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# Killer Combinations

From sudokuwiki.org, the puzzle solver's site

Each clue in each cage will give rise to a certain set of combinations of numbers for each cell. It is a convention of Killer Sudoku that no cage will duplicate a number (even if it dog-legs through several 3 by 3 boxes).

Some cages will not be very helpful but certain combinations, such as these listed here mean that the total number of possible numbers is equal to the cage size. In the first, for example, a clue of 3 in a cage of size 2 can only have 1 and 2 in it. These means you can remove 3 to 9 from those cells.

I have written an example of the use of 'hard' Killer combinations on this page.

 Clue Cage Size Combinations 3 2 1,2 4 2 1,3 16 2 7,9 17 2 8,9 6 3 1,2,3 7 3 1,2,4 23 3 6,8,9 24 3 7,8,9 10 4 1,2,3,4 11 4 1,2,3,5 29 4 5,7,8,9 30 4 6,7,8,9 15 5 1,2,3,4,5 16 5 1,2,3,4,6 34 5 4,6,7,8,9 35 5 5,6,7,8,9 21 6 1,2,3,4,5,6 22 6 1,2,3,4,5,7 38 6 3,5,6,7,8,9 39 6 4,5,6,7,8,9 28 7 1,2,3,4,5,6,7 29 7 1,2,3,4,5,6,8 41 7 2,4,5,6,7,8,9 42 7 3,4,5,6,7,8,9 Any 8 1-9 except 45minus clue value Any 9 1-9 inclusive

The absolutely exhaustive combination of all cage clues, cage sizes
and the combination of combinations for those cage size/values is:

 Cage Size Cage Value Combination Combos Combined 2 3 1/2 Unique 2 4 1/3 Unique 2 5 1/4 2 5 2/3 1/2/3/4 2 6 1/5 2 6 2/4 1/2/4/5 2 7 1/6 2 7 2/5 2 7 3/4 1/2/3/4/5/6 2 8 1/7 2 8 2/6 2 8 3/5 1/2/3/5/6/7 2 9 1/8 2 9 2/7 2 9 3/6 2 9 4/5 1/2/3/4/5/6/7/8 2 10 1/9 2 10 2/8 2 10 3/7 2 10 4/6 1/2/3/4/6/7/8/9 2 11 2/9 2 11 3/8 2 11 4/7 2 11 5/6 2/3/4/5/6/7/8/9 2 12 3/9 2 12 4/8 2 12 5/7 3/4/5/7/8/9 2 13 4/9 2 13 5/8 2 13 6/7 4/5/6/7/8/9 2 14 5/9 2 14 6/8 5/6/8/9 2 15 6/9 2 15 7/8 6/7/8/9 2 16 7/9 Unique 2 17 8/9 Unique 3 6 1/2/3 Unique 3 7 1/2/4 Unique 3 8 1/2/5 3 8 1/3/4 1/2/3/4/5 3 9 1/2/6 3 9 1/3/5 3 9 2/3/4 1/2/3/4/5/6 3 10 1/2/7 3 10 1/3/6 3 10 1/4/5 3 10 2/3/5 1/2/3/4/5/6/7 3 11 1/2/8 3 11 1/3/7 3 11 1/4/6 3 11 