2]-9-[r5c2]-1-[r5c9]-3-[r4c9]-4

implying r4c7, r8c9<>4, solving the puzzle.

Or finally, if you're really lazy, you might find this move before the Skyscraper:

Code: Select all

 *--------------------------------------------------*
 | 2    4    7    | 68   36   38   | 5    1    9    |
 | 19   6    19   | 5    2    7    | 3    4    8    |
 | 8    5    3    | 1    9    4    | 2    6    7    |
 |----------------+----------------+----------------|
 | 67A  2    8    | 37-  1    5    | 46   9    34   |
 | 179  19A  5    | 4    37   6    | 8    2    13   |
 | 16A  3    4    | 9    8    2    | 16   7    5    |
 |----------------+----------------+----------------|
 | 4    19B  2    | 37B  5    13B  | 79   8    6    |
 | 3    8    6    | 2    47   9    | 147  5    14   |
 | 5    7    19   | 68   46   18   | 49   3    2    |
 *--------------------------------------------------*


A={1,6,7,9} on r46c1, r5c2
B={1,3,7,9} on r7c246
x=9
z=7

ALS xz-rule gives r4c4<>7, solving the puzzle.