Limited Pattern Overlay

Efforts to combine patterns to solve Sudoku puzzles started with the Pattern Overlay Method, or POM, a computer solving method which exhaustively discovers pattern conflicts to eventually find the true pattern of each value. A shadow of POM is now offered in both solvers Sysudoku uses to illustrate humanly practical methods. These options make candidate removals only, and  are labeled here as Limited Pattern Overlay, or LPO.

In Heine’s ultrahardcore 355 post of 11/24/20, the Beeby POM option note suggests a 3 step LPO process. The West to East freeforms in the 9 panel reveal two cells that are in every 9-pattern.  That’s step one.

Step two is to find another X-panel covering 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.

For step 3, Beeby’s POM note states that we can delete all 3-patterns that include both of these cells, and that, as a result, 3r8c6 can be removed. Looking at the three freeform combinations of South to North patterns crossing both cells, the removal of 3r8c6 alone isn’t justified,

because its  removal leaves 3 patterns that include both cells.

The Beeby note suggests we make the 3r8c6 removal by finding all of the 3 patterns, and after eliminating all of those crossing both offending cells, remove all orphans of the remaining patterns.  But there is an easier and more direct way to find those removals that eliminate conflicting patterns, if they exist.

That is to use the freeform direction that limits patterns crossing the conflict cells.

In this example, by starting West to East, and limiting columns to the two cells, the candidate in all such patterns is shown to be 3r9c7, not 3r8c6.

An LPO example detectable on X-panels occurred in Heine’s ultrahardcore 3, where Beeby’s POM option  announces the removal of 3r4c3 by overlay of 3 and 7 candidates. 

Any South to North 7 freeform starting in r8c8 must cross r6 at c9, leaving it to  cross r4 at c3. So every 7 pattern contains  r8c9 or r4c3. No patterns of another value can contain both of these cells.

But 3r4c3 => 3r8c9, so no 3 pattern can contain r4c3.

For comparison of spotting difficulty, in the ultrahardcore 3 post of 6/9/20 the LPO removal of 3r4c3 is also attributed to grouped shortcut boomer 4.

KrazyDad Insane 445, originally posted 8/13/13, can demonstrate how an LPO pair may be revealed by a slink AIC on an X-panel. We update the post with some outstanding fishing examples to get there. Returning to the grid before the Single Alternate Sue de Coq,

replace all the 6 candidates and return to the grid before the Single Alternate Sue de Coq. At that point. There’s a sashimi 3-wing.

If r3c7 contained a 3 candidate, the 3-wing would remove the two victims. As it is, if one of the victims is true, it removes the fins, 3r3c1, and 3r6c7, leaving no 3-candidate for c7.

Now we get NW3, W3, and a c3457 jellyfish.  The 4 columns each get a row position

The next Beebe find is a finned franken swordfish of 1s (c137\r15SW), fin at r2c7, eliminating 1 from r1c9. A franken fish is a  mutant fish with a single box unit.  The fin is a single uncovered base candidate.

In finned mutant fish, any victim must see all fins, so that If the victim were true and the fins are removed,  the base will be covered, making all victims false.  Strange-ly, if r2c7 were discovered to be false, 3 morer covered candidates outside of base units can be removed.

Now after an ANL and Sc4 boxline,

and a simple 1-way,

A simple ANL presents a Limited Pattern Overlay opportunity on the 2 panel. The covering value is 3.

Since at least one of the terminal candidates 2r1c5 and 2r6c8 must be true, the true 3-pattern cannot cover both cells. 3r2c7 is removed because it enables a S to N freeform  including both cells

3r2c9 escapes removal because it is not an orphan when all covering 3-patterns are removed.

The 7 candidates also cover those two cells, but there are no freeforms in either direction that include both of them, thus making an LPO removal.

An extension of the 2-AIC gives another pair of LPO cells, but no 7-pattern includes them.

The 7 candidates do cover the terminals of one more 2 AIC and removal of 7r9c1 precludes every 7-pattern covering r5c3 and r7c4.

After the LPO removals, we can finish Insane 445 in a typical way, by coloring.

The cluster started by SW7 and the  shortcut 2 ANL, the Er4 boxline and the naked pair 15

expands to reach a green wrap and a blue solution.