Heine’s left page ultrahardcore 45 offers a host of examples illustrating when a Beeby “discontinuous nice loop” is and isn’t a 1-way. Coloring closes the class.
UHC 45 has a remarkable number of bv and line slinks, given the monster-like cloud of candidates on the line marked grid.
The basic trace shows a good number of box slinks as well, to go with the two bypass clues.
The solvers start teaching early. Sudokuwiki finds an APE new to me. I once thought “aligned” in “Aligned Pair Exclusion” meant the elimination cells are in the same line or box, and had never seen a counter example. Actually, APE “aligned” means the elimination cells are both aligned with a set of ALS that collectively exclude at least one combination of candidates in the pair.ypass clues.
So you look for ALS aligned with two cells, with values sets matching the cell values. Then look for a value in one cell whose every combination with values in the other cell is contained in one of the aligned ALS. You can ignore 4 and 5 combinations, they are absent from both ALS.
Beeby duplicates the Sudokuwiki APE in an ALS_72. The ALS has common value X = 7 and the victim sees all Z = 2 candidates in both ALS.
Beeby next finds this X, XZ based ANL, which I would get to in AIC building.
The removal allows an XY ANL onto the railway.
Next is a series of “discontinuous nice loops”. Some end with 1-way logic, but many become 2-way almost nice loops, or ANL. The Beeby’s notes describe this grouped ANL as a discontinuous loop, type 2, starting at 7r6c6. In AIC building, I would have started at the 7 slink in r8, seeing the c6 7 group as a link to 7r6c6 to complete the ANL. The removal brings a host of naked pair removals within the box.
In Beeby’s next, starting from 7r5c3, the 7-group is the end a 7-slink, and an ANL terminal seen by the victim. Or is it one of several ways the AIC started by assuming 7r5c3 is false can prove 7r5c3 is false? That’s a simple 1-way.
In both of the last two examples, a group is taken as a slink partner. That is, a slink combined with a group functions as a slink.
The next “discontinuous” is also a simple grouped 9-chain ANL.
OK, here is an authentic simple 1-way. You start with a possible target which is false if the starting candidate is true, and the AIC started with a false starting candidate makes the target false, not true, and not by making a third candidate seeing the target true.
Now two true 1-ways starting on 8r4c4.
The first winks in to 1-way target. That is extended to another target. The extended AIC AIC makes the target false by making another candidate in the target cell true.
The second 1-way triggers the Nc5 boxline.
With the next boomer from 6r9c5, we add a blue green cluster.
Then we see the red orange 1-cluster wrapping blue.
Red orange expands, and SW9 => S9 => orange,
and we have a green and orange solution, without trial.
Next time, it’s left side ultrahardcore 89.
Systematic Sudoku 