This post introduces nice loop coloring, with a very special nice loop. Pattern slicing has a cameo role. The solution comes in a color trial suggested by nice loop extensions.
Stingiest on the bypass, 465 gives no more clues in basic, but does allow a 7-wing on the last line marked.
That leaves a very busy grid.
The 7 candidates remaining form a cluster around three sides of the grid. The cluster expands logically to complete itself. If r1c8(orange) is false, the red 7r3c7 and 7r9c8 are true, making 7r5c9 true, and red. Now 7r5c7 is orange, and 7r8c7is trapped.
Then in AIC building, a hinge and a reversed bv allows a boomerang ANL. Starting at 1r6c1, a hinge and the bv later, the 4-group sees the 4 winking at the starting 1. It has to be a group, by the way. And the other 4 in the N box is an innocent bystander. Never saw the incoming boomerang.
Sudokuwiki closes the store after this boomerang, without closing the 7 cluster as we do above. At this point in the update I had managed a clue S2 at r8c5 with an irregular XYZ, but watchful reader Dov Mittelman pointed out that a wing linking chain of the XYZ was faulty.
Knowing what that unearned clue leads to, I doubled down on the AIC building effort to justify it, with this happy result. The ALS in cells r239c4 with value groups 1,2,4, and 8 completes a chain from 2r5c8 to the second terminal, the ALS 2-group r23c4, removing three 2’s, and leaving 2r5c8 as that highly prized S2.
Now we come to the highlight of this post, a nice loop with two ALS nodes serving as XY links. Andrew’s solver sees it, but doesn’t say anything because this nice loop has no victims. But we appreciate nice loops for another reason. They define coloring clusters.
The alternating inferences of a nice loop makes alternating coloring links. The ends of a weak link are the terminals of a slink chain as well. Going clockwise, the far end of each slink is green. Going counterclockwise the far ends are blue. It means that nice loops have a true direction, and a false direction.
The nice loop candidates form a coloring network that can be mixed and merged with any other coloring. The closed loop structure means that for each wink link in the chain , there is a corresponding slink chain in the opposite direction around the loop.
But there’s more. Once the nice loop establishes the colors, AIC branching off the loop(red) carry ANL power with them. The coloring rides for one slink in the AIC direction.
However if blue is true, the 4 group in C is gone and 4r4c2 is confirmed. Besides that, there is also a grouped slink loop with two groups extending the cluster. A trial of blue is likely to be decisive.
Next, to carry the theme of pattern analysis, the only pattern slicing opportunity is the 3-panel, and the two mapping choices of the independent slink in c4 fail to provide a single, or a cluster pair.
Since we have the two clusters, there is a possibility of conflict of both cluster patterns with panel derived patterns.
Pattern conflict analysis is the other side of pattern analysis. In Sysudoku, it’s called Limited Pattern Overlay, or LPO. Conflict analysis is easy to code, but inherently hard for humans to do. It is no fun, and that is a distraction. Tabulation is the most systematic and reportable way to examine conflicts.
On the pink side, red candidates conflict with two of the five patterns, while orange ones conflict with a different two. The table lists, the conflict cells. More decisively, both red and orange conflict with two of three olive patterns, leaving oly the solid one, with orange.
Unfortunately, the reduction of olive patterns nets only the one orphan 3r1c8. Pink patterns provide a home for all other candidates abandoned by departing olive patterns. A deeper LPO tabulation is made in KD Insane 4X5, the tenth and last puzzle of this review.
For now, I’m ready to put blue on trial, including 4r4c2 as W4. The follow up soon confirms orange, and the solution emerges. The true 3-pattern turns out to be the short dash pink.
To guard against multiple solutions, green is tested as well. Since blue implied orange, red implies green. The test of red and green fails.
Next we embark on Insane 475, so if you want to get started, it’s not too soon.
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Very nice article, just what I wanted to find.
A nice loop coloring comes up later as a polarity almost nice loop elimination in the review of Bob Hanson’s explanation on Sudoku Assistant.