Guides

Killer Sudoku combinations: use cage sums without guessing

Killer Sudoku cages are useful because a total plus a number of cells sharply limits the possible digits. The order of those digits is not known yet, and a digit may not repeat inside a cage. Treat each cage as a small set problem: list only the distinct digit sets that reach its sum, then use the row, column and box to remove sets that cannot fit. This is much safer than trying numbers or memorising a long list of sequences.

Read a cage as a set, not as an order

A two-cell cage totalling 3 must contain 1 and 2, but either cell may hold either digit until another rule decides. A two-cell cage totalling 15 must contain 6 and 9; 7 and 8 are impossible because they total 15 too, but 7+8 is also 15—so both sets must be considered. Always write digit sets in ascending order and do not duplicate a set by reversing it. The cage rule bans repeated digits, so 5+5 is never a valid two-cell 10.

Start with totals at the extremes

Extreme sums have the fewest combinations. For two cells, 3 gives only 1+2 and 17 gives only 8+9. For three cells, 6 is 1+2+3 and 24 is 7+8+9. For four cells, 10 is 1+2+3+4 and 30 is 6+7+8+9. These cages are powerful because they give exact candidates immediately. Mark them first, then look for row, column or box clues that decide their order.

Use the grid to eliminate an entire combination

Suppose a three-cell cage totals 15. One possible set is 1+5+9, another is 2+4+9, and others may exist. If a row already contains 9, every set containing 9 disappears for the cage cell in that row. If a box cannot accept 1 in any cage cell, remove every set containing 1. Do not cross out a digit from just one cell without a unit-based reason; cross out a whole set only when that set cannot be placed anywhere in the cage.

Keep notes compact enough to compare

For a small cage, record the remaining sets beside the cage or as candidates in its cells. For a larger cage, begin with the possible sets, not all their permutations. Then reduce the sets using the cells' rows, columns and boxes. If only one set remains, its digits are locked into that cage and can sometimes be removed from other cells of a shared unit. The aim is a short, auditable record—not a calculation spreadsheet.

Know when a sum alone is not enough

Middle sums often have several valid sets, so a cage total may not place a digit by itself. That is normal. Pair it with the no-repeat rule, nearby given digits, another cage crossing the same unit, or the rule of 45 for a complete row, column or box. A useful Killer deduction combines constraints. When the combination list stays broad, leave the cage marked and work on a more restrictive one rather than forcing a choice.

Turn a sum into a proven set of digits

Open the cage-combinations lesson for a clickable proof and targeted practice, then try an easy Killer puzzle where small cages make the first deductions visible immediately.

Killer Sudoku Learning path Play Easy

Cage combinations No repeats in a cage Cage elimination Cage-Line Reduction

Frequently asked questions

Can a digit repeat in a Killer Sudoku cage?

No. Digits inside one cage must all be different, in addition to the normal row, column and box rules. This is why a two-cell cage totalling 10 cannot be 5 and 5.

Does a cage sum tell me the order of its digits?

Usually not. A sum gives a set of possible digits. The row, column, box and other cage constraints decide where each digit goes. Treating a set as an order is a common source of mistakes.

Which Killer Sudoku cages should I inspect first?

Start with single-cell cages and small cages with extreme totals, such as two cells totalling 3 or 17 and three cells totalling 6 or 24. They have the fewest possible digit sets.

Should I memorise every cage combination?

Memorise the most restrictive extremes first. For the rest, calculate or check the remaining sets carefully, then use the grid to reduce them. Understanding why a set is removed is more useful than memorising a long table.