Adjacent Line Freeforms

In Heine’s ultrahardcore 355, the Beeby POMs option demonstrates another way to exploit very limited patterns. The West to East freeforms in the 9 panel reveal a small set of cells such that every pattern contains one of them. Beeby finds a set of two.  Between them, r5c9 and r6c5 are on all 9 patterns. Beeby’s notes don’t explain why two cells on adjacent lines r56 are chosen.

Now look for other panels with values on both of these cells. The 3-panel qualifies.

No 3-pattern that contains both of these two cells can be the true pattern, because that would leave no pattern for 9. We can delete all 3-patterns that include both of these cells.

That is the basis for the Beeby POMs option illustrated in the 11/24/17 post, where the reports that 3r8c6 can be removed, because any panel including it also includes both forbidden cells. This removal was indecisive, and uhc 355 was eventually solved by a Single Alternative Sue de Coq trial.

Finding all patterns on the busy 3-panel is not easy, but freeforms make it much easier to find all 3-patterns that include both r5c9 and r6c5, especially because the two cells are on adjacent lines, and a vertical freeform must go between them.

The choice of South to North limits the number of starting combinations, but there are 3 combinations of two cells each on rows r89  that may start prohibited patterns.

The combination r9c7 and r8c6 starts three prohibited patterns. And combinations 2 and 3 match that.

The three starting combinations and their patterns reveal that the removal of 3r8c6 alone does not remove all 3-patterns that take all patterns of 9, as Beeby’s note claims.  But in a trial of 3r9c5, 3r8c9, and 3r5c4, we are including all South to North freeform patterns made up of starting candidates and those seeing both forbidden cells.

It’s enough, as the trial trace below confirms.