From the rulebook
How to Solve Kakuro
Kakuro, also called cross sums, is a crossword built from numbers. Every white cell takes a digit from 1 to 9, and each horizontal and vertical run must add up to the clue at its head without repeating a digit. There are no given numbers to start from, only the clues, so the sums are your way in. Once you can read them they give you plenty to work with. This guide takes you from the rules to the techniques that crack hard grids.
The rules of Kakuro
A Kakuro grid looks like a crossword. Black cells block off the grid, and the white cells form runs that read left to right and top to bottom. Each run is headed by a clue, the small number printed in the black cell before it, which is the total the digits in that run must reach. A clue in the upper part of a split black cell belongs to the run going down, and one in the lower part belongs to the run going across.
Fill every white cell with a digit from 1 to 9 so that each run adds up to its clue, and no digit repeats within a single run. Unlike Sudoku there are no rows or columns of fixed length and no given digits, so the clues are your only starting information. A digit may appear more than once in the grid, just never twice in the same run.
Start with the magic combinations
The fastest opening move is finding runs whose clue can be made only one way. These are the magic combinations. A two-cell run of 3 must be 1 and 2. A two-cell run of 4 must be 1 and 3, since 2 and 2 would repeat. At the top end a two-cell run of 17 must be 8 and 9, and one of 16 must be 7 and 9. A three-cell run of 6 must be 1, 2 and 3, and a three-cell run of 24 must be 7, 8 and 9.
These locked runs do not tell you the order of the digits yet, but they tell you exactly which digits live in the run. That lets you eliminate those digits from any crossing run, which is usually enough to start placing numbers with certainty.
Work the crossings
Every white cell sits where a horizontal run crosses a vertical run, so each digit you place has to satisfy both totals at once. This crossing is the heart of Kakuro. When a magic combination fixes the digit set for one run, check what the crossing run allows in the shared cell, and keep only the digit that appears in both.
A cell where a tight horizontal run meets a tight vertical run often has just one surviving digit. For example, if an across run can only hold 1 and 3 and the down run through one of its cells can only hold 3 and 5, that shared cell must be 3, since 3 is the only value both runs allow.
Use the box-sum trick for stray cells
When a block of runs is nearly complete, totals pin down the leftovers. Add up the clues of the runs that feed into a region, then subtract the digits you have already placed, and the remainder is the sum the empty cells must reach. If only one cell is left, that subtraction gives its exact value.
This works because every run total is fixed. Where several runs overlap in a corner of the grid, comparing what the crossing runs demand against what each run still needs often forces a cell that neither run could pin down on its own.
Combine combination logic with elimination
When the obvious moves run out, write candidate pencil marks driven by the run maths. For each run list the digit combinations that reach its clue without repeating, then cross off any combination blocked by digits already placed in the crossing runs. The surviving candidates in a cell are the intersection of what its across run and its down run still permit.
Apply the techniques in order: lock the magic combinations first, then work the crossings, run the box-sum checks, then narrow candidates by elimination, looping back as the grid fills. On Puzzle Lair every Kakuro is guaranteed solvable by logic alone, so if you are stuck there is always a deduction waiting, never a guess. Longer runs and larger grids are what make the hard set a genuine workout.
Frequently asked questions
- What is Kakuro?
- Kakuro, also called cross sums, is a crossword built from numbers. The grid is split into horizontal and vertical runs of white cells, each headed by a clue. You fill every white cell with a digit from 1 to 9 so that each run adds up to its clue, and no digit repeats within a single run. There are no given digits, so the clues are your only starting information.
- What are magic combinations in Kakuro?
- Magic combinations are run totals that can be made only one way for a given length. A two-cell run of 3 is always 1 and 2, a two-cell run of 17 is always 8 and 9, and a three-cell run of 6 is always 1, 2 and 3. They fix exactly which digits live in a run, even before you know their order, so you can eliminate those digits from any crossing run.
- How do you start a Kakuro puzzle?
- Begin with the magic combinations, the runs whose clue can only be made one way, usually the shortest runs with very low or very high totals. Place what they force, then work the crossings, since every white cell must satisfy both its across run and its down run. The digit in a shared cell is the one value both runs allow.
- Can a digit repeat in Kakuro?
- Not within a single run. A digit can never appear twice in the same horizontal or vertical run, which is why a two-cell run of 4 must be 1 and 3 rather than 2 and 2. The same digit can appear elsewhere in the grid, as long as it does not repeat inside any one run.
- Does solving Kakuro ever require guessing?
- No. A well-made Kakuro has exactly one solution reachable by pure logic. Every Puzzle Lair Kakuro is generated and verified to be solvable without guessing, so being stuck means there is a deduction you have not spotted yet, often a magic combination or a box-sum check you have not run.