How a Kakuro Solver Reasons

Read this to understand how a kakuro puzzle is solved by pure logic: how a sum and a run length pin down a small set of digit combinations, how the across and down runs sharing a cell narrow each other, and how to read the technique log the solver prints.

What kakuro is, with one hook example

Kakuro (also called cross sums) is a grid of black and white cells. White cells hold digits from 1 to 9. Black cells carry clues: a number in the upper right is the sum of the white run to its right, and a number in the lower left is the sum of the white run below it. Inside any single run, no digit repeats.

Here is the whole game in one clue. Suppose a black cell shows an across sum of 16 and the run to its right is exactly two cells long. The two digits must be distinct, from 1 to 9, and add to 16. The only unordered pair that works is 7 and 9. So before you look at anything else, those two cells are limited to \{7, 9\}. That single fact, repeated and combined across the grid, is most of kakuro.

The solver at /games/kakuro-solver/ makes that reasoning explicit. Every time it fills or trims a cell, it names the clue and the rule that justified the move.

When to reach for the solver, and when not

Use the solver when you want to check that a puzzle you built has exactly one solution, when you are stuck and want the next logical step named rather than the full answer dumped on you, or when you are teaching and need a clean record of which technique cracked which clue.

Do not use it as a substitute for learning the combinations. The value of kakuro comes from recognizing that 3 in two cells is always 1+2, or that 45 in nine cells is the full set 1 through 9. If you let the solver do every step, you never build that memory. Run it after you attempt a section, then compare its log to your own reasoning.

The combination formula and why it works

A run is defined by two numbers: its length n (how many white cells) and its target sum S. Because the digits are distinct and drawn from 1 to 9, the count of valid digit sets is a bounded combinatorial problem. The number of white cells fixes the smallest and largest sums a run can hold.

Here n is the run length, S_{\min} is the smallest reachable sum (using the lowest digits 1, 2, up to n), and S_{\max} is the largest (using the highest digits). For n = 2, that gives S_{\min} = 3 and S_{\max} = 17. Any two-cell clue outside the range 3 to 17 is impossible, and the solver rejects it.

The most useful clues are the ones with a unique combination: exactly one set of digits fits. These are the anchors. In two cells, 3 (1+2), 4 (1+3), 16 (7+9) and 17 (8+9) are unique. In three cells, 6 (1+2+3), 7 (1+2+4), 23 (6+8+9) and 24 (7+8+9) are unique. The table below lists the extreme sums for each length.

Cross-reference: how two runs trim each other

Every white cell sits in one across run and one down run. Its digit must be legal in both. So the candidate set for a cell is the intersection of the digits allowed by its across run and the digits allowed by its down run.

Take a cell where the across run is 16 in two cells, so its candidates from across are \{7, 9\}. Suppose the same cell's down run is 4 in two cells, whose combination is 1+3, giving candidates \{1, 3\}. The intersection of \{7, 9\} and \{1, 3\} is empty. That means the puzzle is contradictory at that cell, and the solver reports it. When the intersection has exactly one member, the cell is solved on the spot.

A worked example using the demo puzzle

Reading the technique log

The log lists moves in the order the solver made them, and it distinguishes several techniques. Learn to tell them apart:

Unique sum combination

A run's length and sum allow only one digit set, so every cell in that run is limited to those digits at once. Example: "17 across in two cells must be 8+9."

Cross-reference intersection

A cell's across candidates and down candidates share only one digit, which becomes the answer. This is the workhorse move.

Naked single

Inside one run, a cell already has just one candidate left after eliminations, so it is filled.

Hidden single

Inside one run, some digit can go in only one of the run's cells even though that cell still shows several candidates. That digit is placed.

Uniqueness elimination

Once a digit is placed in a run, the no-repeat rule removes it from every other cell of the same run.

Common mistakes when building and reading puzzles

The errors below account for most rejected or multi-solution puzzles.

• Sum outside the legal range. A two-cell run summing to 2 or 18 is impossible; the minimum is 3 and the maximum is 17. The solver rejects it rather than guess your intent.
• A run with no clue. Every white run needs a sum clue on its black cell. A run left blank has no constraint, which usually creates many solutions.
• Contradictory intersections. Two clues can each be valid alone yet share no digit at the cell where they cross, as in the earlier 16-across and 4-down example. That yields an empty candidate set.
• Under-constrained grids. If the solver warns of several solutions, add or tighten a clue. It still shows one valid grid so you can see where the ambiguity sits.

A quick self-check: count your white cells and the total of all across clues. Those two totals must be equal, because every white cell belongs to exactly one across run and the across clues cover every white cell once. The same holds for the down clues.

How many combinations each sum has

The chart shows, for two-cell runs, how many valid digit pairs each sum admits. The count peaks in the middle and falls to exactly one at the extremes. Those single-count sums (3, 4, 16, 17) are your anchors.

To read a value: sum 7 shows 3 pairs (1+6, 2+5, 3+4). Sum 10 shows 4 pairs (1+9, 2+8, 3+7, 4+6); note 5+5 is not allowed because digits cannot repeat. This is why an experienced solver hunts for the 3s, 4s, 16s and 17s first.

Related solvers on this site

The same "name the technique for every move" approach appears in two neighboring tools. If you like kakuro for its digit logic, the Sudoku solver uses naked and hidden singles in the same sense, applied to rows, columns and boxes. If you prefer deducing cells from run-length clues, the Nonogram solver reasons line by line and explains each filled or crossed cell.

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