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What completes this 3-cell renban?
Level 3 · Lines
How to solve renban lines
A purple line holds a run of consecutive digits in any order, with no repeats.
A renban line is usually purple or pink. Its digits form a run of consecutive numbers with no repeats, in any order. A 4-cell line holding 3, 4, 5 and 6 could read 5, 3, 6, 4.
The line's length fixes its spread. On a line of n cells, the highest digit is exactly n - 1 above the lowest, which limits how far apart any two cells can be.
How setters write it
The same rule, worded the ways you’ll see it on puzzles.
- Digits on a purple line form a set of consecutive digits in any order.
- Renban lines.
- Digits on a pink line are a non-repeating consecutive set.
Worth memorizing
- A line of n cells holds n consecutive digits. A 3-cell line could be 4-5-6 but never 4-5-7.
- Every renban of 5 or more cells contains a 5. Any run of five from 1–9 passes through 5.
- If a line holds a 1, it runs from 1 up to its length.
Solving tips
Long lines contain the middle
Every renban of 5 or more cells contains a 5, since any run of five digits from 1 to 9 passes through 5. A 6-cell line must hold 4, 5 and 6.
Length caps the spread
On a 3-cell line, no two digits can be more than 2 apart. If one cell is 8, the other two come from 6, 7 and 9.
1 or 9 pins the run
If a line holds a 1, it runs from 1 up to its length, so a 4-cell line with a 1 is 1, 2, 3 and 4. A 4-cell line with a 9 is 6, 7, 8 and 9.
Fill gaps in the run
No digit appears twice on a renban, even when the line bends into another box. A 3-cell line with a 4 and a 6 must have a 5 as its third digit.
A worked example: a 4-cell renban with a 7
A 4-cell purple line sits in one row. One of its cells is a given 7. Elsewhere in the row there is a 5.
- List the runs of four that contain a 7. From the table, those are 4 to 7, 5 to 8 and 6 to 9. Any other run of four either stops below 7 or starts above it.
- The row already has a 5 in a cell off the line. The line is in the same row, so it can't hold another 5. That removes 4 to 7 and 5 to 8, which both contain 5.
- Only 6 to 9 is left. The line holds 6, 7, 8 and 9, and the three empty cells hold 6, 8 and 9 in some order.
- Now the rest of the row gets simpler. Its other five cells hold 1, 2, 3, 4 and 5. The 5 is already placed, so those four empty cells are exactly 1 to 4.
- To place the line digits, check the columns. If one line cell's column already holds an 8 and a 9, that cell must be the 6.
Practice
2 of 3
What goes next to the 9?
3 of 3
This renban has five cells. Which digit must be on it?
Reference table
| Cells | Sets | Possible sets | Always on the line | Sums |
|---|---|---|---|---|
| 2 | 8 | 1 to 2, 2 to 3, 3 to 4, 4 to 5, 5 to 6, 6 to 7, 7 to 8, 8 to 9 | none | 3, 5, 7, 9, 11, 13, 15, 17 |
| 3 | 7 | 1 to 3, 2 to 4, 3 to 5, 4 to 6, 5 to 7, 6 to 8, 7 to 9 | none | 6, 9, 12, 15, 18, 21, 24 |
| 4 | 6 | 1 to 4, 2 to 5, 3 to 6, 4 to 7, 5 to 8, 6 to 9 | none | 10, 14, 18, 22, 26, 30 |
| 5 | 5 | 1 to 5, 2 to 6, 3 to 7, 4 to 8, 5 to 9 | 5 | 15, 20, 25, 30, 35 |
| 6 | 4 | 1 to 6, 2 to 7, 3 to 8, 4 to 9 | 4, 5 and 6 | 21, 27, 33, 39 |
| 7 | 3 | 1 to 7, 2 to 8, 3 to 9 | 3, 4, 5, 6 and 7 | 28, 35, 42 |
| 8 | 2 | 1 to 8, 2 to 9 | 2, 3, 4, 5, 6, 7 and 8 | 36, 44 |
| 9 | 1 | 1 to 9 | 1, 2, 3, 4, 5, 6, 7, 8 and 9 | 45 |
Going deeper
A 5 outside the line splits it
Runs of 5 or more cells always include 5. Shorter runs can avoid it, but only by staying on one side. A 4-cell run without 5 is either 1 to 4 or 6 to 9. A 3-cell run without 5 is 1 to 3, 2 to 4, 6 to 8 or 7 to 9.
