COMBINATION NARROWING — BEYOND THE FORCED CASES
Most solvers memorise the fully forced combinations — the 2-cell runs summing to 3 (only {1,2}) or 17 (only {8,9}), the 3-cell run summing to 6 (only {1,2,3}). But the technique extends to nearly-forced combinations that don't get memorised because they have two or three valid sets instead of one.
A 3-cell run summing to 7 has only two options: {1,2,4} and {1,3,3} — but the no-repeat rule eliminates the second, leaving only {1,2,4}. That's forced. A 3-cell run summing to 24 has only {7,8,9}. A 4-cell run summing to 11 has only {1,2,3,5}. These aren't zero-choice but they restrict the candidate set to a tight union.
When a run has only two or three valid combinations, the union of those combinations is itself a constraint. A 3-cell run summing to 8 has three options: {1,3,4}, {1,2,5}, {2,3,4} — but note that 6, 7, 8, 9 appear in none of them. Every crossing run that shares a cell with this run can have 6–9 removed from that cell's candidates.
Combination narrowing is the same logic as forced combinations, just applied to the union rather than a single set. It does substantial work in medium puzzles where nothing is fully forced.
LOCKED SETS — THE KAKURO EQUIVALENT OF NAKED PAIRS
In a run, if two cells are constrained to exactly the same two candidates — say both can only be {2, 7} — those two digits are locked into those cells. Every other cell in the same run can have 2 and 7 removed from its candidates. This is the Kakuro equivalent of a naked pair in Sudoku.
The logic extends to triples: if three cells in a run can only hold candidates from the same set of three digits, those three digits are locked to those cells. The rest of the run can eliminate all three.
Locked sets appear frequently in medium and hard Kakuro where the initial combination narrowing leaves runs with multiple valid combinations but cells with restricted candidates. Look for cells where cross-referencing has narrowed options to two or three digits — groups of those cells forming locked sets are where the cascade restarts.
SUM-REGION ARITHMETIC — THE KAKURO VERSION OF THE 45 RULE
In Killer Sudoku, every row sums to 45. Kakuro has no equivalent global structure — but it has a local one. Within any rectangular region of the grid, the clue sums constrain what must flow in and out of that region.
Here's the technique: select a set of runs where most cells fall inside a defined boundary, with only a few cells crossing out. Sum all the clues for runs entirely inside the boundary, then consider the runs that straddle it. The arithmetic tells you exactly what the crossing cells must contribute — often narrowing them to a forced combination without scanning the cells directly.
A practical version: find a row-like band of white cells where all horizontal runs are entirely contained, and one vertical run exits the band. Sum the horizontal clues. The cells in the vertical run inside the band must contribute the remainder of that region's implied sum. This often pins a 2- or 3-cell section of the vertical run directly.
This is the most underused technique in Kakuro. Most solvers treat each run independently. But runs share a grid, and the grid imposes arithmetic constraints on groups of runs just as it does in Killer Sudoku.
CASCADE ORDERING — WHICH RUNS TO RESOLVE FIRST
Hard Kakuro is solvable by constraint elimination, but the order in which you apply eliminations matters. Processing the wrong run first leaves you with a minimal narrowing; processing the right run first triggers a cascade that resolves five cells in a row.
The general rule: target runs with the fewest valid combinations first, not the runs with the most cells placed. A long run with one combination is more powerful than a short run with three combinations, because every crossing run benefits immediately.
Concretely: scan all runs and compute how many valid combination sets remain. Order them ascending. Work from the most constrained. When you resolve or narrow a run, recalculate the constraint count for all crossing runs. The most constrained run in the updated grid may be different from the most constrained run a moment ago.
This sounds mechanical, but it reflects something real: the cascade in Kakuro is not uniform. Some cells are hubs — four runs cross through a dense section of the grid, so resolving one propagates to all four. Find those hubs and prioritise the runs that run through them.
HOW THESE TECHNIQUES COMBINE
In practice, medium and hard Kakuro is solved by cycling through all four tools in order: first, apply combination narrowing across every run to establish the union of valid digits per cell. Second, look for locked sets within each run and propagate their eliminations. Third, apply sum-region arithmetic to any boundaries where clues cluster inside. Fourth, pick the most constrained run and fully resolve it, then re-examine everything that run crosses.
Repeat the cycle. Each pass reduces the candidate lists, which tightens the combination counts, which surfaces new locked sets and new forced cells. Hard Kakuro puzzles are typically three or four passes from start to finish — the cascade isn't instant, but it is steady.
The only Kakuro position that genuinely requires trial-and-error is a puzzle with two symmetric solutions — which a well-constructed puzzle won't have. If you reach a point where no technique applies, re-check combination narrowing. A skipped near-forced run is almost always the bottleneck.
THE COMBINATIONS WORTH MEMORISING
You don't need to memorise all valid combinations — there are hundreds. But the combinations that appear repeatedly in constrained positions are worth having automatic:
- 2 cells: sums 3 (1+2), 4 (1+3), 16 (7+9), 17 (8+9).
- 3 cells: sums 6 (1+2+3), 7 (1+2+4), 23 (6+8+9), 24 (7+8+9).
- 4 cells: sums 10 (1+2+3+4), 11 (1+2+3+5), 29 (5+7+8+9), 30 (6+7+8+9).
- 5 cells: sums 15 (1+2+3+4+5), 16 (1+2+3+4+6), 34 (4+6+7+8+9), 35 (5+6+7+8+9).
For the intermediate sums — the ones with several valid combinations — what matters is not the full list but the union of which digits can and cannot appear. 3-cell sum 15 can use any digit 1–9. 3-cell sum 7 can only use digits 1–4. Knowing the digit range for common sums is faster than listing combinations.