2/3/6 3 11 2/4/5 1/2/3/4/5/6/7/8 3 12 1/2/9 3 12 1/3/8 3 12 1/4/7 3 12 1/5/6 3 12 2/3/7 3 12 2/4/6 3 12 3/4/5 1/2/3/4/5/6/7/8/9 3 13 1/3/9 3 13 1/4/8 3 13 1/5/7 3 13 2/3/8 3 13 2/4/7 3 13 2/5/6 3 13 3/4/6 1/2/3/4/5/6/7/8/9 3 14 1/4/9 3 14 1/5/8 3 14 1/6/7 3 14 2/3/9 3 14 2/4/8 3 14 2/5/7 3 14 3/4/7 3 14 3/5/6 1/2/3/4/5/6/7/8/9 3 15 1/5/9 3 15 1/6/8 3 15 2/4/9 3 15 2/5/8 3 15 2/6/7 3 15 3/4/8 3 15 3/5/7 3 15 4/5/6 1/2/3/4/5/6/7/8/9 3 16 1/6/9 3 16 1/7/8 3 16 2/5/9 3 16 2/6/8 3 16 3/4/9 3 16 3/5/8 3 16 3/6/7 3 16 4/5/7 1/2/3/4/5/6/7/8/9 3 17 1/7/9 3 17 2/6/9 3 17 2/7/8 3 17 3/5/9 3 17 3/6/8 3 17 4/5/8 3 17 4/6/7 1/2/3/4/5/6/7/8/9 3 18 1/8/9 3 18 2/7/9 3 18 3/6/9 3 18 3/7/8 3 18 4/5/9 3 18 4/6/8 3 18 5/6/7 1/2/3/4/5/6/7/8/9 3 19 2/8/9 3 19 3/7/9 3 19 4/6/9 3 19 4/7/8 3 19 5/6/8 2/3/4/5/6/7/8/9 3 20 3/8/9 3 20 4/7/9 3 20 5/6/9 3 20 5/7/8 3/4/5/6/7/8/9 3 21 4/8/9 3 21 5/7/9 3 21 6/7/8 4/5/6/7/8/9 3 22 5/8/9 3 22 6/7/9 5/6/7/8/9 3 23 6/8/9 Unique 3 24 7/8/9 Unique 4 10 1/2/3/4 Unique 4 11 1/2/3/5 Unique 4 12 1/2/3/6 4 12 1/2/4/5 1/2/3/4/5/6 4 13 1/2/3/7 4 13 1/2/4/6 4 13 1/3/4/5 1/2/3/4/5/6/7 4 14 1/2/3/8 4 14 1/2/4/7 4 14 1/2/5/6 4 14 1/3/4/6 4 14 2/3/4/5 1/2/3/4/5/6/7/8 4 15 1/2/3/9 4 15 1/2/4/8 4 15 1/2/5/7 4 15 1/3/4/7 4 15 1/3/5/6 4 15 2/3/4/6 1/2/3/4/5/6/7/8/9 4 16 1/2/4/9 4 16 1/2/5/8 4 16 1/2/6/7 4 16 1/3/4/8 4 16 1/3/5/7 4 16 1/4/5/6 4 16 2/3/4/7 4 16 2/3/5/6 1/2/3/4/5/6/7/8/9 4 17 1/2/5/9 4 17 1/2/6/8 4 17 1/3/4/9 4 17 1/3/5/8 4 17 1/3/6/7 4 17 1/4/5/7 4 17 2/3/4/8 4 17 2/3/5/7 4 17 2/4/5/6 1/2/3/4/5/6/7/8/9 4 18 1/2/6/9 4 18 1/2/7/8 4 18 1/3/5/9 4 18 1/3/6/8 4 18 1/4/5/8 4 18 1/4/6/7 4 18 2/3/4/9 4 18 2/3/5/8 4 18 2/3/6/7 4 18 2/4/5/7 4 18 3/4/5/6 1/2/3/4/5/6/7/8/9 4 19 1/2/7/9 4 19 1/3/6/9 4 19 1/3/7/8 4 19 1/4/5/9 4 19 1/4/6/8 4 19 1/5/6/7 4 19 2/3/5/9 4 19 2/3/6/8 4 19 2/4/5/8 4 19 2/4/6/7 4 19 3/4/5/7 1/2/3/4/5/6/7/8/9 4 20 1/2/8/9 4 20 1/3/7/9 4 20 1/4/6/9 4 20 1/4/7/8 4 20 1/5/6/8 4 20 2/3/6/9 4 20 2/3/7/8 4 20 2/4/5/9 4 20 2/4/6/8 4 20 2/5/6/7 4 20 3/4/5/8 4 