Techniques / Uniqueness

Uniqueness techniques: unique rectangles and BUG

Arguments from the fact that the puzzle has exactly one solution.

21 techniques in this family, introduced at tier 6, 2 with a worked example from a real board. How to read a sudoku grid explains the notation used below.

What these have in common

Every move in this family reasons about the puzzle as a made thing, not just as a grid. They ask what layout would give the puzzle two answers. Then they rule it out, because a well made puzzle has one.

Why the logic holds

Picture four cells at the corners of a box shape. They span two rows, two columns and two boxes. They hold only the same two digits. You could fill them two ways, and both obey every rule. So a puzzle with that shape in it has at least two answers. A well made puzzle does not, so something must break the shape. That something is your elimination. The Bivalue Universal Grave runs the same idea over the whole grid. If every open cell held just two digits, and each digit showed up twice in every house, the grid would have two answers. So the one candidate that breaks the pattern is true.

When to reach for it

On a grid that has gone quiet and is full of two digit cells, which is where these shapes live. Scan for two cells holding the same pair in the same two columns. That is usually enough to turn one up.

How the members differ

The Unique Rectangle types are a list of what else sits in the fourth corner. Type 1 keeps the spare digits in one cell, which is the tidiest case. Types 2 through 6 spread them out, and each gets its own move. Hidden and Extended rectangles bend the shape. Avoidable Rectangles run the same idea using digits you have already placed. The BUG forms count how far the grid sits from that flat state.

The mistake to avoid

This is the one family that can hand you a wrong answer rather than no answer. Every move here assumes the puzzle has just one solution. If that is false, so is the conclusion. So never use one on a grid you made up, or on a puzzle from a source you do not trust, or to test whether a puzzle is sound. Each page in this family carries that warning for the same reason.

Every technique in this family

• Avoidable RectangleFour squares can form a rectangle across two boxes.
• Avoidable Rectangle Type 1Three corners of a rectangle were worked out by you, not printed in the puzzle.
• Avoidable Rectangle Type 2The same idea, but two corners are still open and share an extra number.
• Bivalue Universal GraveSay every open square had just two numbers left.
• BUG-LiteThe pattern applied to part of the grid rather than all of it.
• BUG+1Say every open square had two numbers left, and each number had two homes in every group.
• BUG+2The same idea when two squares break the pattern instead of one.
• BUG+3The same again with three odd squares.
• Deadly PatternA shape that would give the puzzle two answers if it ever appeared.
• Extended Unique RectangleA bigger version using six squares and three numbers instead of four and two.
• Hidden Unique RectangleThe rectangle trick again, but the giveaway is hidden.
• Reverse BUGThis one uses the numbers you have already filled in, not the ones still open.
• Unique LoopLike a rectangle but longer, a ring of squares all sharing two numbers.
• Unique RectangleFour squares in a rectangle across two boxes, all holding the same two numbers.
• Unique Rectangle Type 1Three corners of the rectangle hold only the two numbers.
• Unique Rectangle Type 2Two corners hold only the pair.
• Unique Rectangle Type 3Two corners hold the pair.
• Unique Rectangle Type 4Two corners hold only the pair.
• Unique Rectangle Type 5Like Type 2, but the extra number sits in corners that sit across from each other rather than side by side.
• Unique Rectangle Type 6Like Type 4, but the two numbers form an X across the rectangle instead of sitting along one side.
• Unique Rectangle with Missing CandidatesA rectangle where one corner has already lost one of the two numbers.