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Cage Unit Overlap

This is an important Killer Sudoku strategy which I have placed at the start of all the more complex strategies in the solver because it is so useful and very easy to spot. It is related to Intersection Removal. Whereas IR is the overlap of rows/columns with boxes, this is the overlap of 'cages' with rows, columns and boxes.

Each 'cage' is made up of one or more 'combinations' - sets of numbers that total the cage clue. If you can find a candidate number inside that cage that is not found elsewhere on the row, columns or box the cage is aligned on, then you know that number must appear in the cage. That part is self-evident since, in the solver at least, these numbers are displayed. Given that number is true, we can remove all combinations which omit that number. Often that means we can remove a bunch of numbers.

Example 1 (Link)
In the first example below, two such Cage/Unit Overlaps occur. The 2-cell cage with the red box has a clue of 14, which means the two combinations (visible if you hover over the cage on the solver) are {5,9} and {6,8}. The 6s in the cage are unique to that cage and the cage is entirely inside the box. So the only combination that fits is {6,8}. Hence we can remove 5s and 9s from the cage. In fact, since {6,8} is the only combination left both those numbers must fit in the cell and ALL other candidates can be removed, so the 4s and 3s can also go.

Cage/Unit Overlap Example 1
The blue ringed cage, a 3-cell with a clue of 19 gives us five different combinations. But the 9 in that cage is unique to both the cage and row H, so only the combinations with 9 in them are valid. {4,7,8} and {5,6,8} are not possible. Of the remaining candidates, the 5s can be removed.

Example 2 (Link)
Cage/Unit Overlap Example 2
In this second example, five Cage/Unit Overlaps have been found. Taking just the centre one as an example, the red ringed 4-cell cage has a clue of 28 - the combinations being {4,7,8,9} and {5,6,8,9}. 7 is, however, unique to the cage and the centre box, so only the first combination can be valid. 5s, 6s and other numbers not in that combination (the 1s) can be removed.

I'll leave it to you to show how the other four cages have similar eliminations.

Overall, most Killer Sudoku puzzles will have at least one example of this strategy so they are well worth looking out for, and often you can reduce the puzzle with this method while looking into the cages with multiple combinations. Keep an eye out on the rows, columns and boxes you are studying.




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Article created on 22-March-2011. Views: 14376
This page was last modified on 23-March-2011.
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Copyright Andrew Stuart @ Syndicated Puzzles Inc, 2011