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Combination Culling

This is perhaps the most important Killer technique, and is not shared with regular Sudoku at all. The technique allows us to eliminate digits from entire cages. Simply put if including a particular digit in a cage means that the sum for the cage cannot be reached, then that digit cannot go in the cage. To see how this works, here is an example:

The cage highlighted in red has a sum of "4". So clearly, we can't put any digit larger than a "3" anywhere in the cage, because the other digit would have to be 0 or less. Similarly, we can't put a "2" in the cage, because the other digit would have to be a "2", and we can't repeat the same digit twice. So the only options left for the cage are "1" and "3". Here is another example:

We can remove the candidates 7, 8 and 9 from the red highlighted cells. So far, two of the cells in the highlighted cage have already been filled. The cage has a sum of 18, but because there are already a "3" and a "4" in the cage, the two remaining empty cells must add up to 11. Normally, the possible ways of making 11 are {2, 9}, {3, 8}, {4, 7} and {5, 6}. In this case, however, neither cell can be a "3" or a "4", because they already share a cage with these digits, and neither cell can be a "2", because they are in the same box as one. So the only valid combination left is {5, 6}. This means that the candidates "7", "8" and "9" can be removed from both cells.

In general, the quickest way to solve a Killer puzzle is to look for small cages with either very high or very low sums, because these have the fewest possible combinations, and therefore give you the most information.

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