It is the same mathematics as the printed killer sudoku cheat sheet, with the crossing-out done for you. That makes it most useful on the busy middle totals, where a four-cell cage might start with a dozen possible sets.
How to use it
- Pick the cage size: count the cells inside the dotted outline.
- Pick the sum printed in the cage's corner.
- Tick a digit as included only when you have proved the cage contains it, for example because it is the only place left for that digit in a row, column or box.
- Tick a digit as excluded only when no cell of the cage can hold it. A 7 elsewhere in the row of one cage cell is not enough if another cage cell sits in a different row.
- Read the list. Digits that appear in every remaining set are certain; digits that appear in none can come out of every cell in the cage.
Worked example: a four-cell 18
A four-cell cage of 18 has 11 possible sets, which on its own is useless. Now suppose the box it sits in has only one spot left for its 9, and that spot is inside this cage. Tick 9 as included and the list drops to three: 1+2+6+9, 1+3+5+9 and 2+3+4+9.
Next, say the box already holds a 2 elsewhere. Since the whole cage sits inside that box, tick 2 as excluded and only 1+3+5+9 survives. The cage's four digits are now fixed, and the cage's neighbors in the rows and columns usually place them within a move or two.
Worked example: trimming a three-cell 15
A three-cell 15 can be made eight ways. If the column it sits in already has a 5, excluding 5 removes four of them and leaves 1+6+8, 2+4+9, 2+6+7 and 3+4+8. No digit is certain yet, but 5 is gone for good and your pencil marks get shorter.
Smaller cages trim the same way. A two-cell 10 is 1+9, 2+8, 3+7 or 4+6. Exclude 1 and 5 and three pairs remain, 2+8, 3+7 and 4+6. The 5 never mattered, since 5+5 repeats a digit and was never allowed.
How combinations meet rows, columns and boxes
When a digit appears in every surviving set, the cage must contain it. If the cage lies wholly in one row, that digit can then be crossed out of the rest of the row, and the same goes for a column or box. A three-cell 22 in one column is 5+8+9 or 6+7+9, so the 9 in that column is inside the cage and nowhere else.
It works the other way too. If the only cells in a box that can take a 6 all belong to one cage, the cage must hold a 6, and you can tick it as included. Our grader calls these moves cage/house interaction, and they are the first technique beyond simple combinations that medium puzzles ask for.
Keep an eye on cages that bend across two boxes or rows. Exclusions apply to the cage as a whole only when every one of its cells is blocked from the digit. When just one cell is blocked, leave the box unticked and make the note in that cell's pencil marks instead.
Where to practice
Easy puzzles mostly need the unique cages, so the calculator matters more as you climb. Try it on the hard killer sudoku puzzles, where the rule of 45 produces leftover sums of three cells or more and the middle totals rarely give anything away at first glance. If you are new to cages, how to play killer sudoku covers the rules and first moves.