## Sudoku Assistant Finds Locked Sets

This posts discloses several significant omissions in Hanson’s discussion of locked sets, including his examples, in the report, Sudoku Assistant – Solving Techniques.

A subtle and sometimes bungled task in basic solving is the discovery of locked sets or subsets. Bob Hanson defines locked sets by the following two rules. The naked subset rule is “When n candidates are possible in n cells of [a unit], and no other candidates are possible in those cells, then those candidates are not possible elsewhere in the unit.

In these rules, “is possible” can be made more straightforward, now that we have established that, following the completion scan, all unassigned cells contain at least the true candidate. Now we can paraphrase Bob’s naked subset rule as

When n cells of a unit contain the candidates of n numbers, and no other candidates, then the candidates of those n numbers can be removed from the remaining unresolved cells.

Bob’s example of a naked subset is the naked pair r1np37 in the puzzle we just solved. The picture is a bit distracting, because the naked pair is not determined until the contents of r1c9 are known. In our basic solving, N6 =>r1np37, not the r1np37 => N6 of Bob’s naked subset rule.

The second naked triple we displayed last post makes a much more arresting example.

The r136c7nt237 removes two 7-candidates, leaving only NW 7-candidates in c7, and a boxline removing two more 7-candidates in NW, and deriving N7 by hidden dublex. It’s right there in the trace.

The corresponding hidden subset rule is

When n cells of a unit contain all candidates of n numbers, then the candidates of the remaining numbers can be removed these n cells.

Bob’s report has no examples of hidden subsets. You’ll find good ones in Subsets and Susets, posted 10/25/11, with a very tough homework example from the Sudocue site. The Suset algorithm explained in that post is a scratchpad version of the “analyzeX” Bob mentions later, which is commonly used in computer solvers to “see” locked and almost locked sets. Suset building is necessary for human solvers only when a multitude of candidates creates uncertainty about the locked sets. Naked subsets are easily recognized. The hidden subset rule describes the best human approach, which is to look for combinations of n numbers confined to n cells.

Bob doesn’t press the point by showing how his naked and hidden subset rules specialize the generalized method. I think it’s important, to emphasize the theme of Bob’s outstanding work on the SA.

For naked subsets, A is n cells of a unit containing candidates of n numbers, and no other candidates. B is all cells of the unit. The B cells outside of A cannot contain candidates of A’s n numbers.

For hidden subsets , it’s a little more subtle. Let N be the number of unsolved cells of the unit. A has all N cells, with n cells containing all candidates of n numbers, and N – n cells containing only other numbers. B contains the candidates of the other N-n numbers. But by the generalized rule, B can contain no candidates of the n numbers, and is therefore identical to A! The remainder of a hidden subset of n cells is a naked subset of N-n cells!

My Subsets by Susets post also covers naked and hidden locked sets occuring together, and that the Suset algorithm need only explore cell combinations numbering half of the unassigned cells to detect any locked set, naked or hidden. In c1 above, we spotted naked triple 237 more easily that the hidden pair 19 that was also there. Sometimes, the smaller number hidden subset is easier to spot than the complementary larger number naked subset.

Bob invites readers to find additional 1, 4, and 7 locked subsets in his completion grid. Your inventory from basic solving expands to include 2 and 3. I challenge you to find them as a Hanson reader by the subset rules, after Bob’s cross-hatch and range checking eliminations. Is it easier than in Sysudoku basic? If you believe so, then you have to stay after school and do the completion exercise I did to derive that grid, to see if the effort is worth it.

It reminds me of copying the geography notebook in Miss Bryan’s fourth grade class. I did it more than once. And Miss Bryan was my next door neighbor!  Now since you were so attentive in his locked subsets class, we get to go fishing with Bob next week.