20 3/4/6/7 1/2/3/4/5/6/7/8/9 4 21 1/3/8/9 4 21 1/4/7/9 4 21 1/5/6/9 4 21 1/5/7/8 4 21 2/3/7/9 4 21 2/4/6/9 4 21 2/4/7/8 4 21 2/5/6/8 4 21 3/4/5/9 4 21 3/4/6/8 4 21 3/5/6/7 1/2/3/4/5/6/7/8/9 4 22 1/4/8/9 4 22 1/5/7/9 4 22 1/6/7/8 4 22 2/3/8/9 4 22 2/4/7/9 4 22 2/5/6/9 4 22 2/5/7/8 4 22 3/4/6/9 4 22 3/4/7/8 4 22 3/5/6/8 4 22 4/5/6/7 1/2/3/4/5/6/7/8/9 4 23 1/5/8/9 4 23 1/6/7/9 4 23 2/4/8/9 4 23 2/5/7/9 4 23 2/6/7/8 4 23 3/4/7/9 4 23 3/5/6/9 4 23 3/5/7/8 4 23 4/5/6/8 1/2/3/4/5/6/7/8/9 4 24 1/6/8/9 4 24 2/5/8/9 4 24 2/6/7/9 4 24 3/4/8/9 4 24 3/5/7/9 4 24 3/6/7/8 4 24 4/5/6/9 4 24 4/5/7/8 1/2/3/4/5/6/7/8/9 4 25 1/7/8/9 4 25 2/6/8/9 4 25 3/5/8/9 4 25 3/6/7/9 4 25 4/5/7/9 4 25 4/6/7/8 1/2/3/4/5/6/7/8/9 4 26 2/7/8/9 4 26 3/6/8/9 4 26 4/5/8/9 4 26 4/6/7/9 4 26 5/6/7/8 2/3/4/5/6/7/8/9 4 27 3/7/8/9 4 27 4/6/8/9 4 27 5/6/7/9 3/4/5/6/7/8/9 4 28 4/7/8/9 4 28 5/6/8/9 4/5/6/7/8/9 4 29 5/7/8/9 Unique 4 30 6/7/8/9 Unique 5 15 1/2/3/4/5 Unique 5 16 1/2/3/4/6 Unique 5 17 1/2/3/4/7 5 17 1/2/3/5/6 1/2/3/4/5/6/7 5 18 1/2/3/4/8 5 18 1/2/3/5/7 5 18 1/2/4/5/6 1/2/3/4/5/6/7/8 5 19 1/2/3/4/9 5 19 1/2/3/5/8 5 19 1/2/3/6/7 5 19 1/2/4/5/7 5 19 1/3/4/5/6 1/2/3/4/5/6/7/8/9 5 20 1/2/3/5/9 5 20 1/2/3/6/8 5 20 1/2/4/5/8 5 20 1/2/4/6/7 5 20 1/3/4/5/7 5 20 2/3/4/5/6 1/2/3/4/5/6/7/8/9 5 21 1/2/3/6/9 5 21 1/2/3/7/8 5 21 1/2/4/5/9 5 21 1/2/4/6/8 5 21 1/2/5/6/7 5 21 1/3/4/5/8 5 21 1/3/4/6/7 5 21 2/3/4/5/7 1/2/3/4/5/6/7/8/9 5 22 1/2/3/7/9 5 22 1/2/4/6/9 5 22 1/2/4/7/8 5 22 1/2/5/6/8 5 22 1/3/4/5/9 5 22 1/3/4/6/8 5 22 1/3/5/6/7 5 22 2/3/4/5/8 5 22 2/3/4/6/7 1/2/3/4/5/6/7/8/9 5 23 1/2/3/8/9 5 23 1/2/4/7/9 5 23 1/2/5/6/9 5 23 1/2/5/7/8 5 23 1/3/4/6/9 5 23 1/3/4/7/8 5 23 1/3/5/6/8 5 23 1/4/5/6/7 5 23 2/3/4/5/9 5 23 2/3/4/6/8 5 23 2/3/5/6/7 1/2/3/4/5/6/7/8/9 5 24 1/2/4/8/9 5 24 1/2/5/7/9 5 24 1/2/6/7/8 5 24 1/3/4/7/9 5 24 1/3/5/6/9 5 24 1/3/5/7/8 5 24 1/4/5/6/8 5 24 2/3/4/6/9 5 24 2/3/4/7/8 5 24 2/3/5/6/8 5 24 2/4/5/6/7 1/2/3/4/5/6/7/8/9 5 25 1/2/5/8/9 5 25 1/2/6/7/9 5 25 1/3/4/8/9 5 25 1/3/5/7/9 5 25 1/3/6/7/8 5 25 