So once you know where the 5 goes in a row, column or box that holds the whole line, the line becomes all low or all high. One low or high digit on it then settles which. This is also why a 4-cell renban and the 5 in its row, column or box are worth checking together early in a solve. The same logic works for 1 and 9 on longer lines: a 6-cell line without a 1 is 2 to 7, 3 to 8 or 4 to 9.
Two renbans in one box
When two lines fill a box between them, they split its digits. A 4-cell line and a 5-cell line in the same box use all nine digits. The 5-cell line must hold the 5, so the 4-cell line is 1 to 4 or 6 to 9.
That leaves only two ways to fill the box. Either the short line is 1 to 4 and the long line is 5 to 9, or the short line is 6 to 9 and the long line is 1 to 5. Any single digit on either line picks one.
Odd and even counts on a renban
A run of consecutive digits alternates odd and even. An even-length line always holds the same number of each. A 4-cell renban has two odd and two even digits, whatever the run.
An odd-length line has one extra of one type. A 3-cell run starting on an odd digit, like 3, 4, 5, has two odds. One starting on an even digit, like 4, 5, 6, has two evens. This helps when the puzzle also has odd and even cells or parity lines.
Common mistakes
Treating the ends as the smallest and largest
Renban digits can go in any order. The ends of the line don't have to hold the lowest and highest digits. A 3-cell line could read 5, 3, 4.
Allowing a repeat across a box border
No digit repeats on a renban, even when two cells are in different rows, columns and boxes. This makes renban stricter than most lines, where repeats are allowed if sudoku rules permit.
Forgetting that 1 and 9 fix the run
A 1 on a line of n cells means the run is 1 to n. A 9 means it ends at 9. New solvers often keep extra candidates after spotting a 1 or 9, and miss an easy fill.
In mixed-rule puzzles
Renbans work well with sums. Each run of a given length has its own total, so a renban inside a killer cage often names its set. A 3-cell renban in a cage of 15 must be 4, 5, 6. With thermos, a renban that is also a thermo must rise by exactly 1 each step, since the digits are consecutive and all different. Two cells joined by a white Kropki dot are consecutive and could sit on any renban containing both. A 2-cell renban is the same as a white dot. A 3-cell renban can never lie along three cells in a row of an entropic line, since three consecutive digits always come from at most two of the low, middle and high groups.
What the video shows
- A purple line holds a run of consecutive digits, in any order.
- 3, 4, 5, 7 skips 6, so it isn’t a run.
- 3, 4, 5, 6 is a run. Order doesn’t matter.
Questions solvers ask
What is a renban line in sudoku?
A line whose digits are a set of consecutive numbers, like 2, 3, 4 and 5, placed in any order.
Do renban digits have to be in order?
No. The digits can appear in any order along the line, as long as together they form a consecutive run.
Can digits repeat on a renban line?
No. Each digit on a renban appears once, even when the line crosses into another box.
What color are renban lines?
They are usually purple or pink. The rules text will say which color the puzzle uses.
How many combinations can a 4-cell renban have?
Six sets: 1 to 4, 2 to 5, 3 to 6, 4 to 7, 5 to 8 and 6 to 9. Each set can be arranged in any order along the line.
Can a renban line have 9 cells?
Yes. A 9-cell renban holds every digit from 1 to 9 once. It adds to 45, like a row.
What is the sum of a renban line?
Multiply the number of cells by the average of the lowest and highest digit. A 4-cell line of 4 to 7 adds to 4 times 5.5, which is 22. The table on this page lists every total.
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