Sudoku Assistant’s ALS Toxic Sets

This post reviews Bob Hanson’s illustrations on ALS toxic sets, in Sudoku Assistant – Solving Techniques. For these illustrations, Hanson uses the bent naked n-set examples of the previous post. Strangely enough, this extreme economy brings insights.

Bob Hanson designed his Sudoku Assistant solver to bypass by default the enumeration and analysis of Almost Locked Sets. His reasons are similar to my reasons for placing this task well back in the Sysudoku Order of Battle. For one thing, as Bob points out, there are so many ALS in the typical puzzle. They slow SA down so much, that unless the user requests it, SA bypasses them. Also, they produce toxic sets of various sizes, and the larger they are, the fewer removals they produce.

Hanson wing ALSBob re-uses the 145-wing of the previous post to illustrate the standard restricted common form of the ALS toxic set. You’ll find examples that are not regular XYZ-wings in my ALS Toxic Sets post of two years ago. If two ALS share a number and all candidates of that number see each other, the number is called a restricted common. Shared candidates of any other number in the two sets form a toxic set.

Bob points out that an XYZ-wing is also a pair of simple, related ALS with a toxic set of three candidates. The 1-pair is the restricted common, making the other 4-candidates of the two ALS a toxic set. Of course, we don’t look for this in a cloud of other ALS. It’s much easier to find as an XYZ-wing.

The example shows clearly why the other common candidate must be shared by the two ALS. One of the ALS ultimately gets the restricted common number, but we don’t know which one. The ALS that doesn’t get it cannot give up another number. The victim would take a number from both ALS.

Hanson dbl rc ALSThe second BNS example adds another ALS toxic set wrinkle. Bob uses it to illustrate something new: that a pair of ALS can share two restricted common links and thus become a single locked set, in which every number defines a toxic set. Bob doesn’t say why, but it’s because, in the solution, each ALS loses one restricted common number and gains the other. We don’t know which is which, but we know that neither ALS can give up a number, so the two restricted commons are toxic. Also since the other numbers in each ALS are locked within their containing units, the 5’s, 8’s, and and 9’s of the W ALS are toxic sets..

In this case, the 5 eliminations are relevant, being directly attributed to the double link of the ALS. The 7-candidate that Bob left out also gets clobbered. As a glider pilot, Bob is a lot more careful.

In the next posts, I will be reviewing more ALS applications from Bob’s report. His interest in them may come from the fact that enumeration of ALS comes as a byproduct of the computer algorithm finding locked sets. That scratchpad list leading up to the locked set, or failing to find one, contains all of the possible ALS of the unit. It’s all there in Subsets by Susets. Pretty neat, eh?

About Sudent

I'm John Welch, a retired engineering professor, father of 3 wonderful daughters and granddad to 7 fabulous grandchildren. Sudoku analysis and illustration is a great hobby and a healthy mental challenge.
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