Sudoku techniques · Extreme
BUG+1 in Sudoku
BUG+1, short for Bivalue Universal Grave plus one, is a uniqueness argument. If every unsolved cell had exactly two candidates and every number appeared exactly twice in every unit, the grid would have two solutions, which a valid puzzle never does. So when the board reaches that state except for one cell with three candidates, the candidate that breaks the pattern must be the answer.
Why it works
In a puzzle with a unique solution, suppose every unsolved cell has two candidates except one cell with three. If removing one candidate from that cell would leave every digit appearing as a candidate either zero or two times in each row, column and box, that candidate must be true: otherwise, any solution could be mirrored by swapping each unsolved cell to its other candidate.
How to spot it
You do not hunt for this; you notice it late in a hard puzzle when the board goes uniformly bivalue.
- Check that every unsolved cell has exactly two candidates, except one with three.
- In that one cell, find the candidate that appears an odd number of times in its row, column and box. That is the answer.
- Like Unique Rectangle, this only holds because the puzzle is guaranteed to have one solution: it is not pure logic on the grid alone.
Builds on
Worked example
These are the app's own lesson boards. Step through them the same way a hint does.
- 1
Every unsolved cell left on the board is bivalue: it holds exactly two candidates. The only exception is the highlighted cell, which holds three: 2, 4, 6.
- 2
Now look at the highlighted cell's row, column and box: the candidate 6 appears three times in each of them, while every other candidate appears exactly twice, if at all. If this cell were not 6, every candidate would appear exactly twice everywhere, a deadly pattern that could be completed in two different ways. The puzzle has only one solution, so 6 must go here, and 2 and 4 can be removed.