Rooted: Created page with "A Killer Sudoku contains cages and cage sums. Knowing the number of cells in a cage and its sum, we can determine the possible '''cage combinations''', i.e., the p..."

2020-05-31T19:02:49Z

Created page with "A Killer Sudoku contains cages and cage sums. Knowing the number of cells in a cage and its sum, we can determine the possible '''cage combinations''', i.e., the p..."

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A [[Killer]] Sudoku contains [[cage]]s and cage sums. Knowing the number of [[cell]]s in a cage and its sum, we can determine the possible '''cage combinations''', i.e., the possible combinations of [[digit]]s that can go into the cage. For example, if we have a cage of 2 cells whose sum is 5, written 15[2], then either it contains the digits 1 and 4, or the digits 2 and 3. The cage combinations are written in set notation as {1,4} or {2,3}, or simply {14} or {23}.

Manually enumerating the cage combinations for arbitrary cages can be very tedious, but some [[Sudoku Programs|helper programs]] such as [[SumoCue]] can automatically compute the cage combinations for the player. However, for some cages, the cage combinations are easy to enumerate and can lead to [[candidate]] [[elimination]]s, either simply because the candidate is not part of any combination, or in conjunction with some other [[technique]]. A suggested way of enumerating the combinations is to start from the largest possible digit and then working systematically downwards, so a cage 11[3] would be enumerated as {8,2,1} or {7,3,1} or {6,4,1} or {6,3,2} or {5,4,2}. One should be careful to list all combinations.

== Example 1 ==
For a cage 24[3], its only combination is {7,8,9}. Then we can eliminate the digits 1 to 6 from all three cells in the cage. Further, if all the three cells all belong to the same [[line]] or [[nonet]], we can then apply [[Naked Triple]] to that line or nonet.

== Example 2 ==
Given a cage 28[4], there are two possible cage combinations: {4,7,8,9} or {5,6,8,9}. Either way, the cage must contain the digits 8 and 9 in 2 of the 4 cells. Knowing this, we might be able to apply the [[Locked Candidates]] technique. Further, if a cell whose candidates are 8 and 9 sees all 4 cells, we may be able to apply the [[Killer Subset]] technique.

== Example 3 ==
Suppose we have a cage 15[3] such that
* Cell A has candidates 3, 4, 5.
* Cell B has candidates 7, 8.
* Cell C has candidates 1, 2, 3, 4, 5, 6.

We could enumerate the permutations (3+7+5 or 3+8+4 or 4+8+3 or 5+7+3 or 5+8+2) and conclude that 1 and 6 can be eliminated from the cell C.

Alternatively, we can also consider the minimum and maximum possible values of the sum A + B. Since 3 <= A <= 5 and 7 <= B <= 8, we have 10 <= A + B <= 13. Due to the constraint A + B + C = 15, we have 2 <= C <= 5. This means that we can eliminate 1 and 6 from cell C.

[[Category:Killer Sudoku]]
[[Category:Solving Techniques]]

Cage combinations - Revision history

Rooted: Created page with "A Killer Sudoku contains cages and cage sums. Knowing the number of cells in a cage and its sum, we can determine the possible '''cage combinations''', i.e., the p..."