Kakuro Strategy: Magic Sums First, Then Intersections

🪄The magic sums — runs with exactly one combination

Some sums in a given length can be built in only one way (digits never repeat inside a run). Those runs hand you their complete digit set for free:

• 2 cells: 3 = {1,2} · 4 = {1,3} · 16 = {7,9} · 17 = {8,9}
• 3 cells: 6 = {1,2,3} · 7 = {1,2,4} · 23 = {6,8,9} · 24 = {7,8,9}

Sweep the board for these first and pencil their sets into every cell of the run. Just as useful: the near-magic sums with exactly two combinations (2-cell 5, 6, 14, 15; 3-cell 8 and 22) narrow a run to at most six candidate digits before you've solved anything.

🧪Worked example — a whole block from one intersection

From the Wednesday, October 14, 2026 DailyGrid board. In the top block, the second row's across-run sums to 23 in 3 cells — magic: it is exactly {6, 8, 9}. The row beneath runs 7 in 3 cells — magic again: exactly {1, 2, 4}. Neither row says which digit goes where. The columns decide:

1. The third column of the block sums down to 9 in 2 cells. Its possible pairs are {1,8}, {2,7}, {3,6}, {4,5} — so its top cell (in the 23-row) and bottom cell (in the 7-row) split 9 between them.
2. The bottom cell must come from the 7-row's set {1, 2, 4}. Which partners make 9? 8+1, 7+2, 6+3, 5+4 — only one of those top digits (8) is in the 23-row's set {6, 8, 9} and leaves a bottom digit from {1, 2, 4}: top = 8, bottom = 1. Two cells solved.
3. Now the middle column, down 11 in 2: pairs {2,9}, {3,8}, {4,7}, {5,6}. The top cell is one of the 23-row's remaining digits {6, 9}; the bottom is one of the 7-row's remaining {2, 4}. 9 pairs with 2 — top = 9, bottom = 2. (6 would need a 5 below, and there is no 5 in the 7-row.)
4. Elimination finishes it: the first column gets top = 6, bottom = 4 (and the down-10 run agrees: 6 + 4 = 10 ✓).

Six cells, zero guesses — two magic rows, three ordinary columns, and every step forced by the intersection of the two runs each cell belongs to. That is the entire game; everything else is scale.

✖️The method, generalised

• Pencil candidate sets, not hunches. Every cell's candidates are its across-set ∩ its down-set. When a cell drops to one candidate, write it in ink and remove that digit from both runs' other cells.
• Recompute after every placement. A placed digit reduces its runs' remaining sums and lengths — a 3-cell run with one cell placed is a 2-cell run on the remainder, often a magic one.
• Mind the 45 rule at block scale. A full 1–9 set sums to 45; when a region's runs overhang by one cell, the difference between the run totals and 45 (or the relevant subset sum) is that cell's digit. Powerful late, useless early.

⚠️Common mistakes

• Repeating a digit inside a run because it appeared in the crossing run — the no-repeat rule applies per run, not per board. (The same digit can repeat in one row of cells if a black cell splits them into two runs.)
• Ignoring the two-combination sums. A 14-in-2 ({5,9} or {6,8}) carries almost as much information as a magic sum once it crosses a constrained run — the shared cell must sit in both runs' sets at once. Keep the small sets pencilled, even the two-way ones.
• Guessing early. Kakuro boards are designed to unwind by intersections; if you feel forced to guess, a magic sum somewhere is unharvested.

❓FAQ

Which sums have only one combination?

2 cells: 3, 4, 16, 17. 3 cells: 6, 7, 23, 24. Learn those eight and every board starts moving immediately.

How do you solve Kakuro without guessing?

Intersect each cell's across and down candidate sets; place singles; shrink the runs; repeat.

Where are the basic rules?

In the companion guide: How to Play Kakuro.

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