Hidden Subsets

fig8Behind Naked Subsets, the next most difficult solving strategy is Hidden Subsets. This strategy is reminiscent of Naked Subsets, but is based on the exact opposite philosophy. The idea is that if a group of cells of size x in a row, column, or block is the only group of cells to contain a certain x candidates, though not necessarily only those candidates, it is a hidden subset (Armstrong). Within this hidden subset, all candidates can be removed that are not one of the x candidates in question. This sounds a lot like Naked Subsets, but Figure 8 helps to display the difference. The highlighted cells are the only cells to contain 1, 6, or 8 in their candidate sets. Because there are exactly three cells fostering these values, there is a hidden triple, and any other candidate values in these three cells can be eliminated. The middle-left cell now only contains 1, 6 and 8 as candidates, the bottom-center 6 and 8, and the bottom right 1, 6, and eight.

Programming Hidden Subsets, unsurprisingly, is very similar to programming naked subsets. Again, a common framework will be applied to different subset sizes. The strategy will again loop through all columns, blocks, and rows, and obtain all possible combinations of x cells in these columns, blocks, or rows. The difference is that now, of all of these combinations, the algorithm is going to find the number of candidates in these x cells, removing duplicates, and compare this to the number of candidates outside of the x cells, removing duplicates. If these two numbers subtracted equal x, then a hidden subset has been found. Now it is just a matter of seeing if any candidate removals can be made in those x cells. If none can be made, then it is time to move onward to the truly advanced strategies.

Posted in Sudoku Solving Strategies and Techniques at April 26th, 2009. Trackback URI: trackback

One Response to “Hidden Subsets”

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