Killer Sudoku Cages: Sums, 45s and Combinations

1. Keep the Sudoku spine under the sums

Killer Sudoku never abandons its roots: rows, columns and 3x3 boxes each take 1 to 9 exactly once, and the cage totals sit on top of that skeleton. A one-cell cage is simply a locked given digit. Because the two cells of a two-cell cage always lie side by side, they can never repeat either — so a pair that must total 10 is never 5 and 5.

2. Memorise the one-way cage totals

Small sums leave the solver no choice, and recognising those on sight is the first skill to drill. Two cells adding to 3 can only be 1+2; to 4, only 1+3; to 16, only 7+9; to 17, only 8+9. Three cells that total 6, 7, 23 or 24 are just as forced — 1+2+3, 1+2+4, 6+8+9 and 7+8+9 respectively.

• Jot the surviving combination into notes and let the surrounding house decide which digit lands in which cell.
• A three-cell cage totalling 8 still has options such as 1+2+5 or 1+3+4, so open with the forced totals above before the flexible ones.

3. Apply the 45 rule to whole houses

Every row, column and box contains the digits 1 to 9 exactly once, so each of them sums to a fixed 45. When a cluster of whole cages sits inside a single row and covers eight of its cells, the one cell left uncovered equals 45 minus the total of those cages. That lone leftover cell is called an innie, and it is often the first digit you can place on a hard board.

4. Read the outies that poke out of a house

The mirror trick — the outie — pays off when cages refuse to stay inside one house. If a collection of cages covers an entire row and exactly one of them reaches a single cell beyond it, add every cage total in the collection and subtract 45: the remainder is the digit sitting outside the row. Boxes obey the same arithmetic, and that is where the hardest boards hide their first placements.

5. Treat red as early-warning radar

The game paints a cell red when its digit repeats in a row, column or box, or when its cage has already overshot its total or finished on the wrong sum. Red is free information: it catches an overspent cage while you still have room to backtrack. Check the extremes yourself as you fill — with e empty cells left in a cage, the target must sit between the filled total plus e and the filled total plus 9 times e, or the cage is already dead.

6. Open from the boxes that are nearly claimed

Look first for a 3x3 box that is almost entirely swallowed by whole cages, then let the 45 rule resolve its leftovers and feed the neighbouring boxes. Difficulty comes mostly from how many locked digits you are handed: Easy typically opens with eight to fourteen single-cell givens, Medium with four to nine, and Hard with as few as one to seven, so hard boards lean far more heavily on cage arithmetic from the first move.

7. Return to pure Sudoku when the sums stall

Every digit you place is simultaneously a Sudoku digit, so the classic toolkit still works when the cages go quiet: a hidden single in a near-full row, a candidate that fits only one cell of a box. Solve the placement first and the cage totals usually fall into line. Stuck anyway, Hint places the next correct digit — pause long enough to see the reasoning, because that insight carries over to the next cage and the next board.