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Reverse BUG

In a 9 by 9 sudoku, a Reverse BUG is a state of the grid where every solved cell of two particular digits form an unavoidable set of less than 18 cells, without the presence of hidden singles.

If we assume that a valid sudoku must contain a unique solution, then a Reverse BUG cannot occur. Thus, as a solving technique, we can eliminate candidates from cells that cause a Reverse BUG to appear.

Contents

Proof

For any two digits, we have an unavoidable set of 2n cells involving these two digits if and only if these cells occupy exactly n rows, n columns, and n boxes. Suppose we have a Reverse BUG of 2n cells with n < 9, then we have an unavoidable set occupying n rows, n columns, and n boxes. Now, the remaining 9 - n rows, 9 - n columns, and 9 - n boxes will contain another unavoidable set of 18 - 2n cells. As this latter unavoidable set contains no placements, it is a deadly pattern which implies that this sudoku puzzle does not have a unique solution. Therefore, a Reverse BUG cannot occur in a valid sudoku puzzle.

Example

This example is taken from the at Sudoku Players' forums. 

*-----------------------------------------------------------*
| 189   7    -18    |*2     4     5     | 3     19    6     |
| 6     3    *2     |*1     8     9     | 4     5     7     |
| 19    5     4     | 3     7     6     | 129   129   8     |
|-------------------+-------------------+-------------------|
| 1238  1289  1378  | 5     6     12    | 129   78    4     |
| 124   1249  6     | 8     12    7     | 5     1239  39    |
| 5     128   178   | 9     3     4     | 6     78    12    |
|-------------------+-------------------+-------------------|
| 7     128   138   | 6     5     12    | 1289  4     39    |
| 123   6     9     | 4     12    8     | 7     123   5     |
| 1248  1248  5     | 7     9     3     | 128   6     12    |
*-----------------------------------------------------------*

If R1C3 = 1, then R12C34 forms a Reverse BUG on the digits 1 and 2. Therefore, 1 can be eliminated from R1C3.

Reference

See Also

This page was last modified 09:22, 29 August 2009.