You shade black squares on a fresh sheet of graph paper, aiming for something that looks like a real Kakuro grid. Halfway through filling in digits you notice three problems. A white cell in the corner has a down clue but nothing across. One row runs ten cells long. And a little 2x2 pocket in the lower left doesn’t touch anything else.
None of those can be fixed with better digits. The layout was broken before you wrote a single number.
The pattern of black and white squares decides most of how a Kakuro plays: where the solver can start, how long each run is, and where a second solution can hide. This article covers the layout rules every published Kakuro follows, how symmetry works, what black cells and run lengths do to the solve, and then builds the 7x7 layout used throughout this series, one pair of black cells at a time.
The Layout Rules
Runs Are 2 to 9 Cells
A run is an unbroken line of white cells under one clue. Two limits apply.
A one-cell run is out. Its clue would just be the answer, printed in plain sight. Published Kakuro doesn’t use them, and a grid full of them would be a fill-in exercise.
A run can’t be longer than nine cells either. Digits can’t repeat within a run, and there are only nine digits. A ten-cell run has no legal fill. A nine-cell run is always 1 through 9 and always sums to 45, which makes it the one place in Kakuro where the number 45 is guaranteed.
The short end of the range does a lot of work. Two-cell runs make good starting points because several of their sums allow only one pair of digits: 3 is {1, 2}, 4 is {1, 3}, 16 is {7, 9} and 17 is {8, 9}.
+-------+-------+-------+
|#######| 4\ |#######|
+-------+-------+-------+
| \4 | |#######|
+-------+-------+-------+
|#######|#######|#######|
+-------+-------+-------+
That white cell is boxed in on all sides. Its across run and its down run are each one cell long, so both clues just say 4. There’s nothing to deduce.
Every White Cell Is Crossed Both Ways, and Everything Connects
Every white cell belongs to exactly one across run and exactly one down run. A cell with black squares directly above and below it has no down run, so nothing checks its digit from that direction. That’s the corner cell from the opening.
All the white cells must also form one connected piece. Cells that touch only at a corner don’t count. A cut-off pocket is a separate puzzle with no way for logic to get in or out, and it usually has several solutions.
180° Rotational Symmetry
How It Works
Turn a symmetric grid upside down and the black-and-white pattern looks the same. Every black cell has a partner on the opposite side of the center.
Symmetry isn’t one of Kakuro’s rules. Nikoli’s rule list has three entries, and none mentions layout. It’s a convention, and a strong one in hand-made grids, because a balanced layout looks deliberate and tends to spread the solving evenly across the page.
Measure symmetry on the interior, not the clue border. The top row and left column are all clue cells. Turn the full grid and they’d land on the bottom and right edges, so no grid would ever pass. Leave the border out.
In the series grid, the full grid is 7x7, the interior is rows 1–6 and columns 1–6, and interior cell (r, c) pairs with (7 − r, 7 − c). In general, for an N×N grid with the border at row 0 and column 0, the partner of (r, c) is (N − r, N − c). Two examples from the series layout: the black cell at r3c3 pairs with r4c4, and r1c5 pairs with r6c2.
The Trade-Offs
Symmetry costs flexibility. Every black cell you add brings a partner, so a fix in the top-left changes the bottom-right too, and the partner might break a run that was fine.
Odd-sized interiors have one more wrinkle. In a 7x7 interior (an 8x8 grid with its border), the center cell pairs with itself. Decide early whether it’s black or white: a white center makes a crossroads in the middle of the grid, while a black one splits the grid into four regions that meet around it.
Some puzzles break symmetry on purpose. An escape-room grid shaped like a keyhole, or a themed puzzle that spells something in black cells, trades the convention for a reason. Do it deliberately, not by accident.
Black Cells
How Many
With too few black cells, runs get long, sums get large, and every clue has dozens of combinations. Long runs also leave room for second solutions. With too many, the grid breaks into short, easy runs, and the puzzle turns choppy.
When you quote a percentage, say what you’re counting. The series grid is 51% black overall: 25 of its 49 cells. But 13 of those are the border, which every grid has. Count only the interior and it’s 12 of 36, or 33%. Both numbers describe the same grid, and only the second tells you much about how it will play.
Where They Go
The top row and left column hold clues. Past that, three placement habits matter:
- Avoid one-cell bridges. If two big sections connect through a single white cell, all the logic has to squeeze through it.
- Keep the middle open enough to carry logic across. A solid wall of black cells through the center makes two separate puzzles that happen to share a page.
- Watch the 2x2 white blocks. Any 2x2 block of white cells is a place where four digits can swap and give a second solution. The series layout has eight of them. They aren’t errors, but each is a spot to check later. How to Make Sure Your Kakuro Has Exactly One Solution covers what to look for.
Run Lengths and How the Solve Moves
Short Runs (2–3 Cells)
Short runs are where solvers start. Four two-cell sums have a single combination (3, 4, 16, 17), and so do four three-cell sums (6 is {1, 2, 3}, 7 is {1, 2, 4}, 23 is {6, 8, 9}, 24 is {7, 8, 9}). Even when a short run’s sum has several options, the options are few enough to compare quickly.
| Run length | Possible sums | Sums with only one digit set |
|---|---|---|
| 2 cells | 3–17 | 3, 4, 16, 17 |
| 3 cells | 6–24 | 6, 7, 23, 24 |
| 4 cells | 10–30 | 10, 11, 29, 30 |
| 5 cells | 15–35 | 15, 16, 34, 35 |
Put a few short runs near the edges and corners, and the solver has somewhere to begin.
Longer Runs (4+ Cells)
Four- and five-cell runs carry the middle of the solve. They connect the short runs and pass information across the grid. They rarely give up a digit on their own, so they rely on the runs crossing them. In the series grid, the row-5 across run is four cells long and is crossed by four down runs, one of them only two cells long. That short crossing is what pins it down.
