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Sudoku BUG Technique: Bivalue Universal Grave and BUG+1 Solution

2025-06-11 · 8 min read
Knowledge Base / Techniques Index / BUG

BUG (Bivalue Universal Grave) is an advanced Sudoku technique based on the unique solution principle. The core idea is: if all unsolved cells have only two candidates (bivalue state), the Sudoku will have multiple solutions. Since a valid Sudoku must have exactly one solution, we can use this principle to determine certain cells.

Core Principle:
The Bivalue Universal Grave (BUG) state leads to multiple solutions, violating the fundamental rule of unique solution. Therefore, when the grid approaches a BUG state, a specific digit must be placed to break this state and ensure uniqueness.
BUG Principle Diagram
BUG Principle: Left shows near-bivalue state, the red cell is the only trivalue cell, right shows the result after placing the digit

What is the Bivalue Universal Grave State?

When solving Sudoku, unfilled cells have candidates. A bivalue cell is a cell with exactly two candidates. If in a Sudoku grid:

  • All unsolved cells are bivalue cells (each cell has exactly 2 candidates)
  • Each candidate appears exactly twice in each row, column, and box

Then the grid is in a BUG state. In this state, all candidates can be swapped pairwise without violating Sudoku rules, resulting in multiple solutions.

BUG+1 Rule

If all unsolved cells except one are bivalue cells,
Then this single non-bivalue cell must contain its "extra" candidate to break the BUG state.

Example Analysis: BUG+1

Let's look at a typical BUG+1 example. In this grid, almost all unsolved cells are bivalue cells, with only one cell having three candidates.

Sudoku BUG Technique Example
Figure: BUG+1 Example - R6C6 is the only trivalue cell
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Current Grid Data

Based on the CSV81 format candidate data, we list all unsolved cells and their candidates:

Bivalue Cells (14):

  • R3C4: Candidates {6, 9}
  • R3C6: Candidates {6, 9}
  • R4C3: Candidates {2, 6}
  • R4C6: Candidates {2, 7}
  • R4C8: Candidates {6, 7}
  • R6C3: Candidates {2, 6}
  • R6C5: Candidates {7, 9}
  • R6C9: Candidates {6, 7}
  • R7C4: Candidates {6, 9}
  • R7C5: Candidates {7, 9}
  • R7C8: Candidates {6, 7}
  • R9C6: Candidates {6, 7}
  • R9C9: Candidates {6, 7}

Trivalue Cell (only 1):

  • R6C6: Candidates {2, 7, 9} ← BUG+1 Cell

Analysis Process

1 Identify Grid State: Check all unsolved cells. Except for R6C6 with 3 candidates, all other unsolved cells have only 2 candidates. This is a typical BUG+1 state.
2 Understand BUG Principle: If R6C6 also had only 2 candidates (e.g., only {2, 9} or {7, 9} or {2, 7}), all unsolved cells would be bivalue cells, leading to multiple solutions.
3 Find the "Extra" Candidate: Among the three candidates {2, 7, 9} of R6C6, we need to find the "extra" one. The method is to check how many times each candidate appears in the relevant row, column, and box:
  • Candidate 2: In Row 6, 2 appears only in R6C3 and R6C6 (twice)
  • Candidate 9: In Row 6, 9 appears only in R6C5 and R6C6 (twice)
  • Candidate 7: In Row 6, 7 appears in R6C5, R6C6, R6C9 (three times)
4 Determine the Answer: Candidate 7 is the "extra" candidate. If R6C6 is not 7, then candidate 7 in Row 6 would only appear twice (R6C5 and R6C9), and combined with all other bivalue cells, would form a BUG state. Therefore, R6C6 must be 7.
Conclusion:
BUG+1: R6C6 is the only trivalue cell (2, 7, 9), 7 must be placed to avoid multiple solutions.
Action: Set R6C6 = 7

BUG Variants

Besides the basic BUG+1, there are other variants:

BUG+1 (Most Common)

Only one cell has more than 2 candidates. The "extra" candidate of this cell is the answer.

BUG+2, BUG+3...

Multiple cells have more than 2 candidates. This requires more complex analysis, usually combined with other techniques.

BUG+1 (Multi-candidate)

The only non-bivalue cell may have 4 or more candidates. Then there are multiple "extra" candidates, and you need to find the one that breaks the BUG state.

Usage Conditions:
  • The BUG technique relies on the assumption of a unique solution. It doesn't apply to puzzles with multiple solutions.
  • Accurate identification of all candidates is required; any omissions or errors will lead to wrong conclusions.
  • This is an advanced technique, typically used when other techniques fail to make progress.

How to Spot BUG Patterns?

1 Check Candidate Counts: Observe the candidate count of all unsolved cells. If most have 2 candidates, a BUG state may be near.
2 Find Exception Cells: Identify cells with more than 2 candidates. If there are only 1-2 such cells, it's likely BUG+1 or BUG+2.
3 Analyze Candidate Distribution: For non-bivalue cells, analyze how many times their candidates appear in rows, columns, and boxes. Candidates appearing more than twice are "extra".
4 Place the Digit: Place the "extra" candidate in that cell to break the BUG state.
Quick Recognition:
When you find that almost all unsolved cells are bivalue cells with only a few having 3 or more candidates, the BUG technique is likely applicable. BUG+1 is the most common and easiest case to recognize and apply.

BUG and Other Techniques

BUG vs Unique Rectangle

Both are based on the uniqueness principle, but with different approaches:

  • Unique Rectangle: Focuses on a specific rectangle pattern of 4 cells
  • BUG: Focuses on the candidate distribution across the entire grid

Advantages of BUG

  • Can quickly locate key cells in complex grids
  • Simple logic: find the only non-bivalue cell and place the "extra" candidate
  • No complex chain reasoning required

Summary

  • Core Concept: BUG state leads to multiple solutions and must be broken
  • Recognition Condition: All unsolved cells are bivalue cells, with only 1 exception
  • Solution Method: Place the "extra" candidate of the non-bivalue cell
  • Use Case: Grid nearly complete with many bivalue cells
  • Note: Puzzle must have a unique solution
Practice Now:
Start an expert-level Sudoku puzzle and try to spot and apply the BUG technique!
Unique Rectangle X-Chain Techniques Index