1 of 1
What completes the second piece of the line?
Level 3 · Lines
How to solve region sum lines
Where a line crosses box borders, each piece of it adds up to the same total.
A region sum line is usually blue. Where it crosses from one 3x3 box into another, the border cuts it into pieces, and every piece has the same sum.
The rule links distant parts of the grid. A short piece limits how big a long piece can be, and a long piece forces the short one to be large.
How setters write it
The same rule, worded the ways you’ll see it on puzzles.
- Box borders divide a blue line into segments with the same sum.
- Region sum lines.
Worth memorizing
- A single-cell segment sets the total for every other segment.
- Segments with more cells need smaller digits to match shorter ones.
Solving tips
One-cell pieces set the total
If a piece is a single cell, its digit is the total for every other piece. A lone 7 means every other piece of the line adds up to 7.
Compare short and long pieces
A three-cell piece in one box adds to at least 1 + 2 + 3 = 6. If another piece is a single cell, that cell is 6 or more.
Four cells rule out singles
Four different digits add to at least 10. A line with a four-cell piece can't also have a one-cell piece, because no single digit reaches 10.
Small totals have short lists
If the total is 5, a two-cell piece must be 1 and 4 or 2 and 3. A total of 3 leaves only 1 and 2.
A worked example: a line across three boxes
A region sum line crosses three boxes. It has three cells in the first box, one cell in the second, and two cells in the third. One cell of the three-cell piece is 4, one cell of the two-cell piece is 7, and the row of the one-cell piece already has a 9.
- The three-cell piece holds three different digits, so it adds to at least 1 + 2 + 3 = 6. The one-cell piece matches that total, so it's 6 or more.
- The three-cell piece holds a 4. Its smallest total is now 4 + 1 + 2 = 7, so the line's total is at least 7.
- The two-cell piece holds a 7, so its smallest total is 7 + 1 = 8. The single cell can't be 9 because its row has one. So the total is exactly 8, and the single cell is 8.
- Fill in the pieces. The three-cell piece needs 4 plus two digits that add to 4. That's 1 + 3, since 2 + 2 repeats. The two-cell piece is 7 + 1.
- Both pieces hold a 1, which is fine because they sit in different boxes. The two 1s only need to be in different rows and columns.
Practice
Reference table
| Cells in piece | Smallest sum | Largest sum | Totals with only one set |
|---|---|---|---|
| 1 | 1 | 9 | all |
| 2 | 3 | 17 | 3, 4, 16, 17 |
| 3 | 6 | 24 | 6, 7, 23, 24 |
| 4 | 10 | 30 | 10, 11, 29, 30 |
| 5 | 15 | 35 | 15, 16, 34, 35 |
| 6 | 21 | 39 | 21, 22, 38, 39 |
| 7 | 28 | 42 | 28, 29, 41, 42 |
| 8 | 36 | 44 | all |
| 9 | 45 | 45 | all |
Going deeper
Matching pieces of different lengths
The total has to fit every piece at once, so take the overlap of their ranges from the table. A one-cell piece allows 1 to 9, and a three-cell piece allows 6 to 24. A line with both has a total of 6 to 9.
A two-cell piece and a four-cell piece share a range of 10 to 17. A one-cell piece and a four-cell piece share nothing, so that line can't appear in a valid puzzle. Each extra cell on the long piece raises the floor, and the short piece has to reach it with large digits.
Long pieces set a high floor. A piece that fills six cells of a box adds to at least 21, so a line with a six-cell piece has a total of 21 or more. A two-cell piece can't sit on that line, since two different digits make 17 at most.
Repeats between pieces
Two pieces can share digits when they sit in different boxes and the matching cells don't share a row or column. Pieces in boxes side by side often share rows, so check each row before copying a set from one piece to the next.
A line can leave a box and come back. Then two pieces sit in the same box and can't share any digit, yet they still have the same total. That pairing often forces very tight sets.
Working with an unknown total
Most lines never state their total. Write the possible range next to the line, such as 7 to 9, and narrow it each time a digit lands.
Once the range is down to one number, treat every piece as a small killer cage with that total and use the cage combination lists.
When one piece's range shrinks, check the other pieces right away. They share the same total, so a change on one end of the line often settles a cell on the other.
Common mistakes
Splitting at the wrong borders
Only box borders split the line. A turn inside a box, or a move from one row to the next, doesn't start a new piece.
Assuming pieces can't share digits
Different pieces in different boxes can hold the same digits. Only cells in the same piece, or in the same box, row or column, have to differ.
Missing a line that returns to a box
If the line leaves a box and comes back, the two pieces in that box can't repeat digits between them. Treat them as one no-repeat group with two equal sums.
In mixed-rule puzzles
Region sum lines work well with killer cages and arrows, since all three are sum rules. A cage that covers a whole piece gives you the line's total at once. Arrows can look similar, but an arrow adds all its cells to one circle. Line colors vary between setters, so read the rules text. When a region sum line overlaps a renban or whisper line, the other rule often decides which of several same-total sets a piece uses.
What the video shows
- Where a line crosses box borders, each piece of it sums to the same total.
- The piece in the first box adds to 9...
- ...so the piece in the next box must too: 4 + 5 = 9.
Questions solvers ask
What is a region sum line in sudoku?
A line split by box borders into pieces, where every piece adds up to the same number.
Do region sum lines tell you the total?
No. You work out the total from the digits you place and the lengths of the pieces.
Can digits repeat on a region sum line?
Not within one piece, because each piece sits inside one box. Different pieces can share digits if their rows and columns allow.
What counts as a region on a region sum line?
In a standard puzzle the regions are the 3x3 boxes. Puzzles with irregular regions use those shapes instead.
What is the difference between a region sum line and an arrow?
An arrow's digits add up to the digit in its circle. A region sum line has no circle. Each piece between box borders has the same total as every other piece.
Can a region sum line stay in one box?
A line that never crosses a box border has only one piece, so there's nothing to match. A useful line crosses at least one border.
How do you find the total on a region sum line?
Compare the smallest and largest sums each piece can make, then narrow the range as digits appear. A one-cell piece, once solved, gives the total directly.
Record Highs
Six Rules Four Clues
Happy Birthday Finnegan!
Increasing Sum Line
Shade Your Own Sum Lines
RAT RUN 4: Borderline