1/4/5/6/9 5 25 1/4/5/7/8 5 25 2/3/4/7/9 5 25 2/3/5/6/9 5 25 2/3/5/7/8 5 25 2/4/5/6/8 5 25 3/4/5/6/7 1/2/3/4/5/6/7/8/9 5 26 1/2/6/8/9 5 26 1/3/5/8/9 5 26 1/3/6/7/9 5 26 1/4/5/7/9 5 26 1/4/6/7/8 5 26 2/3/4/8/9 5 26 2/3/5/7/9 5 26 2/3/6/7/8 5 26 2/4/5/6/9 5 26 2/4/5/7/8 5 26 3/4/5/6/8 1/2/3/4/5/6/7/8/9 5 27 1/2/7/8/9 5 27 1/3/6/8/9 5 27 1/4/5/8/9 5 27 1/4/6/7/9 5 27 1/5/6/7/8 5 27 2/3/5/8/9 5 27 2/3/6/7/9 5 27 2/4/5/7/9 5 27 2/4/6/7/8 5 27 3/4/5/6/9 5 27 3/4/5/7/8 1/2/3/4/5/6/7/8/9 5 28 1/3/7/8/9 5 28 1/4/6/8/9 5 28 1/5/6/7/9 5 28 2/3/6/8/9 5 28 2/4/5/8/9 5 28 2/4/6/7/9 5 28 2/5/6/7/8 5 28 3/4/5/7/9 5 28 3/4/6/7/8 1/2/3/4/5/6/7/8/9 5 29 1/4/7/8/9 5 29 1/5/6/8/9 5 29 2/3/7/8/9 5 29 2/4/6/8/9 5 29 2/5/6/7/9 5 29 3/4/5/8/9 5 29 3/4/6/7/9 5 29 3/5/6/7/8 1/2/3/4/5/6/7/8/9 5 30 1/5/7/8/9 5 30 2/4/7/8/9 5 30 2/5/6/8/9 5 30 3/4/6/8/9 5 30 3/5/6/7/9 5 30 4/5/6/7/8 1/2/3/4/5/6/7/8/9 5 31 1/6/7/8/9 5 31 2/5/7/8/9 5 31 3/4/7/8/9 5 31 3/5/6/8/9 5 31 4/5/6/7/9 1/2/3/4/5/6/7/8/9 5 32 2/6/7/8/9 5 32 3/5/7/8/9 5 32 4/5/6/8/9 2/3/4/5/6/7/8/9 5 33 3/6/7/8/9 5 33 4/5/7/8/9 3/4/5/6/7/8/9 5 34 4/6/7/8/9 Unique 5 35 5/6/7/8/9 Unique 6 21 1/2/3/4/5/6 Unique 6 22 1/2/3/4/5/7 Unique 6 23 1/2/3/4/5/8 6 23 1/2/3/4/6/7 1/2/3/4/5/6/7/8 6 24 1/2/3/4/5/9 6 24 1/2/3/4/6/8 6 24 1/2/3/5/6/7 1/2/3/4/5/6/7/8/9 6 25 1/2/3/4/6/9 6 25 1/2/3/4/7/8 6 25 1/2/3/5/6/8 6 25 1/2/4/5/6/7 1/2/3/4/5/6/7/8/9 6 26 1/2/3/4/7/9 6 26 1/2/3/5/6/9 6 26 1/2/3/5/7/8 6 26 1/2/4/5/6/8 6 26 1/3/4/5/6/7 1/2/3/4/5/6/7/8/9 6 27 1/2/3/4/8/9 6 27 1/2/3/5/7/9 6 27 1/2/3/6/7/8 6 27 1/2/4/5/6/9 6 27 1/2/4/5/7/8 6 27 1/3/4/5/6/8 6 27 2/3/4/5/6/7 1/2/3/4/5/6/7/8/9 6 28 1/2/3/5/8/9 6 28 1/2/3/6/7/9 6 28 1/2/4/5/7/9 6 28 1/2/4/6/7/8 6 28 1/3/4/5/6/9 6 28 1/3/4/5/7/8 6 28 2/3/4/5/6/8 1/2/3/4/5/6/7/8/9 6 29 1/2/3/6/8/9 6 29 1/2/4/5/8/9 6 29 1/2/4/6/7/9 6 29 1/2/5/6/7/8 6 29 1/3/4/5/7/9 6 29 1/3/4/6/7/8 6 29 2/3/4/5/6/9 6 29 2/3/4/5/7/8 1/2/3/4/5/6/7/8/9 6 30 1/2/3/7/8/9 6 30 1/2/4/6/8/9 6 30 1/2/5/6/7/9 6 30 1/3/4/5/8/9 6 30 1/3/4/6/7/9 6 30 