Jigsaw Sudoku

Click a cell and type 1–9, or use the pad. Backspace clears. Press N to toggle pencil marks. Thick lines mark the nine regions. Everything runs in your browser; nothing is uploaded.

Irregular regions need fewer clues, and the two do not overlap

The natural guess is that squiggly regions make a puzzle harder, so a jigsaw grid ought to need more clues to pin one answer. We took 200 grids of each kind and removed clues in random order for as long as the puzzle still had exactly one solution, stopping when nothing more could go. The standard grids stopped at 24.35 clues on average. The jigsaw grids stopped at 16.53 — a gap of 7.8 clues.

It is not just a difference of averages. The worst jigsaw puzzle in the sample still needed fewer clues (20) than the best standard one (21). In 318 puzzles the two ranges never touch:

Standard sudoku has a proven minimum of 17 clues, established by exhaustive search. Jigsaw puzzles here went down to 12. That is a floor for one removal order rather than a claim about the true minimum — a different order reaches a different floor, which is exactly why the same procedure landed anywhere between 12 and 20.

The obvious explanation is wrong: irregular regions barely constrain more

The tempting reason is that an irregular region overlaps its own rows and columns less than a box does, so it ties down more cells. A 3×3 box sits inside exactly three rows and three columns: of the eight other cells in your box, two share your row and two share your column. Those four get counted twice, so every cell in standard sudoku has 8 + 8 + 8 − 4 = 20 peers — the same for all 81 cells, with no exceptions.

Irregular regions do spread wider: across 200 random layouts a region spans 4.15 rows on average against a box's exact 3, reaching as far as 9, and 67% of them cover more than three rows. So they should double-count fewer peers. They do — by almost nothing:

The real reason: there are far fewer grids to tell apart

Clues do not rule out cells, they rule out whole grids. A puzzle is finished when its clues single out one complete grid from every grid the layout allows — so what matters is how many there were to start with. Fewer grids, less to distinguish, fewer clues needed to distinguish it.

A 9×9 board has far too many to count. A 6×6 board — six regions of six — does not, so this can be settled exactly instead of sampled. The standard 2×3 boxes admit 28,200,960 complete grids, a figure published long before this page and the reason to trust the counter that produced the rest. Every one of 24 irregular layouts admits fewer — about 608,306 on average, 46× fewer.

What makes that a mechanism rather than a coincidence is that it holds inside the irregular group. Sort those layouts by how many grids they admit and the half with fewer grids needs 6.47 clues, the half with more needs 7.8. The count of grids predicts the count of clues.

Pushed to its end, it reaches zero

If irregular layouts admit fewer grids, some should admit none — and they do. A layout can be perfectly well formed, every region connected and the right size, and still have no valid filling whatsoever. Counting exhaustively, 10 of 300 irregular 6×6 layouts are impossible. The generator on this page simply moves on to another layout when it meets one.

That impossibility claim rests on counting every grid, which only the small board allows. The same question at 9×9 cannot be answered this way: a search that gives up has not shown a grid does not exist, only that it did not find one. That is why 82 of the 200 jigsaw seeds above are reported as skipped rather than as impossible. Skipping them also tilts the result — layouts that yield a grid easily tend to be the ones admitting more grids, and more grids means more clues, so the measured gap is if anything an understatement.

So is jigsaw sudoku harder?

Needing fewer clues is not the same as being easier, and the two senses pull opposite ways. Fewer clues needed means the layout carries more of the information, so less of it has to be printed. For a person at the grid that is the harder thing to sit in front of: a 17-clue start gives you less to work from than a 24-clue one. What is measured on this page is how few clues can still pin one answer — a fact about the layout, not about the solving.