How do you solve a Kakuro puzzle?

Start by scanning every run for forced combinations — runs where only one digit set is mathematically possible for the given sum and length. Fill those cells, mark candidates for the rest, then apply cross-run intersection to eliminate digits that cannot satisfy both the across and down constraints at each cell. Repeat until the puzzle is solved.

What is the first step in solving Kakuro?

The first step is identifying forced combinations — runs with a unique valid digit set. A 2-cell run summing to 3 must be {1, 2}; a 3-cell run summing to 24 must be {7, 8, 9}. These are free placements that cost no analysis and tighten the rest of the grid immediately.

Do you need to guess to solve Kakuro?

No. Every well-formed Kakuro puzzle has a unique solution reachable entirely by logic. If you feel stuck, it usually means a technique has not been fully applied — try cross-run intersection more carefully, or check whether residual sums after recent placements have opened a new forced combination.

What is cross-run intersection in Kakuro?

Cross-run intersection means using the fact that each white cell belongs to two runs simultaneously — one across and one down. Any digit placed in that cell must satisfy both run's constraints. If a digit is ruled out by one run, it is ruled out for that cell entirely, eliminating it from the other run's candidate set as well.

What are residual sums in Kakuro?

A residual sum is the remaining target for a partially filled run. If a 4-cell run has a clue of 20 and you have already placed a 5 in one cell, the remaining 3 cells must sum to 15 with no repeats. That sub-run often has far fewer combinations than the original, unlocking new forced moves.

How do I solve harder Kakuro puzzles?

Hard and ultra-hard Kakuro puzzles require advanced techniques beyond basic intersection. Locked sets identify groups of cells whose candidates form a closed subset, allowing those digits to be removed from other cells in the same runs. Combination pruning eliminates digit combinations that cannot coexist with placements in crossing runs. The technique library at freekakuro.com/techniques/ covers each method in detail.

6-step method · Works on all difficulty levels

How to Solve Kakuro

Every Kakuro puzzle has exactly one solution reachable by logic alone — no guessing required. This guide walks through the six steps of a complete solving method, from the first forced combination to the final cell. Use it alongside Free Kakuro to practice each step on a real puzzle.

Solve an easy puzzle Combination reference

The short answer

To solve a Kakuro puzzle: (1) find forced combinations — runs with only one valid digit set. (2) Mark candidates for every remaining cell. (3) Use cross-run intersection to eliminate candidates that cannot satisfy both the across and down constraints. (4) Place cells with one candidate left. (5) Recalculate residual sums after each placement. (6) Apply locked sets or combination pruning if steps 1–5 stall.

6-step Kakuro solving method

1
Scan for forced combinations

Before marking anything, look up every run in the combination reference. Some runs have exactly one valid digit set for their sum and length — fill those immediately. They are free points that cost nothing to figure out.

Example

A 2-cell run summing to 3 can only be {1, 2}. A 3-cell run summing to 6 can only be {1, 2, 3}. A 2-cell run summing to 17 can only be {8, 9}.
2
Enable candidate mode and mark every cell

Turn on candidate mode in Free Kakuro (or pencil small digits into corners on paper). For each empty cell, list which digits from its run's valid combinations remain available after excluding digits already placed in the same run.

Example

If a 3-cell run summing to 16 has possible sets {2,5,9}, {3,4,9}, {3,6,7}, {4,5,7} and one cell in that run already has a 4, eliminate every set that does not contain 4 — leaving {3,4,9} and {4,5,7} — and update the remaining cells' candidates accordingly.
3
Apply cross-run intersection

Each empty cell sits at the crossing of an across run and a down run. The digit must satisfy both. If a digit is not in the candidate set for the across run, remove it from the down run at that cell, and vice versa. Repeat across all intersections — this cascade of eliminations resolves most medium-difficulty grids.

Example

An across cell's candidates are {2, 5}. The crossing down run's candidates for that cell are {3, 5, 7}. The intersection is {5}, so the cell must be 5. That placement then removes 5 from every other cell in both runs.
4
Place single-candidate cells

After each intersection pass, look for cells where only one candidate remains. Place those digits. Each placement may eliminate candidates in adjacent cells of the same runs, triggering a chain of forced moves.

Example

After removing conflicting candidates from step 3, a cell whose candidates narrowed to {3} can be placed immediately. That 3 changes the residual target of both its runs, often revealing another single-candidate cell.
5
Recalculate residual sums

As cells fill in, recompute the remaining sum for each run's unfilled cells. Subtract placed values from the run's clue. The shorter sub-run often has a much tighter combination space — even a new forced combination may appear.

Example

A 5-cell run with clue 25 has had two cells filled with 3 and 6. The residual is 16 across 3 cells. The only 3-cell combinations summing to 16 are {1,6,9}, {2,5,9}, {3,4,9}, {3,6,7}, {4,5,7}. Since 3 and 6 are already used in this run, all sets containing 3 or 6 are eliminated — leaving {2,5,9} and {4,5,7}. The 5 is now forced in one of the remaining cells based on which candidate the crossing runs allow.
6
Use advanced techniques when stuck

If steps 1–5 do not resolve a cell, apply locked sets, combination pruning, or min/max boundary forcing. These are detailed in the technique library.

Example

If two cells in a run have candidates {2,7} and no other cell in the run can hold 2 or 7, those two cells form a locked set — remove 2 and 7 from every other cell in that run. This often unblocks cells that seemed stuck.

Tools that make solving easier

Combination reference

Every valid digit set for every sum–length pair. Check it before step 1 to find forced combinations quickly, and return to it during step 5 for residual sums. Open the full chart →

Kakuro helper calculator

Enter a run length, clue sum, required digits, and excluded digits to instantly filter valid combinations. Ideal when crossing runs have eliminated several options. Open the helper →

Candidate mode

Free Kakuro's built-in candidate toggle displays a 3×3 digit grid inside each cell showing which values remain possible. This automates step 2 so you can focus on the logic rather than bookkeeping.

Hint button

When stuck, press Hint to reveal the next forced cell and a brief explanation of why. This is useful for understanding which step of the method applies without spoiling the rest of the solve.

Common mistakes when solving Kakuro

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Repeating a digit within a run. The no-repeat rule applies per run, not globally. The same digit can appear in different runs, but never twice in the same across or down sequence.
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Forgetting to update candidates after a placement. Each time a digit is placed, remove it from the candidates of every other empty cell in the same across and down run. Stale candidates cause missed deductions.
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Skipping residual sums. Solvers who feel stuck often have not recalculated what the partially filled runs still need. A 2-cell residual often has a forced combination not visible when the run was longer.
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Guessing when deduction stalls. If the standard steps have not resolved a cell, the next move is a technique, not a guess. Try applying locked-set analysis before assuming the puzzle requires trial and error.

Advanced techniques for harder puzzles

The six-step method above resolves most easy and medium puzzles. Hard and ultra-hard grids require additional tools. Each technique below is covered in depth in the technique library.