One post wasn’t enough, so this is the second featuring highlights from the sysudokie review of Frank Longo’s Absolutely Nasty Level 4 collection.

X-chains are normally denizens of the X-panels, but in AbsNasty4 #21, one was caught in line marking close, the stage when marked lines in one direction cover the grid, and locked sets, including line slinks are sought in the other direction. It’s a simple X-chain, and of a type called a skyscraper.

Earlier in the blog I complained about a special name like “skyscraper” being accorded an X-chain and thus confusing beginners into thinking of it as a special pattern.  Then, by way of revenge I think,  skyscrapers started turning up everywhere in the blog.

One victim of this particular skyscraper generates a boxline, demanding a 5-candidate in NWc1. The boxline removes two 5-candidates, one of which was not removed by the skyscraper.  A collapse followed.

Adventures continued with AbsNasty4 #31.  At the end of line marking, the unfortunate events were an X-wing and a naked triple on successive lines, the latter triggering another X-wing. The puzzle entered the bv scan covered in bv.  Only two cells with three candidates each stood between #31 and a Bi-value Universal Grave, or BUG, with two solutions, or no solutions.  Then the depicted 169-wing pushed the lead domino, and the BUG dissolved into a solution.

Now let me explain how I even got into seeking multiple solutions for AbsNasty4 #61, the culprit I mentioned earlier. Basic solving was typical of the Level 4 collection, and here are the 79417-chain(purple) and the swordfish(red)  or the nice loop(blue) that followed. The swordfish and nice loop generate the same removal, and the removal enables a naked pair for another two removals.

Finding no other fish or productive X-chains I moved on to coloring. The first step is to extend one or more clusters over most of the slinks and bv, looking for color trap eliminations. The result was three clusters, with color traps on the third one, and bridging possibilities.

Since orange and green 7-candidates see each other – or because the two colors occur in the same cell – one of the  partnering colors, blue or red, is true.  The similar conflict between orange and aqua means that red or tan is true.  The significance is that pairs of blue and red candidates are toxic to candidates of the same number.  Same for aqua and red.  Unfortunately, only the 7 candidates inhabit more than one cluster.

If you conclude from the bridging that red is more likely, I agree.  I brought up my AIC hinges again, and started looking for an AIC to confirm a red candidate.  If you don’t do coloring, get with it.  It’s powerful.  So is the human mind when it has a definite goal and the tools to work with. This time the tool ware is the awareness of the many forms of Alternative Inference Chains (AIC) that can be cobbled together.

Does that AIC look like a key?

Well, it didn’t unlock much and that is surprising.  One cluster gone, and the other two don’t seem to be related at all. Also, although I didn’t impose the details on you, my ALS and AIC constructions were unproductive.

Here’s one more thing.  Really tough puzzles don’t usually have extensive clusters, and a bv cloud this large.  The logical pieces fit together too well. This is about the time I start looking for that written guarantee of unique solutions.

Here’s my plan.  We can safely assume that aqua or tan is true,  and even more safely, that blue or green is true.  On the laptop, it’s easy to copy the slide, make a color true, and see what happens.  I go with the more extensive cluster, and usually find a color of that cluster will imply a color of the other.

What happened in this case is that three out of four cluster color combinations are compatible, and that the remaining cells can be colored to identify even more solutions.  It’s astounding when you’ve been working on the puzzle, to see 9 solutions arrayed together.

Each solution is displayed in a different keypad position. Position 1 is the solution given in Frank Longo’s book Level 4 collection book.  Solutions 1 and 2 interchange 1 and 4 in one of the six-cell uniqueness patterns cited from the Sudocue Guide.  A 1/4 interchange in UR r13c17 occurs in many solution pairs.  A few candidates proved true in all solutions.  I haven’t tried to account for the fact that clues were not discovered for these cells.  Doing so would not have affected the multiplicity of solutions.

Wow! That concludes our snapshots of the Absolutely Nasty Level 4 Review. Buy the book. Work the puzzles.  Share the adventures.

The next puzzle collection review is for the Super Tough puzzles that can be downloaded for free from www.krazydad.com, and you have worked out the box marking for v.5, b.1 #5 haven’t you? And here is your 2D checkpoint.

Line mark it, and look for a break,  but go no farther than coloring.