## UHC 397 Fills the Stocking

After two “simple” ANL, Beeby finds a candidate removal by pattern overlay analysis. On PhilsFolly.com, this is requested under “POMs”.  Then a pattern analysis for The Guide offers many more removals using freeforms.   A finned swordfish gets us to coloring, with two clusters wrapped by a DIY unique rectangle. .   The review table for uhc 221 -397 is included

Following the complex 1-way of the previous post, Beeby  continues with this very productive ANL.

The follow up includes a NEr1 boxline,

Then a simple ANL removes two 2 candidates.  With fish and other AIC requests denied, Beeby does a pattern analysis removal of 6r1c9, which triggers a NEc3 boxline removing 6r6c7.

Beeby explains the removal this way:

Between them r3c3 and r7c6 include all patterns of 3, so patterns of 6 which include both cannot be true. As a result, no pattern of 6 includes r1c9.

So how does Beeby know that? And WTH is a pattern?

A pattern is a set of remaining candidates of one value, with one member candidate in every row, column and box. You can mark a pattern by drawing a freeform, a connected set of lines from one side of the pattern to the opposite side, going through exactly one candidate per column and box. Similarly, by a freeform from top to bottom or bottom to top, through exactly one row and box.

A first step in duplicating Beeby’s pattern analysis with freeforms is to draw all patterns of 3. I would start them on c6, going left or r8, going up. From c6, we enumerate 6 patterns. Any 3 candidate not on any pattern, an orphan, is removed. On the other hand, any candidate on all patterns is a clue. All patterns go through r6c6 or r6c3, but we want rows and columns to be different, so that patterns of other values crossing all of them can be discarded, removing candidates. The fewer the cells other values must avoid, the more of their patterns we can discard, and the more of their orphans we can remove.  Beeby finds r7c6 as a second cell other value freeforms must avoid, because freeforms through r6c3 start from r7c6.

Now operating DIY, we would have to examine all freeform patterns of 8 other values. We don’t have to find all patterns of these values, just those that don’t include both r3c3 and r7c6.

Starting r1c9, ffs must hit r7c6, r6c7 and r5c4, then avoid r3c3, but then they can’t cover r8. The two patterns hitting both red cells are discarded. That’s Beeby’s removal of 6r1c9

All ff from r5c9 must hit r6c6, so they avoid r7c6, and don’t have to avoid r3c3.

Finally, the 6 freeforms from r6c9 must hit r7c6 and then r5c4. The one that supplies a 6 to r1 can’t get to c1. The one that hits r1c3 and avoids6r3c3 can’t get to c1. And the one that hits r3c3 and gets across hits both the red cells and is eliminated. So our freeform analysis finds another  removal, 6r6c9, plus all the orphans left out of the successful freeforms starting from 6r5c9.

We have only verified a particular removal and showed how freeform analysis can generate related removals. Finding all such removals would take magnitudes more calculations.

Here are the freeform analysis removals, along with a Beeby simple 1-way and its  naked pair r8s16 =>W3.

Clue W3 creates naked pair r5s48

A finned swordfish is enabled. Victim 4r6c3 is in the fin box. Potential kraken victims r3c36, r6c7 and r7c3 don’t see the fin, so for them, the swordfish isn’t there.

You know progress is being made when an XY ANL comes in. It’s also time to get get out the crayons.

After

two small clusters are started.

Starting an AIC from 5r3c5, when red 7 is encountered, any orange 5 on the red/orange cluster is a potential ANL terminal. We know there’s an XY removing chain to 5r7c9 without completing it.

Remarkably, the matter is settled when  a DIY unique rectangle removes 7r3c6, because it confirms 5r3c5 and 2r1c5, stripping  rectangle corners of extra candidates. Removing the offender,

N14 => C7 => orange and blue.

The orange and blue solution is immediate.

This post completes the review of 10 left page ultrahardcores, the number selected in all reviews prior to the right page ultrahardcores.

In the multiple cluster solution paths of both reviews, there’s certainly evidence of interlocking slink networks in ultrahardcores that have to be resolved, one by one.

Kudos to Stefan Heine for these superbly constructed puzzles, and to the authors of the two DIY oriented solvers essential to my getting my reviews done

Here is the review table for ultrahardcore 221 – 397.

Next week, we look at the first two Sudoku presented on the Enjoy Forum to introduce a solving technique proposed by correspondent shye at http://forum.enjoysudoku.com/fireworks-t39513.html on Nov. 3, 2021. Thanks to my friend Gordon Fick for alerted me to this thread.

Forum correspondent shye calls methods based on this technique “fireworks”. Both puzzles are immediately collapsed by a single application of fireworks, but the second one, shown here, turns out to be the equal of a Stefan Heine ultrahardcore right page without it.

Next week’s post illustrates how the fireworks method is supported by Sysudoku X-panels, and why it is best applied immediately after line marking.