On a small grid, keep runs short. The series layout tops out at four cells, which suits a 7x7.
Practical Example: Drafting the Series 7x7 Layout
You’re building the blank layout used for the rest of this series. The goals:
- 7x7 including the clue border, leaving a 6x6 interior
- 180° symmetry on the interior: (r, c) pairs with (7 − r, 7 − c)
- No one-cell runs, every white cell crossed both ways, everything connected
- A mix of 2-, 3- and 4-cell runs
Step 1: Black out the border. Row 0 and column 0 become clue cells.
Step 2: Black out the top-right and bottom-left corners. Shade r1c5, r1c6, r2c5 and r2c6, then their partners r6c2, r6c1, r5c2 and r5c1. That cuts off the interior’s top-right and bottom-left corners and leaves a wide band of white cells running diagonally from top-left to bottom-right.
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######|#######|#######|#######|#######|#######|
+-------+-------+-------+-------+-------+-------+-------+
|#######| | | | |#######|#######|
+-------+-------+-------+-------+-------+-------+-------+
|#######| | | | |#######|#######|
+-------+-------+-------+-------+-------+-------+-------+
|#######| | | | | | |
+-------+-------+-------+-------+-------+-------+-------+
|#######| | | | | | |
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######|#######| | | | |
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######|#######| | | | |
+-------+-------+-------+-------+-------+-------+-------+
Rows 3 and 4 still run six cells wide, which is too long for a grid this size.
Step 3: Split the middle. Shade r3c3 and its partner r4c4. Rows 3 and 4 now break into shorter runs, and columns 3 and 4 split top from bottom.
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######|#######|#######|#######|#######|#######|
+-------+-------+-------+-------+-------+-------+-------+
|#######| | | | |#######|#######|
+-------+-------+-------+-------+-------+-------+-------+
|#######| | | | |#######|#######|
+-------+-------+-------+-------+-------+-------+-------+
|#######| | |#######| | | |
+-------+-------+-------+-------+-------+-------+-------+
|#######| | | |#######| | |
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######|#######| | | | |
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######|#######| | | | |
+-------+-------+-------+-------+-------+-------+-------+
Step 4: Trim the edges. Shade r3c6 and its partner r4c1. That shortens the row-3 and row-4 runs at the edges and keeps the column-1 and column-6 runs to three cells.
Step 5: Check everything. Trace every row and every column for a one-cell run. Confirm every white cell has both an across and a down run. Walk the white cells to confirm they connect. Then turn the page upside down and compare the interior.
Every pair you place goes through the same loop:
The finished layout:
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######|#######|#######|#######|#######|#######|
+-------+-------+-------+-------+-------+-------+-------+
|#######| | | | |#######|#######|
+-------+-------+-------+-------+-------+-------+-------+
|#######| | | | |#######|#######|
+-------+-------+-------+-------+-------+-------+-------+
|#######| | |#######| | |#######|
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######| | |#######| | |
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######|#######| | | | |
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######|#######| | | | |
+-------+-------+-------+-------+-------+-------+-------+
You should get 24 white cells and 25 black ones (13 border, 12 interior), and 16 runs: six of 2 cells, four of 3 and six of 4. There’s a printable version, kakuro-series-layout-7x7.pdf, with the clue cells marked and the center of rotation shown.
Printable Templates
Graph paper works, but it doesn’t mark the border or show you where the center is. These US Letter PDFs do:
- Series layout, 7x7: the layout above, clue cells marked
- Design grid, 7x7: small, for quick puzzles and practice
- Design grid, 10x10 and 12x12: typical puzzle-book sizes
- Design grid, 15x15: for large, expert-level grids
- Series puzzles, Medium and Hard: the finished series grids, ready to solve
- All templates in one file (7 pages)
Each design grid has the clue border pre-marked, row and column numbers that match the (r, c) notation used here, a dot at the center of rotation, and the pairing rule (r, c) ↔ (N − r, N − c) printed at the bottom.
Best Practices
Do:
- ✓ Place black cells in pairs, and check the partner every time.
- ✓ Put at least two short runs near the corners so the solver has a way in.
- ✓ Trace every row and column for one-cell runs after each change.
Don’t:
- ✗ Trap a white cell between black squares in one direction.
- ✗ Wall off a section of the grid.
- ✗ Quote a black-cell percentage without saying whether the border counts.
Troubleshooting Layout Problems
A white cell has no down run. There’s a black cell directly above it and another directly below. Move one of them, or turn the cell black (along with its partner). After any change, check each column top to bottom.
The layout stops being symmetric after an edit. You changed one cell and forgot its partner. Always edit in pairs, (r, c) and (N − r, N − c). Keeping a list of pairs next to the grid while you work makes this hard to forget.
Moving a black cell created a one-cell run somewhere else. Shifting a black cell to shorten one run can leave a one-cell stub in the crossing direction. Check both the row and the column after every move, not just the run you were fixing.
What Comes Next
Layout comes first, because most layout problems can’t be fixed later with digits. Keep runs between two and nine cells, cross every white cell both ways, connect everything, and measure symmetry on the interior. Black cells decide where the solver starts and where swaps can hide.
Print the series layout, or grab a blank design grid and sketch your own. The next step is filling the layout with digits and making sure only one solution comes out: How to Make Sure Your Kakuro Has Exactly One Solution.
The Kakuro Puzzle Design series:
- How to Make a Kakuro Puzzle From Scratch
- How to Make Sure Your Kakuro Has Exactly One Solution
- How to Set Kakuro Difficulty
- Designing Kakuro Grids That Hide an Answer