Cage combinations

A killer cage is a small addition problem with two extra rules: the digits are distinct, and they also have to satisfy sudoku. That second rule is what makes a short list of combinations so powerful. Before you write a digit, write the list.

Two cells summing to 3 can only be 1 and 2. Two cells summing to 4 can only be 1 and 3, because 2 and 2 would repeat. At the other end, 16 is 7 and 9, and 17 is 8 and 9. Three cells have the same kind of unique ends: 6 is 1+2+3, 7 is 1+2+4, 23 is 6+8+9, and 24 is 7+8+9. Four cells start getting unique at 10 (1+2+3+4) and 30 (6+7+8+9).

Those unique cages are the opening of almost every easy puzzle. Pencil both digits into both cells. You have not solved the cage yet, but every other digit is gone, and the rest of the row, column, and box can feel that absence. If one of those two cells already sees a 1, the other digit is placed at once.

Most cages are not unique, and that is where the work is. A three-cell 13 can be 1+3+9, 1+4+8, 1+5+7, 2+3+8, 2+4+7, 2+5+6, or 3+4+6. The moment the cage already contains a 1, every combination without a 1 drops out, and the other two cells are adding to 12. If the column already holds a 4, combinations that need a 4 in that column drop out too. You are not guessing. You are deleting combinations that the grid has made impossible.

Keep the list short enough to see. If a cage still has more than four combinations, leave it and clear a unique cage somewhere else. Cleared digits come back as new deletions. On iLoveSudoku the selected cage shows this list beside the board, with impossible combinations struck through, and the full table lives on the combinations page.

A useful habit: when a digit appears in every remaining combination, it is locked inside the cage. If those possible cells all sit in one row, the digit is also locked in that row, and you can erase it from the rest of the row. That single observation is the bridge from beginner combinations to the harder cage-locking ideas.

Use this technique on a puzzle

Keep the combination table open while you read, and the rules if the cages themselves are still new.