Law of Leftovers
The Law of Leftovers (LoL) is the signature technique of Jigsaw Sudoku — the variant where the nine regions are irregular shapes instead of neat 3×3 boxes. It uses the mismatch between those wobbly regions and the straight rows and columns to place and eliminate digits, and nothing else quite replaces it.
The idea
Take any block of rows — say the top three. Those three rows contain exactly three of every digit (three full rows, 1–9 each). Now look at the irregular regions that lie mostly within those three rows. Some region cells poke out below the block; and to make up the numbers, cells of other regions poke in from below. The law: the cells sticking out and the cells sticking in must contain exactly the same digits. They're the "leftovers", and they balance.
Because the regions in Jigsaw don't align with the grid lines, these leftovers are usually just one or two cells — small, precise sets whose contents you can pin down. Match a single outie against a single innie and you often place a digit outright.
The top three rows (above the bold line) form a band. The blue jigsaw region has one cell poking below the band — an outie — and the green region has one poking in — an innie. The two pink leftovers must hold the same digit (here 5).
Step by step
- Pick a block of rows or columns — one, two or three.
- Find the leftovers. Region cells poking out of the block, and region cells poking in.
- Match them. The outies and innies hold the same digits.
- Place or eliminate — a single-cell match is solved outright; larger matches constrain candidates.
Why only Jigsaw?
In Classic Sudoku the regions are the 3×3 boxes, which line up perfectly with rows and columns — nothing ever "leaks", so there are no leftovers to exploit. The Law of Leftovers exists precisely because Jigsaw's regions are irregular. It's closely related in spirit to Killer's innies and outies, but where those balance sums, the Law of Leftovers balances digit sets.
Common mistakes
- Wrong block. The rows or columns you choose must be complete bands, or the counting breaks.
- Miscounting leftovers. Every cell poking out must be matched by one poking in — get the balance exact.
Which Sudoku types is this best for?
The same logic applies across variants, but it pays off more in some than others.