1/3/5/6/7/8 6 30 2/3/4/5/7/9 6 30 2/3/4/6/7/8 1/2/3/4/5/6/7/8/9 6 31 1/2/4/7/8/9 6 31 1/2/5/6/8/9 6 31 1/3/4/6/8/9 6 31 1/3/5/6/7/9 6 31 1/4/5/6/7/8 6 31 2/3/4/5/8/9 6 31 2/3/4/6/7/9 6 31 2/3/5/6/7/8 1/2/3/4/5/6/7/8/9 6 32 1/2/5/7/8/9 6 32 1/3/4/7/8/9 6 32 1/3/5/6/8/9 6 32 1/4/5/6/7/9 6 32 2/3/4/6/8/9 6 32 2/3/5/6/7/9 6 32 2/4/5/6/7/8 1/2/3/4/5/6/7/8/9 6 33 1/2/6/7/8/9 6 33 1/3/5/7/8/9 6 33 1/4/5/6/8/9 6 33 2/3/4/7/8/9 6 33 2/3/5/6/8/9 6 33 2/4/5/6/7/9 6 33 3/4/5/6/7/8 1/2/3/4/5/6/7/8/9 6 34 1/3/6/7/8/9 6 34 1/4/5/7/8/9 6 34 2/3/5/7/8/9 6 34 2/4/5/6/8/9 6 34 3/4/5/6/7/9 1/2/3/4/5/6/7/8/9 6 35 1/4/6/7/8/9 6 35 2/3/6/7/8/9 6 35 2/4/5/7/8/9 6 35 3/4/5/6/8/9 1/2/3/4/5/6/7/8/9 6 36 1/5/6/7/8/9 6 36 2/4/6/7/8/9 6 36 3/4/5/7/8/9 1/2/3/4/5/6/7/8/9 6 37 2/5/6/7/8/9 6 37 3/4/6/7/8/9 2/3/4/5/6/7/8/9 6 38 3/5/6/7/8/9 Unique 6 39 4/5/6/7/8/9 Unique 7 28 1/2/3/4/5/6/7 Unique 7 29 1/2/3/4/5/6/8 Unique 7 30 1/2/3/4/5/6/9 7 30 1/2/3/4/5/7/8 1/2/3/4/5/6/7/8/9 7 31 1/2/3/4/5/7/9 7 31 1/2/3/4/6/7/8 1/2/3/4/5/6/7/8/9 7 32 1/2/3/4/5/8/9 7 32 1/2/3/4/6/7/9 7 32 1/2/3/5/6/7/8 1/2/3/4/5/6/7/8/9 7 33 1/2/3/4/6/8/9 7 33 1/2/3/5/6/7/9 7 33 1/2/4/5/6/7/8 1/2/3/4/5/6/7/8/9 7 34 1/2/3/4/7/8/9 7 34 1/2/3/5/6/8/9 7 34 1/2/4/5/6/7/9 7 34 1/3/4/5/6/7/8 1/2/3/4/5/6/7/8/9 7 35 1/2/3/5/7/8/9 7 35 1/2/4/5/6/8/9 7 35 1/3/4/5/6/7/9 7 35 2/3/4/5/6/7/8 1/2/3/4/5/6/7/8/9 7 36 1/2/3/6/7/8/9 7 36 1/2/4/5/7/8/9 7 36 1/3/4/5/6/8/9 7 36 2/3/4/5/6/7/9 1/2/3/4/5/6/7/8/9 7 37 1/2/4/6/7/8/9 7 37 1/3/4/5/7/8/9 7 37 2/3/4/5/6/8/9 1/2/3/4/5/6/7/8/9 7 38 1/2/5/6/7/8/9 7 38 1/3/4/6/7/8/9 7 38 2/3/4/5/7/8/9 1/2/3/4/5/6/7/8/9 7 39 1/3/5/6/7/8/9 7 39 2/3/4/6/7/8/9 1/2/3/4/5/6/7/8/9 7 40 1/4/5/6/7/8/9 7 40 2/3/5/6/7/8/9 1/2/3/4/5/6/7/8/9 7 41 2/4/5/6/7/8/9 Unique 7 42 3/4/5/6/7/8/9 Unique 8 36 1/2/3/4/5/6/7/8 Unique 8 37 1/2/3/4/5/6/7/9 Unique 8 38 1/2/3/4/5/6/8/9 Unique 8 39 1/2/3/4/5/7/8/9 Unique 8 40 1/2/3/4/6/7/8/9 Unique 8 41 1/2/3/5/6/7/8/9 Unique 8 42 1/2/4/5/6/7/8/9 Unique 8 43 1/3/4/5/6/7/8/9 Unique 8 44 2/3/4/5/6/7/8/9 Unique