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Superset Culling

The Superset Culling technique is complementary to the Combination Culling technique. If you don't already know what combination culling is, you can find out about it here.

When using the combination culling technique, it is often the case that all the empty cells in the cage being culled are contained entirely within one or more groups (rows, columns and boxes). The Superset Culling rule allows us to remove possibilities from the other cells of these groups. The rule for superset culling is if the empty cells in a cage must contain one or more digits, then these digits can be removed from any groups which contain all the cells. Here is an example of this working:

Here, we can eliminate the candidates "7" and "9" from the cells highlighted in red. The cage highlighted in orange has only one possible combination {7, 9} and must therefore contain both "7" and "9". So these digits can be eliminated from any groups which contain this cage. In this case, there are two: the box in the bottom left corner, and the bottom row. Of course, these eliminations could also have been made using the Naked Pairs technique, but this isn't always the case:

In this example, the candidate "9" can be eliminated from all the cells highlighted in red. The possible combinations for the cage highlighted in orage are {5, 8, 9} and {6, 7, 9}. Although there is more than one possible combination, both of them contain the digit "9", so we know there must be a "9" somewhere in this cage. This means that there cannot be a "9" in any of the cells highlighted in red.

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