Kakuro Combinations

The magic block table — clues with exactly one combination

Some clue-and-length pairs allow exactly one set of digits. Solvers call them magic blocks (or unique combinations), and they are the entry points of every Kakuro: you may not know the order of the digits yet, but you know precisely which digits are present. Two rules generate the long tail of the table: an 8-cell run's missing digit is always 45 minus the clue, and a 9-cell run is always all nine digits summing to 45.

How do you use combinations to solve Kakuro?

The tables above are not trivia — they are the whole strategy of the game. Every technique below is a different way of asking "which combinations are still alive?". These are also exactly the techniques our grader replays to assign every puzzle its easy, medium, or hard label.

Unique combinations (magic blocks)

Scan any new board for the starred clues first: 3 and 4 in two cells, 6 and 7 in three, 16 and 17 in two, 23 and 24 in three, and their friends. A magic block tells you the exact digit set immediately; the crossing clues then tell you the order. Two magic blocks crossing each other usually resolve a cell on the spot — the intersection of {1,2} and {1,3} can only be 1.

Cross-referencing

Every white cell answers to two clues. List (or just picture) the digits its across clue allows and the digits its down clue allows — the cell can only hold the intersection. This is the bread-and-butter move of Kakuro, and it is what the in-game helper automates: tap either clue and see the live lists side by side.

Min/max bounding

When a run is partially filled, its remaining cells must cover exactly the remaining sum. Each empty cell is bounded by what the others can still contribute: in a 12-over-two run where one cell is at least 3, the other can be at most 9 and at least... whatever the first cannot cover. Formally: a cell's maximum is the remaining sum minus the minimum of its partners, and its minimum is the remaining sum minus their maximum. Applied run by run, this trims candidates surprisingly fast.

Combinatoric elimination

As digits land, revisit each clue's combination list and strike the sets that are no longer possible — because they omit a digit already placed in the run, or demand a digit no empty cell can take. Whatever digits appear in every surviving combination must be placed somewhere in the run; a digit appearing in no surviving combination can be erased from all its cells.

The box technique

An area-sum trick for hard boards: pick a region and add up all the across clues whose runs live entirely inside it, then all the down clues the same way. If one run pokes a single cell out of the region, the difference between the two totals is exactly that cell's value. It feels like magic the first time — it is just arithmetic over the region's cells counted two ways.

Naked and hidden pairs

Sudoku players know these: if two cells of a run hold exactly the same two candidates, those digits are locked into that pair (naked pair) and vanish from the run's other cells. Conversely, if two digits can only live in the same two cells of a run, everything else in those cells goes (hidden pair). On our boards, needing pair logic is precisely what earns a puzzle the hard label.

Ready to apply them? Play today's daily Kakuro, or pick a size from the puzzle library.