Kakuro: The Complete Strategy Guide

Kakuro is what happens when a crossword and a Sudoku have a baby. The grid looks like a crossword — black blocks separating white cells — but each "word" is a run of digits that must sum to a target, with no digit repeating in any run. The result is a deeply satisfying number puzzle where every cage is a small constraint puzzle, and the cages interlock like crossword answers.

This guide walks through the rules, the indispensable "magic sum" combinations, and the techniques that transform a daunting blank grid into a sequence of forced placements.

How Kakuro Works

The grid is a mix of black blocks and white cells. Black blocks may contain one or two numbers: a number in the upper-right corner is the sum for the across run starting to its right; a number in the lower-left corner is the sum for the down run starting below it.

Each run is a horizontal or vertical sequence of white cells bounded by black blocks or the grid edge. Your job is to fill each white cell with a digit 1-9 so that every run sums to its target, and no digit repeats within a run.

The two constraints that matter

Two rules give Kakuro its character: the sum constraint (the run totals to the target) and the no-repeat constraint (each digit appears at most once per run). The no-repeat rule is what makes the puzzle solvable through logic — it dramatically restricts which combinations of digits can fill a given run length.

The Magic Sums (Memorize These)

Some sum-and-length combinations have only one possible set of digits. These are called "magic sums" and they're the bread and butter of Kakuro solving. Memorize the most common ones and you'll find forced placements all over the grid.

Notice the symmetry — the lowest sum for any length uses 1,2,3,… and the highest uses 9,8,7,… back down. Once you internalize these, you'll spot them at a glance.

Reading the Grid Like an Expert

Beginner Techniques

The intersection lock

If the across run's possible digits are {1, 2, 4} and the down run's possible digits are {2, 5, 7}, the intersecting cell must be 2 — the only digit in both sets.

Forced-fill via magic sums

When a magic sum's digit set is locked, every cell of that run can only be one of those digits. Cross-reference with intersecting runs to lock specific cells.

Eliminate by length

A run of length 2 cannot contain 0, and the digits sum to a number between 3 and 17. If the sum is 5 with 2 cells, possible sets are {1, 4} or {2, 3}. Many cells are obviously not 5 or higher — eliminate them aggressively.

Intermediate Techniques

The shared-cell narrowing

Two runs that share a cell put a joint constraint on that cell. If the across run is sum 6 over 3 cells ({1, 2, 3}) and the down run is sum 17 over 2 cells ({8, 9}), the shared cell would need to be in both sets — impossible. This means your placement assumption was wrong somewhere; check the surrounding runs.

Sum partitioning

For non-magic sums, list the possible digit sets (e.g., sum 11 over 3 cells: {1,2,8}, {1,3,7}, {1,4,6}, {2,3,6}, {2,4,5}, {3,4,4}-invalid, so 5 valid sets). The intersecting runs usually rule out most sets quickly.

The Sudoku-style scan

Once a digit is placed, scan to see if it can be eliminated from other cells in the same run (it cannot repeat) and other cells in intersecting runs (depending on their constraints). This works like a mini-Sudoku at each run.

Advanced Techniques

Forced singletons from sum gaps

If a run of length 4 sums to 30 ({6,7,8,9}) and three of the cells are placed as 6, 7, 8, the fourth is forced to be 9 even though you didn't know its value from any other clue. Always check whether the placed digits constrain the unfilled cells through the sum.

Combination forcing

If multiple intersecting runs share their candidate sets, you can sometimes deduce all cells of a junction simultaneously by enumerating the valid combinations and finding which agree.

Contradiction by chain

When stuck, assume one of two possible digits in a cell, then propagate. If a contradiction surfaces (a sum becomes impossible, a digit duplicates), the other choice is forced.

Common Mistakes to Avoid

• Forgetting digits can't repeat. A run summing to 8 with 4 cells cannot be 1+1+3+3 — it must be 4 unique digits.
• Ignoring 0. Kakuro uses digits 1-9 only. There's no 0.
• Not memorizing magic sums. Magic sums are the fastest way into a Kakuro puzzle. Without them you're searching blind.
• Solving runs in isolation. Every cell is in two runs. Always check both.

How Kakuro Compares to Other Number Puzzles

Kakuro is the most arithmetic-heavy puzzle in the Puzzle Page lineup. Cross Sum is a lighter cousin: similar idea of equation targets, but without the unique-digit-per-run rule that makes Kakuro feel like Sudoku. Sudoku shares the unique-digit constraint but drops the sums.

If you like Kakuro's intersection-based logic, you'll probably enjoy Crossword — same crossword grid structure, words instead of numbers.

Frequently Asked Questions