# Techniques for Killer Sudoku

All techniques for vanilla Sudoku also applies to Killer Sudoku since the underlining grid follows all rules for vanilla Sudoku.

This page describes techniques specific to Killer Sudoku and other variants with constraints over combinations: (non)-consecutive, square wisdom, ratio, bossdoku, renban.... If you are not familiar with the Sudoku techniques, please read the pages for vanilla Sudoku.

Several solvers for vanilla Sudoku have been adapted or extended to Killer Sudoku. They usually have the same name as the vanilla technique prefixed by 'Complex'. Other solving techniques are specific to Killer Sudoku and will be disabled for vanilla Sudoku.

Each time a cell is set to a value either by a solver or manually, all cages containing the newly solved cell will have their combinations restrained to those having the solved values. You may turn off this automatic filtering in Options > Update Combinations after Solve

### Complex Hidden Single

For Killer Sudoku, if some cage must include some value and only one cell in the cage may have that value, then this cell must have that value. In this example, the cell in blue must be 1 because a cage 8/3 must include a 1 and this is the only cell where it may go in the cage.

### Complex Intersections alias Row-Column Block Interactions, Locked Candidate, Pointing...

For Killer Sudoku, this solver will also detect intersections of cages and houses: rows, columns, nonets, diagonals... If some cage must include some value and all the cage’s cells where that value may go are locked within one house, then no other cells in the house may contain that value. In this example, one of the cell in blue must have a 9 since a cage 22/3 must include a 9 which cannot go in the bottom cell. Therefore the 9 is locked in the blue cell and nowhere else in the row and nonet of the blue cells.

And the other way: If a value for some house is locked within some cage which cannot hold repeated values, then the cells of that cage which are outside of the house may not have that value.

In this example, the 8 of C3 is locked in R56C3 which are all in the cage 30/4. Since the cage cannot hold repeated values, no other cells of the cage may have a 8. Consequently R6C4 is forced to 7.

Note: We may also infer that the cage must include that value, which may restrict its possible combinations (not the case in this example). The Intersections solver will not detected such restricted combinations, but the Mandatory Inclusions will find them.

### Unique Pair, Triplet... Combinations

This simple solver will search for cages with unique combinations either solving the cage or forming a naked subset. This solver comes early, since a human will typically spot these early.

### Odd Pairs, Triplets... Combinations alias Orphan Possibilities

While solving a grid, possibilities will be removed from cells which may result in cages containing odd combinations or orphan possibilities. This solver will detect and remove them.

In this example, when R1C3 was set to 1, only the 1 was removed from the cage 5/2, leaving the 4. But the cage 5/2 in R1C12 can't have a 4 because it can't have a 1, so it must be {23}.

In this second example, the cage 3/2 forms a naked pair on {12}, so these possibilties were removed from the top cell of cage 5/2 leaving {34} without affecting the bottom cell of cage 5/2. But the bottom cell can only be {12} since the top cell is {34}.

The solver will search for cages in order of increasing complexity, from cages with least number of unsolved cells and least number of possibilties to cages with more cells and possibilities. It will stop searching at the first odd combination found. All cages will be checked whether visible or not, simple cages which cannot have duplicate numbers or complex cages resulting from innie-outies where duplicates may occur and even the more complex cages with differences or multiple count of cells.

### Naked Subset

For Killer Sudoku, the vanilla solver will also detect naked subsets within a cage which cannot hold repeated values. In this example, the two cells in blue forms a naked pair on {89} within the cage 30/4. Since the cage cannot hold repeated values, no other cell in the cage may have {89}. R6C4 (with 6) could not hold {89} either.

For Killer Sudoku, the vanilla solver will remove the naked possibilities from all buddies in common to every cell forming the naked subset. This may also include some cells which are not within the same house as the naked cells, but which is a buddy of all of them. In this example from the Killer #7 by Nate Dorward, the four cells in blue R45C7+R6C89 forms a naked quad on {6789} within N6. Since R6C6 is a buddy of all four cells (either in same row or in same cage), then it may not have {6789}.

Note: The vanilla Naked Subset solver will remove the naked possibilities from all common buddies in one go.

The Generalized naked subset will find the counterpart case of the Killer #7 by ND. Here is an example of such a generalized naked quad. The four cells in blue R4C7+R6C689 forms a naked quad on {6789} since each of them is a buddy of the other 3 (either in same row, same nonet or same cage). The two cells at R56C7 are buddies of all the four cells forming the naked quad (either in same nonet or in same cage). Therefore they may not have {6789}.

### Hidden Subset

For Killer Sudoku, the vanilla solver will also detect hidden subsets within a cage. The logic is more complex than for regular naked subsets since only the mandatory inclusions should be taken into consideration. In this example, the cage 12/4 can be {1236|1245} = {12(36|45)}. Whichever the case, it must include {12}. Because {12} cannot go in the bottom row of the cage 12/4, there are only two cells where {12} may go. Two cells for two mandatory values, this forms a hidden pair. So the two cells in the top row of cage 12/4 must be {12} and cannot have any other value.