All Sudoku Variants
9×9 · Arithmetic

Region Sum Lines Sudoku

Also known as Region Sum Sudoku

Blue lines cross box borders; the digits within each box the line passes through sum to the same total.

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An example Region Sum Lines puzzle: the line's cells within each box share a total — 4 + 5 = 9 in the left box, 1 + 2 + 6 = 9 in the right box.

Region Sum Lines Sudoku draws blue lines that cross box borders. The rule looks at each box the line passes through: the cells the line covers within one box must sum to the same total as the cells it covers within every other box. Different boxes may hold different numbers of line cells, yet each segment's sum has to match.

The rule: every box segment sums the same

Split the line at the box borders. Each resulting segment — however long — adds up to one shared value. Below, a line crosses two boxes: the left segment (4 + 5) and the right segment (2 + 3 + 4) both total 9.

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The box border (black) splits the line: left 4+5 = 9, right 2+3+4 = 9.

A worked example, step by step

Take a line crossing two boxes, where the left segment already reads 4 and 5.

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Step 0. The left box segment is 4 and 5; the right segment (three cells) is empty.

Step 1 — total the known segment. The left segment adds to 4 + 5 = 9. Every other box segment on this line must therefore also total 9.

Step 2 — constrain the other segment. The right segment has three cells that must sum to 9. Three different digits adding to 9 are limited to {1,2,6}, {1,3,5} or {2,3,4} — a big cut before any other clue.

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Step 2. The three right-hand cells must also total 9: {1,2,6}, {1,3,5} or {2,3,4}.

Segments of different lengths

The clever part is that the segments need not be the same length. A two-cell segment summing to 9 forces fairly large digits (like 4 and 5), while a three-cell segment summing to 9 forces smaller ones (like 1, 3, 5). Matching those two facts across the border often pins several cells: the short segment can't use the small digits the long one needs, and vice versa.

Solving strategy

  • Find the total from the shortest segment. A fully-known segment fixes the sum for the whole line.
  • Treat each segment as a cage. Once the total is set, every box segment is a small sum — use combination logic.
  • Play lengths against each other. Short segments need big digits, long ones need small — the mismatch narrows both.
  • Feed the grid. Each digit a line fixes behaves like a given for its row, column and box.

Common mistakes

  • Summing the whole line. It is each box segment that shares a total, not the line as a whole.
  • Assuming equal lengths. Segments can differ in length and still share the same sum.
  • Splitting at the wrong place. Break the line only at box borders, not at every cell.
  • Forgetting ordinary Sudoku. Line cells still can't repeat within a row, column or box.

Can I play it here?

Not yet — Region Sum Lines Sudoku isn't one of the puzzle types SudokuStreak generates for now. But you can learn its rules above, and there's plenty here to sink your teeth into meanwhile: