NRCZ CHAINS SUBSUME BROKEN WINGS
I already posted this in the fully supersymmetric chains thread, but participants in the Broken Wings discussion may have missed it.
When re'born asked me (at the bottom of this page:
http://forum.enjoysudoku.com/viewtopic.php?t=5591&postdays=0&postorder=asc&start=45) whether there was "a nice way of either subsuming broken wings into nrczt-chains, or melding them into a even slightly bigger theory", I concentrated on Rod Hagglund's definition of Broken Wings (here:
http://forum.enjoysudoku.com/viewtopic.php?t=2666&highlight=) and I noticed the logic for justifying the eliminations was very different from the logic of the nrczt-chain rules.
But, recently I was again asked the same question by a friend. As I looked at Broken Wings again, instead of reading the proof, I tried to translate them into my approach.
The result is obvious and I don't understand how I can have missed this in my answer to re'born:
nrcz-chains subsume broken wings. What Rod Haglund calls guardian cells can be understood as mere additional z-candidates in my approach.
There's no need of the t-extension.
Even without the t-extension, nrcz-chains are much more general than broken wings (and each of the following reasons may have misled me the first time I looked at them = in addition to the different logic involved):
- there's no need for a closed loop
- there`s no need for all links to be conjugacy links (modulo the guardians): the constraint bears only on even links;
- the length of the `loop` can be odd or even, no matter (this is a consequence of the previous point);
- the number of additional z-candidates in any link is completely irrelevant.
I wish this may help users of Broken Wings generalise this technique and widen their arsenal of techniques.
