Sudoku Uniqueness Techniques: Rectangles, BUG and Caveats

Why Sudoku uniqueness techniques work

Sudoku uniqueness techniques lean on a promise: a properly made puzzle has exactly one solution. If a pattern on the grid would allow two different solutions, that pattern can never fully form, so you may eliminate whatever candidate would complete it. This is powerful and fast, but it comes with a warning. The logic is only valid when the puzzle truly has a single solution. Hand-made puzzles, some competition grids, and certain generators do not guarantee that, and if you use uniqueness on a multi-solution grid you can derive a false result. Shapedoku validates every puzzle to have exactly one solution before it ships, and its free solver checks the same, so uniqueness moves are safe there. Elsewhere, treat them as a helpful shortcut rather than gospel.

The unique rectangle

The classic pattern is the unique rectangle. Four cells sit at the corners of a rectangle spanning exactly two rows, two columns and two boxes, and all four share the same two candidates, say a 3 and an 8. If those four cells held only 3 and 8, the puzzle could swap them in two valid ways, creating two solutions. Since that is forbidden, at least one corner must contain a third candidate. Suppose three corners are clean 3-8 pairs but r2c7 also holds a 5. Then r2c7 must be that extra digit, or the deadly pattern would form, so you can remove 3 and 8 from r2c7, often solving it outright. That is a type one unique rectangle; other types let you eliminate the shared digits from cells that see the two floor corners instead.

Hidden and avoidable rectangles

A hidden unique rectangle is the same trap seen from the other side, where the extra candidates are buried among many and you use a strong link on one of the shared digits to force the elimination. An avoidable rectangle looks at already placed digits rather than candidates: if three corners are solved in a way that would let the fourth complete a swappable rectangle, the fourth cell cannot take the value that would close it. Avoidable rectangles are easy to miss because you stop scanning solved cells, yet they can appear late in a solve when the grid is mostly filled and you are hunting for one last crack.

BUG and BUG plus 1

The bivalue universal grave, or BUG, is a whole-grid uniqueness idea. A BUG state is a grid where every unsolved cell has exactly two candidates and every candidate appears exactly twice in each row, column and box. Such a grid has two solutions, so it can never be the true endpoint of a valid puzzle. BUG plus 1 is the useful moment just before it: exactly one cell holds three candidates while everything else is bivalue in that balanced way. The digit that appears three times in that cell, rather than the tidy two, must be the answer there, because taking either other value would drop you into the forbidden BUG. Spotting BUG plus 1 near the end of a hard puzzle can save you a long chain. If you enjoy these patterns, Shapedoku is a comfortable place to practise because every Easy through Extreme grid is validated for a unique solution, and you can double-check any tricky position on the free solver at shapedoku.com before trusting a uniqueness move.

• Only use uniqueness when the puzzle is guaranteed to have one solution.
• Never apply it to hand-checked or unknown-source grids you cannot trust.
• Unique rectangles need two rows, two columns and exactly two boxes.
• BUG plus 1 requires one tri-value cell and a perfectly bivalue remainder.