Renban Sudoku: Rules & How to Solve
Each purple line is a set of consecutive numbers in any order. Flexible, fast, and great for chaining.
The rule
Digits on a renban line form a set of consecutive numbers in any order, with no repeats. A three-cell renban could be 4-6-5 or 7-5-6, but always a run.
Renban run table
A renban line holds a set of consecutive digits in any order, with no repeats. So a line of length L has only 10 − L possible sets — and the longer the line, the fewer choices remain.
| Line length | Possible runs | The runs |
|---|---|---|
| 2 | 8 | {1,2}, {2,3} … {8,9} |
| 3 | 7 | {1,2,3}, {2,3,4} … {7,8,9} |
| 4 | 6 | {1,2,3,4}, {2,3,4,5} … {6,7,8,9} |
| 5 | 5 | {1,2,3,4,5}, {2,3,4,5,6} … {5,6,7,8,9} |
| 6 | 4 | {1,2,3,4,5,6}, {2,3,4,5,6,7}, {3,4,5,6,7,8}, {4,5,6,7,8,9} |
| 7 | 3 | {1,2,3,4,5,6,7}, {2,3,4,5,6,7,8}, {3,4,5,6,7,8,9} |
| 8 | 2 | {1,2,3,4,5,6,7,8}, {2,3,4,5,6,7,8,9} |
| 9 | 1 | {1,2,3,4,5,6,7,8,9} |
A nine-cell renban is fully determined as a set: it is 1–9 in some order, which makes it behave like an extra region.
VariantDoku shows the same reference inside the app: open any puzzle, tap the rules sheet, and each constraint on the grid gets its own tips card.
How to solve renban sudoku
Length fixes the spread
A four-cell renban spans exactly four consecutive values, so it can't contain both a 1 and a 9 — the extremes are bounded.
Order is free
Unlike a thermometer, renban cells can be arranged any way, so use crossing constraints to pin the order.
Use what every run must contain
Every possible run of a long renban shares its middle digits — a five-cell line always contains 5. If the whole line sits in one row, column or box, those digits are locked onto it and leave every other cell of that house.
Use the length to bound the extremes
On a line of length L the smallest digit is at most 10 − L and the largest is at least L. A six-cell renban must therefore contain both 5 and 6.
Renbans behave like mini-regions
Because a renban never repeats, any cell that sees the whole line cannot hold any digit on it. Short renbans sitting inside a single box are especially strong.
Step by step: techniques from the hint engine
Each example is drawn the way VariantDoku's hints show it: red candidates are eliminated, blue ones are what the step proves, and green cells are solved.
Renban windows
A six-cell renban could be 1–6, 2–7, 3–8 or 4–9. This row already has its 9, so 4–9 is gone, and every remaining window contains 3, 4, 5 and 6. Those four digits are therefore on the line — and because the line lies in this row, they leave every other cell of it.
A worked example
Anchor on the extremes. If a five-cell renban contains a 1, the only run that fits is {1,2,3,4,5} — the line is fully known as a set, and every one of those digits is eliminated from any cell that sees the whole line. A 9 on the same line forces {5,6,7,8,9} instead.
Middle digits are weaker but still useful: a 5 on a five-cell renban leaves five candidate runs, but every one of them contains 5, 4-or-6, and so on — enough to prune the extremes.
On 4×4 and 6×6 boards
Fewer digits mean fewer windows. On an N×N board a line of length L has N + 1 − L possible runs, and every run contains the digits from N + 1 − L up to L. So on a 6×6 board a four-cell renban always contains 3 and 4 and a five-cell renban always contains 2–5; on a 4×4 board a three-cell renban always contains 2 and 3.
| Board | Length | Possible runs | Always on the line |
|---|---|---|---|
| 4×4 | 2 cells | 1–2 · 2–3 · 3–4 | none |
| 4×4 | 3 cells | 1–3 · 2–4 | 2, 3 |
| 4×4 | 4 cells | 1–4 | 1, 2, 3, 4 |
| 6×6 | 2 cells | 1–2 · 2–3 · 3–4 · 4–5 · 5–6 | none |
| 6×6 | 3 cells | 1–3 · 2–4 · 3–5 · 4–6 | none |
| 6×6 | 4 cells | 1–4 · 2–5 · 3–6 | 3, 4 |
| 6×6 | 5 cells | 1–5 · 2–6 | 2, 3, 4, 5 |
| 6×6 | 6 cells | 1–6 | 1, 2, 3, 4, 5, 6 |
Try one from the catalog
Real VariantDoku puzzles with this rule. Each one opens in the app.
Frequently asked questions
What is a renban line?
A line whose cells hold a set of consecutive digits in any order, with no repeats.
Can a renban line repeat digits?
No. The set is consecutive and distinct, like {5,6,7}.
How is renban different from a thermometer?
Both use consecutive runs, but a thermometer is strictly ordered low-to-high while a renban can be in any order.
Does the order matter on a renban line?
No. A renban holds consecutive digits in any order — the set is constrained, the sequence is not.
Can a renban line repeat a digit?
No. The digits form a set of consecutive values with no repeats, regardless of whether the cells see each other.
Play Renban Sudoku in VariantDoku
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