Expert Guide

Uniqueness Techniques: Unique Rectangle & BUG+1

Two expert techniques that solve cells not from the digits on the grid, but from a promise every published puzzle makes: that it has exactly one solution.

6 min read·← Not there yet? Read the Advanced Guide

A Different Kind of Deduction

Every technique in the Advanced Guide — pairs, X-Wing, colouring — reasons only from the digits already placed and the three basic rules. Uniqueness techniques add one more premise: a properly set puzzle has exactly one solution. If assuming a candidate would allow the finished grid to be completed in two different ways, that candidate must be wrong — not because the rules forbid it, but because the puzzle's setter guaranteed a single answer.

That premise is worth being explicit about. These are not "validation" techniques — applying them never checks or proves anything about the puzzle. They simply spend the uniqueness guarantee the same way other techniques spend the row, column, and box constraints. On any properly published puzzle the deductions they produce are exactly as sound as a naked single.

The Deadly Pattern

Both techniques rest on the same observation. Imagine four cells forming a rectangle — two rows, two columns, and (crucially) spanning exactly two boxes — whose solved values are the same two digits arranged diagonally: A-B on the top edge and B-A on the bottom. Swap all four values (B-A / A-B) and every row, column, and box is still perfectly legal. Two interchangeable arrangements means two complete solutions.

A puzzle with one solution therefore can never finish in that state — it's called the deadly pattern. Any candidate that would force the grid into it can be eliminated on the spot. The two-box condition matters: if the four cells touched four different boxes, swapping the digits would break the box constraint, the arrangements wouldn't be interchangeable, and nothing could be concluded.

Unique Rectangle

The simplest and most common form (Type 1): three corners of such a rectangle are bi-value cells holding exactly the candidates {A, B}, and the fourth corner holds {A, B} plus at least one extra candidate.

Suppose the fourth corner resolved to A or B. The remaining three corners would then be forced into the alternating A-B arrangement — the deadly pattern — and the puzzle would have two solutions. Since it can't, the fourth corner must take one of its extra candidates: eliminate both A and B from it. If only one extra candidate remains, you've just solved the cell outright.

Spotting it: scan for pairs of bi-value cells with identical candidates in the same row or column. When you find two such pairs aligned into a rectangle across exactly two boxes, check the corners — a corner carrying extra candidates is where the elimination lands. Further variants (Types 2–6) handle extra candidates spread over two corners, but Type 1 does most of the work in practice.

BUG+1

BUG stands for Bivalue Universal Grave: a hypothetical grid state where every unsolved cell has exactly two candidates and every candidate digit appears exactly twice in each row, column, and box it belongs to. Like the deadly rectangle, a BUG state can never pin down a single answer — a puzzle that reached it would not have a unique solution.

BUG+1 is the practical payoff. Late in a puzzle you'll sometimes find the whole grid one cell away from a BUG: every unsolved cell is bi-value except a single cell holding three candidates. One of those three candidates appears three times in that cell's row, column, and box — the other two would each leave a perfect BUG behind. To keep the puzzle uniquely solvable, the odd-one-out candidate must be the answer: place it immediately.

BUG+1 is a wonderful endgame technique precisely because it fires when the board is nearly full of slippery bi-value cells — the exact situation where chains are tedious to trace by hand.

When They Apply — and When Our Solver Benches Them

The uniqueness premise is only true of proper puzzles. A board that is still being typed in, or one imported from a photo with a few digits missed, may have many completions — and on such a board a uniqueness argument can place a genuinely wrong digit. Constraint-based techniques never have this problem: a naked single found on a half-entered board stays correct after the missing clues are added.

Our solver handles this automatically. When you ask for the Next Step, the backend checks how many solutions your current board has. If there's more than one, Unique Rectangle and BUG+1 are quietly set aside for that request while every other selected technique stays in contention — and if they were the only techniques that could have helped, the hint tells you so. Once your board has a single solution, they rejoin the roster. You can also enable or disable them yourself in the technique settings behind the gear button next to Next Step.

A note for purists: some solvers prefer never to use uniqueness information, on the grounds that a solution should be derivable from the grid alone. That's a legitimate style choice — untick both techniques in the settings and the solver will respect it. On a proper puzzle, both roads lead to the same single solution; the uniqueness road is often just shorter.

Watch a Unique Rectangle fire in real time

Load a tough puzzle, leave Unique Rectangle and BUG+1 ticked in the technique settings, and let the step-by-step hints show you the deadly pattern being dodged.

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