Sudoku Solving Techniques - Hidden Subset

This technique is known as "hidden pair" if two candidates are involved, "hidden triplet" if three, or "hidden quad" if four. It is sometimes also called "unique subset".

This technique is very similar to naked subsets, but instead of affecting other cells with the same row, column or block, candidates are eliminated from the cells that hold the subset. If there are N cells, with N candidates between them that don't appear elsewhere in the same row, column or block, then any other candidates for those cells can be eliminated.

For example, consider a block that has the following candidates:

{4, 5, 6, 9}, {4, 9}, {5, 6, 9}, {2, 4}, {1, 2, 3, 4, 7}, {1, 2, 3, 7}, {2, 5, 6}, {1, 2, 7}, {8}

(The single {8} indicates that this cell already holds the value 8.) You can see that there are only three cells that have any of the candidates 1, 3 or 7. (These cells have other candidates too, but they're the ones that we can eliminate.) Three candidates with only three possible cells between them means that one of the candidates must be in each of the cells. So, obviously, these three cells cannot hold any other value, meaning we can eliminate any other candidates for these cells.

In this example, we're left with:

{4, 5, 6, 9}, {4, 9}, {5, 6, 9}, {2, 4}, {1, 3, 7}, {1, 3, 7}, {2, 5, 6}, {1, 7}, {8}

Naked subsets and hidden subsets are related - I usually describe them as being opposite sides of the same coin. If a naked subset is present, then so is a hidden one, although it may be longer and so harder to spot. The opposite is also true, if a hidden subset is present, so is a naked one. They obey the following relationship:

NumberOfDigitsInNakedSubset + NumberOfDigitsInHiddenSubset + NumberOfFilledCellsInUnit = 9

or to put it another way:

NumberOfDigitsInNakedSubset + NumberOfDigitsInHiddenSubset = NumberOfEmptyCellsInUnit

Back to Sudoku Solving Techniques