Strategies for Popular Number Puzzles

The Killer Cage Convention

Killer Sudoku 'dog legs'
Killer Sudoku 'dog legs'

In Killer Sudoku clues are placed inside 'cages' - which ring a certain number of cells. The clue number is the sum of the solutions to those cells. It is theoretically possible for a cage to contain two of the same numbers if the cage forms an 'L' shape or 'dog-leg' over two boxes. Examples are to the right in bold red and green.

From the earliest days of Killer Sudoku there has been a convention that no cage should contain duplicate numbers - which applies to these special types of cage. If a cage could contain duplicate numbers then it would greatly increase the number of combinations of solutions.

There are six 'dog-legs' in this example, some have A-A, B-B etc showing where possible duplicate numbers could go - but do not.

All the puzzles on this site and ones I publish conform to this convention, but you might find other publishers ignore this constraint. If you enter such a puzzle into the Killer Solver it will not work.

The convention does not apply to KenKen or Kendoku as designed by the people at



... by: Brett Yarberry

Friday 19-Jun-2015
One strategy I use that relies on this convention is using Cages that from ALSs. Basically, if all the cells in a cage form an ALS, then the candidate equal to the difference between the sum of the numbers and the total in the cage can be removed from all the cells in that cage, reducing it to a locked set. For example, if you reduce a 3-cell cage with a sum of 17 to only the candidates 2,5,6,9, then 5 can be removed because (2+5+6+9)-17=22-17=5. The reason this works is because if that candidate were true, then it would cause at least one of the other candidates to equal another in the cage or, sometimes no combinations that would create the correct sum. The strategy works for both cases. I am not sure how to explain why it always works, but it seams to in all cases that I have tested.

... by: David N. Jansen

Friday 25-May-2012
In the Killer Sudoku of May 25, it seems that the solver does not fully implement this convention. I loaded the Killer Sudoku into the solver and stepped through the solution. At some point, the solver finds out that C3 is 8 and immediately deletes that number from C4, but not from B4.
Andrew Stuart writes:
Interesting point. The puzzles are generated with the convention in mind and the solver is aware of it in terms of the combinations available to that cell, for example in the mouse hover over of that cell its says 7/8/9. (Actually to get 24 you need 9/7/9 as the only possible 'dog leg' cage combo excluded by the convention). But the point you describe is the clear-off of candidates after the placement of a solution. To keep things simple only the candidates in rows, columns and boxes are immediately removed. Those in a cell have to wait for the 'easy combinations' strategy. It's a valid point to assert that cage combinations should also be refreshed but that would require 'easy combinations' to be right at the top of the list. But I suppose if I refreshed the cage combinations just for those cages affected it might make it closer to how a human solves the puzzle.

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