9×9 · Arithmetic

Zipper Lines Sudoku

Also known as Zipper Sudoku

On a zipper line, cells equidistant from the centre add up to the value in the centre cell.

8 3 5 7 9 2 812645

An example Zipper Lines puzzle: on the violet line, digits equidistant from the centre (violet) add to it — here 1 + 5 = 6 and 2 + 4 = 6.

Zipper Lines Sudoku draws a line with a marked middle cell. Imagine folding the line in half at that centre: each pair of cells that meet must add up to the digit in the centre cell. Like a zip pulling two sides together, every equidistant pair shares the same total — the centre value.

The rule: pairs fold to the centre

Take the cell in the middle of the line. Then the two cells one step out from it must sum to that middle value; so must the two cells two steps out, and so on. Below, the centre is 9, and each outer pair — 4 + 5 and 2 + 7 — adds to 9.

24957

Centre 9; each fold pair adds to it: 4 + 5 = 9 and 2 + 7 = 9.

A worked example, step by step

Take a five-cell zipper whose centre cell is already a 9.

9

Step 0. The centre cell is 9; the four outer cells are empty.

Step 1 — each fold pair sums to 9. The two cells either side of the centre add to 9, and so do the two outermost cells. Both pairs share the same target.

9

Step 1. The inner pair sums to 9, and so does the outer pair.

Step 2 — list the pairs. Two different digits adding to 9 are {1,8}, {2,7}, {3,6} or {4,5} — and never a 9, since that is the centre. Each fold takes one of those pairs, so the four outer cells are heavily constrained before any other clue.

1-81-891-81-8

Step 2. Every outer cell is 1–8 (never 9); each fold pair is one of {1,8},{2,7},{3,6},{4,5}.

Reading the centre

The centre digit is always the largest value the line can carry, because it equals the sum of two other positive digits — so a zipper centre is never small. A centre of 9 means every fold is a pair from {1,8} up to {4,5}; a centre of 5 restricts folds to {1,4} or {2,3}, pushing small digits onto the line. Reading the centre first tells you the whole line's flavour.

Solving strategy

  • Find the centre value. It caps the whole line and sets every fold's target sum.
  • Pair from the outside in. Match equidistant cells and treat each as a small two-cell sum.
  • Exclude the centre digit. No outer cell can equal the centre, since two positive digits can't sum to it and reuse it.
  • Feed the grid. Each digit a fold fixes behaves like a given for its row, column and box.

Common mistakes

  • Pairing the wrong cells. Fold pairs are equidistant from the centre, not simply neighbours.
  • Letting an outer cell equal the centre. The centre is the pair total, so it can't reappear as one of the paired digits.
  • Forgetting the centre is fixed. One centre value governs every fold on that line.
  • Ignoring ordinary Sudoku. Line cells still can't repeat within a row, column or box.

Can I play it here?

Not yet — Zipper 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: