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`2.......6.5..8..1...4...9...7.3.1......82.......7.5.3...9...4...8..1..5.6.......2;10;tax;tarek-ultra-0203`

The loop eliminates 13r5c2,1r6c2,37r2c6,37r8c6,8r4c8,7r5c8.

My aim was to test a new analysis tool that finds elimination patterns for a specific candidate. It creates sets of bases ( cells, rows, cols, boxes ) that are non-reducible. It works in an unbiased manner as it uses no predefined Sudoku methods. The above puzzle was used as test case with the candidate 1r5c2 and produced surprising results.

David P Bird wrote:.

2. Domino Loop Definition

Domino Loops consist entirely of arguments for the digits that can occupy a loop of cell pairs.

The patterns can also be expressed as XSudo rank 0 structures or as multi-sector locked sets which allows them to be formulated in a large variety of ways

SK Loops, discovered by Stephen Kurzhals, were the first form of domino loop to be identified.

The sk-loop written in domino terminology:

(69=37)R2C13 - (37=42)R2C78 - (42=78)R13C8 - (78=69)R79C8 -

(69=37)R8C79 - (37=42)R8C13 - (42=13)R79C2 - (13=69)R13C2 - loop

We have 8 pairs of cells and consecutive pairs are double linked and ends connected. To explain the patterns I found, the lo0p is re-written in this form:

{R2C13} - 37R2 - {R2C78} - 42B3 - {R13C8} - 78C8 - {R79C8} - 69B9 -

{R8C79} - 37R8 - {R8C13} - 42B7 - R79C2 - 13C2 - R13C2 - 69B1 - loop

Pattern example found:

{R13C2} - 9R1,6B1 - {9B2,6R2} - R2C46 - {24B2} - 4R1,2R3 - {24C8} - R45C8 -

{6C8,9B6} - 6R7,9C9 - {6B7,9R8} - R8C46 - {2B7,4R8} - 2R7,4B7 - {R79C2} - 13C2 - loop

another without any cells:

{69R2} - R2C46 - {24R2} - 24B3 - {24C8} - R45C8 - {6B6,9C8} - 6C7,9R9 -

{6R8,9B8} - R8C46 - {24R8} - 2C3,4B7 - {4C2,2B4} - R56C2 - {9B4,6C2} - 9C1,6B1 - loop

and many more ...

All these loops are on a certain abstraction level of the same structural kind as sk-loops, whatever name you like to assign them. The construction elements are pairs of bases that are double linked to another pair. In place of cells there is any base type possible. The above notation tries to point this out. OK. Why do I think this is interesting?

1) The sheer number. There are >1000 or more. Very surprising.

2) All elimination patterns found have at least 16 bases and those with 16 are all of 16-matrix type (16 links containing 16 non-overlapping bases) (rank-0 size 16 in your terms) and all eliminate the same 9 candidates.

3) The are even more 17-matrix or 18-matrix ... ( > 10000 ) partially with more eliminations.

4) It seems that the sk-loop (domino loop) is a special case of a much larger class of elimination patterns.

5) All this points to a hidden symmetry property, but I have no idea to get nearer to the mystery.

6) There are other 16-matrix patterns (same elimination set) with a more complex structure but all contain some double linked pairs.

Two double linked pairs consist of four bases diminished by two links. So the remaining candidates at both ends contain together two solution candidates. This property is retained when chaining double linked pairs. The two closing links assure a rank-0.