BUG and BUG+1 in Sudoku X: How Diagonals Change the Grave

What is the BUG pattern?

BUG stands for Bivalue Universal Grave. It describes a specific, unsolvable state in a Sudoku grid. In a true BUG pattern, every unsolved cell has exactly two remaining candidates. Beyond that, every remaining candidate digit appears exactly twice in each row, column, and 3x3 box. This perfect two-candidate balance creates a deadly pattern where the remaining cells could be swapped, resulting in at least two valid solutions for the puzzle. Since a proper Sudoku must have a single unique solution, a true BUG pattern cannot exist in a valid puzzle. Recognizing this pattern is key to using the related BUG+1 Technique to make a logical deduction.

What is BUG+1?

BUG+1 is the logical escape route from a near-BUG situation. It occurs when the grid is one step away from becoming a full BUG. Specifically, all but one of the unsolved cells have exactly two candidates. That single exceptional cell has three candidates. The logic of BUG+1 states that to avoid collapsing into the invalid BUG pattern, the correct digit for that triple-candidate cell must be the one that breaks the perfect two-candidate balance. You identify it by checking each of the cell's three candidates against its row, column, and box. The correct digit will be the one that appears three times in at least one of those units, thereby preventing the symmetrical, two-solution state of a BUG.

How do I spot a BUG+1?

First, examine the grid. If you see that almost every unsolved cell has exactly two pencil marks, you might be close to a BUG+1. Count the cells with three candidates; there should be exactly one. This is your '+1' cell. For each of its three candidate digits, check how many times that digit appears as a candidate in the cell's row, column, and 3x3 box. Two of the digits will appear exactly twice in all relevant units, maintaining the deadly balance. The one digit that appears three times in any of its row, column, or box is the solution for that cell. This deduction often solves very tough puzzles where simpler techniques like Naked Pairs or X-Wing have been exhausted.

• Practice on nearly-solved puzzles. The BUG+1 pattern often appears in the final stages of solving.
• Double-check your candidate notation. Missing a pencil mark can lead to incorrectly identifying a BUG+1.

What changes in Sudoku X?

Sudoku X adds two main diagonals as additional constraint units. This changes the application of BUG and BUG+1 in a foundational way. The core definition of a BUG pattern is expanded: for a position to be a true Bivalue Universal Grave in Sudoku X, every remaining candidate must occur exactly twice not only in each row, column, and box, but also in each of the two main diagonals. Consequently, when applying BUG+1 logic, you must also check the diagonals. The correct candidate in the '+1' cell must be the one that prevents the perfect two-candidate balance across all units, which now includes the diagonals. It may be the digit that appears three times in a row, column, box, or in one of the diagonals.

Can BUG reasoning ever be unsafe?

Yes, BUG and BUG+1 logic is only valid under strict conditions. The most common error is misapplying it to a grid that is not in a near-BUG state. If more than one cell has three or more candidates, the logic does not apply. The technique also relies on the assumption that the puzzle you are solving is well-constructed and has a single unique solution. While this is true for most published puzzles, it's a logical prerequisite. In Sudoku X, failing to account for the diagonal constraints is another way to make an unsafe deduction. Always verify the two-candidate state of all other unsolved cells and the exact candidate counts in all units, including diagonals, before making an elimination. This technique is a powerful last resort, similar in strength to a complex Y-Wing chain.