Jigsaw Sudoku: Rules, Strategy, and the Law of Leftovers

Swap the nine 3×3 boxes for nine irregular shapes and you get jigsaw sudoku, sometimes called irregular sudoku or squiggly sudoku. Every row and column still needs the digits 1 through 9 exactly once. What changes is the third rule: instead of a box always three cells wide and three tall, you get a region that might climb four rows in one spot and duck sideways in another.

That swap does more damage to your solving habits than it looks like on paper. In classic sudoku you can eyeball a box and its two neighboring rows almost without thinking, since box edges line up with tidy three-cell chunks. A jigsaw region ignores that grid, poking one or two cells into a row while stretching through three or four others. That is good news once you know how to read it.

The one rule that changes

Rows and columns behave exactly as in classic sudoku: nine cells, digits 1 to 9, no repeats. Only the third rule moves. Each of the nine regions is a connected, oddly shaped group of nine cells, drawn with a heavy outline that stays fixed for the whole puzzle.

Because a region can wander across several rows and columns instead of sitting inside three of each, it touches more of the grid than a box ever could. A snaking region crossing five rows shares constraints with all five at once, which sounds harder but is a gift to a careful solver. Try it at our free jigsaw sudoku, newly generated with a guaranteed single solution, notes and hints built in, no sign-up required.

Reading the region shapes before you start

Before filling in a single digit, trace every region boundary with your eyes, cell by cell. It costs thirty seconds. Skip it and the single most common jigsaw mistake follows: placing a digit as if a cell belonged to the region next door. Every deduction built on that collapses, and you often will not notice until a contradiction shows up several moves later.

Pay particular attention to regions that snake along a diagonal or double back on themselves, those get misread most. Where a region shares a nearly invisible border with its neighbor, use the notes feature to mark the region number until the shape becomes familiar.

Region scanning walkthrough

Region scanning works the same way box scanning does in classic sudoku, you are just checking more lines. Picture region D, a zigzag dipping through rows 4, 5, and 6, missing the digits 2, 6, and 9. Start with 9. Row 4 already has a 9 in column 8, and row 6 already has one in column 2, ruling out both rows inside region D. Row 5 is still open, but if column 3, one of the columns region D reaches into, already carries a 9 higher up the grid, that cell drops out too. Cross cells off this way and you are usually left with exactly one open spot, here row 5, column 5.

Run the same check for 6. If two of region D’s columns already have a 6 placed further up the board, only its cells in row 6 survive, and the shape leaves just one open cell there, your 6, no guess required.

Region-line interaction

If every remaining candidate cell for a digit inside a region falls on the same row, that digit cannot appear anywhere else on that row outside the region, since it has to go somewhere inside the region and nowhere else is left to hide it.

Take region C, a hook shape spanning rows 2 and 3, missing the digit 4. After marking candidates, only two open cells remain: row 2, column 1, and row 2, column 4. Both sit in row 2, so the 4 for region C must land there, and every other empty cell in row 2 outside region C, say columns 6 and 9, can have the 4 erased. The same logic runs sideways for columns.

The Law of Leftovers

This is the technique jigsaw regulars lean on more than any other, and the one most guides rush through. It comes from puzzle designer Bob Harris, and once it clicks you will start hunting for it on every irregular grid.

Start with the bookkeeping every sudoku shares: a complete row sums to 45, and three complete rows sum to 135. Now watch what happens when regions straddle the edge of those rows. Take rows 1 through 3, twenty seven cells that must sum to 135. Two regions, A and B, sit entirely inside those rows, each contributing a clean 45. A third region, C, mostly sits inside rows 1 through 3, but two of its cells dip down into row 4, call those its outies. A fourth region, D, lives mostly in row 4 and below, but two of its cells reach back up into row 3, its innies.

Rows 1 through 3 still need to total 135. Regions A and B already supply 90 of that. The remaining 45 comes from region C’s seven cells inside those rows plus region D’s two innie cells. Region C also sums to 45 across its own nine cells, so its seven inside cells equal 45 minus its two outie cells. Put those facts together and region C’s outie total has to equal region D’s innie total exactly, even before you know either number.

The cleanest version shows up when one region leaves a single cell sticking out and another leaves a single cell reaching in. Matching totals for two single cells means those digits are identical, not just similar. Spot a lone outie and a lone innie across the same row boundary and you have solved both cells at once. With two or three leftover cells on each side, the totals still have to match, and combined with digits already placed nearby, that usually narrows things to one answer within a move or two. Count the stray cells whenever a boundary almost lines up with a row or column edge, that shared total is the Law of Leftovers at work.

Common mistakes

Beginners lose the most time to three habits. The first is scanning a region as though it were still a 3×3 box, checking only the lines near its center instead of every row and column the shape actually touches.

The second is misreading the boundary itself, especially where two regions interlock along a diagonal. A digit placed one cell into the wrong region quietly corrupts every deduction built on it. If a puzzle stops making sense a few moves after a placement, the boundary is usually where to look first.

The third is guesswork once the easy singles run out, instead of reaching for the Law of Leftovers. Guessing works until it does not, and on a tough grid it usually does not. Our beginner’s guide to sudoku covers the fundamentals that carry over directly, and when you are ready to practice, play jigsaw sudoku for free on relaxed or challenge mode, with hints on hand for moves that do not click yet.

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