The Law of Leftovers: The Key Jigsaw Sudoku Technique

If there's one technique that separates people who breeze through jigsaw sudoku from people who stall on the hard ones, it's the Law of Leftovers. It's the deduction that exists only in irregular-region puzzles — you can't use it in standard sudoku, because the boxes line up too neatly to leave anything over. In jigsaw sudoku, where the regions wander across rows and columns, the Law of Leftovers turns those mismatches into free digits. This guide explains exactly how it works, walks through a clear example, and shows you where to reach for it. Learn it well and most hard jigsaw sudoku puzzles open right up.

New to the puzzle? Read the jigsaw sudoku rules or the beginner walkthrough first, then come back for the technique that levels you up.

The idea in one sentence

A set of complete rows must contain certain digits, and so must the regions that cover most of those rows — so any cells that "stick out" of the regions are forced to hold the same digits as the cells that are "missing" from them.

That's the whole thing. The rest is learning to see it on the grid.

Why it only works in jigsaw sudoku

In a regular sudoku, the three boxes in a band line up exactly with three rows. Three rows hold three complete sets of 1–9; the three boxes hold three complete sets of 1–9; nothing is left over. The geometry is perfectly aligned, so there's nothing to deduce.

Jigsaw sudoku breaks that alignment. A squiggly region might cover most of two rows but poke a cell up into a third row and leave a cell short somewhere else. That mismatch — the "leftover" — is information. Because both the rows and the regions are each guaranteed to contain 1 to 9, the digits in the stick-out cells must exactly equal the digits in the missing cells.

The technique, step by step

Here's how to apply the Law of Leftovers in practice:

1. Pick a band of complete rows — say two adjacent rows (or two columns).
2. Find the regions that lie mostly inside that band. You want regions whose cells are almost entirely within your chosen rows.
3. Spot the leftovers. Some region cells will fall outside the band (the "outies"), and the band will contain some cells that aren't part of those regions (the "innies").
4. Match them. The set of digits in the outies must equal the set of digits in the innies, because removing the shared region cells from both the rows and the regions leaves the same total behind.

When the leftover is a single cell on each side, you get an exact digit. When it's two or three cells, you get a small locked set you can use for elimination.

A worked example

Imagine rows 1 and 2 of a jigsaw sudoku. Together they must contain two full sets of 1–9 (18 digits). Now suppose two regions sit almost entirely inside rows 1 and 2 — but one of those regions dips a single cell down into row 3, and there's a single cell in row 2 that belongs to a different region.

Call the cell that dips into row 3 the outie, and the stray cell in row 2 the innie. The two rows contain everything in the two regions, minus the outie, plus the innie. Since the rows and the regions both total complete 1–9 sets, the outie and the innie must hold the same digit.

So the moment you know the value of that row-3 outie, you instantly know the row-2 innie — and vice versa. One discovered digit hands you a second one across the grid, somewhere a normal scan would never connect.

When to reach for it

The Law of Leftovers shines exactly when ordinary techniques go quiet. Run through your standard moves first — scanning, hidden singles, naked pairs — and when the grid stops giving anything up, go hunting for leftover overlaps. On easy and medium puzzles you may never need it; on hard, expert, and einstein grids, it's often the only thing that breaks a stall.

A practical habit: scan the grid for places where a single region cell pokes out of an otherwise-clean band of rows or columns. Those lone stick-out cells are where the Law of Leftovers pays off fastest.

Tips for spotting leftovers faster

• Look at the most "self-contained" bands. Two rows mostly covered by two regions are the cleanest setups.
• Watch for single-cell leftovers. A region that pokes exactly one cell out of a band gives you an exact digit, not just a set.
• Use the region colors. Tracing the colored boundaries makes the innies and outies jump out.
• Combine with elimination. Even a two- or three-cell leftover set lets you remove those digits from other cells in the band.

Put it into practice

The fastest way to internalize the Law of Leftovers is to find one live overlap on a real grid. Open a hard jigsaw sudoku, look for a band of rows almost fully covered by a couple of regions, and find the cell that pokes out. For the complete method around this technique, see the jigsaw sudoku strategy guide, and for the moves beyond it, the advanced techniques guide.

Frequently asked questions

What is the Law of Leftovers in jigsaw sudoku?

The Law of Leftovers is a technique where you compare a band of complete rows (or columns) with the regions that cover most of them. Because both the rows and the regions must contain 1 to 9, the cells that stick out of the regions must hold the same digits as the cells left over inside the band. It reveals digits no standard technique can.

Why does the Law of Leftovers only work in jigsaw sudoku?

In regular sudoku, the 3x3 boxes line up perfectly with bands of rows, so nothing is ever left over. Jigsaw sudoku's irregular regions break that alignment — a region can poke a cell into a neighboring row — and those mismatched "leftover" cells are exactly what the technique exploits.

How do I use the Law of Leftovers?

Pick a band of complete rows, find the regions that lie mostly within it, then identify the cells that stick out of those regions ("outies") and the stray band cells that don't belong to them ("innies"). The outies and innies must contain the same set of digits, so solving one side reveals the other.

When should I use the Law of Leftovers?

Use it when standard techniques like scanning, hidden singles, and naked pairs stop producing new digits. On hard and expert jigsaw sudoku it's frequently the only move that breaks a stalled grid. On easy puzzles you usually won't need it.

Is the Law of Leftovers the same as innies and outies in killer sudoku?

They share the same underlying idea — comparing a region's contents against a known total — but they're applied differently. Killer sudoku uses cage sums against the number 45; jigsaw sudoku's Law of Leftovers uses the matching digit sets of overlapping rows and irregular regions.