Forcing Chains
Forcing Chains test both candidates in a cell and follow the consequences of each of them.
Learn the techniqueBUG+1
BUG+1 exploits the fact that a valid sudoku puzzle cannot end in a state where all empty cells have exactly two candidates. That state is called a Bivalue Universal Grave.
See the technique in practice
Work through the examples step by step. Each step explains what you see on the puzzle and why the conclusion holds.
Example:
- Look at the whole puzzle. Except for one cell, all empty cells have exactly two candidates left.
- The cell at row 8, column 1 stands out: it has three candidates, 2, 3 and 6.
- A puzzle where absolutely every empty cell had only two candidates would have multiple valid solutions, which is impossible in a real sudoku. This cell must therefore break that pattern.
- Digit 2 appears an even number of times in the cell's row, column and box, while the other two digits appear an odd number of times. Therefore, 2 must go here.
How to recognize the pattern
BUG stands for Bivalue Universal Grave, a state where absolutely every empty cell has exactly two candidates. Such a puzzle cannot have a unique solution, so a valid sudoku can never end there. BUG+1 occurs when the puzzle is one step away: all empty cells have two candidates except one cell that has three.
That one cell must prevent the grave. Among the three candidates, one digit stands out in the number of occurrences in the cell's row, column and box, and it is this digit that must be placed for the puzzle to not collapse into a state with multiple solutions. The pattern often shows up near the end of hard puzzles.
Step-by-step procedure
- Check that all empty cells on the puzzle have exactly two candidates, with one exception.
- Find the exception, that is, the cell with three candidates.
- For each of the three candidates, count how many times the digit appears as a candidate in the cell's row, column and box.
- Place the digit that stands out from the other two in the number of occurrences, because without it the puzzle would end in a state with no unique solution.
Common mistakes
- Using the pattern too early. BUG+1 requires the entire puzzle to be in the state, not just one corner.
- Missing another cell with three candidates somewhere else. If there are two such cells, the puzzle is not a BUG+1, and the conclusion does not hold.
- Forgetting the assumption of a unique solution. The technique rests on the puzzle being a valid sudoku with exactly one solution.
When do you need the technique?
The hardest puzzles require techniques that follow long logical chains through the entire puzzle. They are in practice proof by contradiction: assume something, follow the consequences and see what does not hold. Try your way through the examples below, step by step, using the same tools the solver uses on your own puzzle.
Try it on your own puzzle
Enter your puzzle in the Sudoku Solver and it will find the next step and explain the technique behind it.
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