Kakuro sum combinations

The kakuro sum-combinations table is the foundational reference for kakuro solving. For every possible combination of (number of cells in a run, target sum), the table lists the valid digit sets that can produce that sum without repeating a digit. A 2-cell run with target 3 has only one valid combination: {1, 2}. A 3-cell run with target 6 has only {1, 2, 3}. A 3-cell run with target 12 has several: {1, 2, 9}, {1, 3, 8}, {1, 4, 7}, {1, 5, 6}, {2, 3, 7}, {2, 4, 6}, {3, 4, 5}.

How the table is used

A kakuro solver consults the table at every step. Looking at a run with a known cell count and target sum, the player asks: which digit combinations are valid? If only one combination exists — a unique sum — every cell in the run must be one of those specific digits, and the cross-reference with intersecting runs usually pins them to specific cells. If several combinations exist, the player narrows by combining the run's possibilities with the cells' other runs.

The table's structure makes certain (cell-count, target-sum) pairs especially valuable. Tight targets — sums close to the minimum or maximum — produce unique or near-unique combinations. The minimum sum for N cells is 1+2+...+N (the sum of the first N digits); the maximum is (10-N)+(11-N)+...+9. Sums right at those extremes are forced; sums in the middle of the range have many combinations.

The unique-sum entries are the ones to memorise

Most kakuro experts know the unique-sum entries by rote. They're the cases where the cell count and target sum together force exactly one digit set. Examples:

• 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}).

Higher cell counts have unique sums at the same kind of extremes. Most kakuros lean heavily on these unique-sum entries because they produce eliminations directly without needing further deduction.

The opposite extreme — sums with maximum flexibility

The middle-range targets — for example, 4 cells summing to 22 — have many valid combinations and contribute less directly. Solvers often skip these in early scans and revisit them after the unique-sum runs have placed enough digits to narrow them via cross-reference.

Why this is the centerpiece reference

Most kakuro techniques build on this table. Unique sum is its direct application; partial cage combinations' kakuro analogue tracks how the table's entries narrow as cells get filled. Even advanced moves like kakuro-pair and kakuro-triple use the table as the underlying combination source.

The widget at the top of this entry presents the full lookup. For day-to-day kakuro solving, internalising the unique-sum entries above gets you most of the way; the table is there for the runs that don't fall into the memorised set.