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A '''Bivalue Universal Grave''' or '''BUG''' is a state of the [[grid]] in which every unsolved [[cell]] has 2 [[candidate]]s, without the presence of [[Hidden Single]]s.

This is one of the [[Deadly Pattern]]s that do not occur in a [[valid]] [[Sudoku]].

For some people, it is also a [[solving technique]]. The aim of this technique is to avoid a BUG pattern, by avoiding moves that would cause it to appear. As with other techniques that depend on the assumption that the Sudoku has a unique solution, it is the subject of debate. See [[Uniqueness Controversy]].

== Proof of non-uniqueness ==
In a BUG, each unsolved cell has two candidates, and in each row, column and box [[constraint]], each candidate value occurs exactly twice. If we can find a valid solution using half the number of remaining candidates, all the eliminated candidates would also satisfy each constraint exactly once. This other half will also give us a solution to the puzzle. There is also the possibility that the puzzle does not have a solution at all. In that case, the puzzle is also invalid. The existence of the BUG pattern allows us to conclude that the puzzle is invalid. It has either no solution or 2 solutions.

== Assuming uniqueness ==
When your aim is to prove that the Sudoku has a unique solution, finding a BUG means that you can stop trying. For you, there is no other use for the BUG pattern.

When you trust the maker of the puzzle, who has guaranteed that it has a unique solution, you can use techniques that will prevent the pattern from being formed. Note, however, that since the BUG pattern ''cannot'' occur in a well-formed puzzle, such a configuration can also be solved using [[coloring]] techniques without having to exploit the solution's uniqueness.

== How it works ==
The whole idea of this solving technique is that if we have a grid that is formed by a BUG grid plus extra candidates, then at least one of the extra candidates must be placed. The term '''BUG+''n''''' is used to denote a BUG grid with ''n'' cells with one or more extra candidates. The simplest form is '''BUG+1''', where two BUG candidates may be immediately eliminated from the one cell with extra candidates. If there is only one extra candidate, the result is a placement.

=== BUG+1 example ===
[[Image:BUG.png]]

This example shows a BUG+1, where the extra candidate is the digit 9 in '''r1c5'''. If '''r1c5''' is not 9, then we obtain the BUG pattern. Therefore, 9 can be placed in that [[cell]], and the Sudoku can be solved easily using [[single]]s.

As noted above, the same deduction can be made by coloring. Specifically, the chain
r1c3=9 => r7c3=1 => r7c5=3 => r6c5=9
leaves no candidates for digit 9 in box 2.

=== BUG+2 example ===
The following example is taken from the [http://www.sudoku.com/forums/viewtopic.php?t=4916 Sudoku Players' forum]:

[[Image:BUG-2.png]]

This is a case of BUG+2, where the extra candidates are 7 in '''r7c2''' and 8 in '''r8c3'''. This implies that either '''r7c2=7''' or '''r8c3=8'''. By the following [[Forcing Chain]]s:
r7c2=7 => r3c2=2 => r4c2=8
r8c3=8 => r5c3=2 => r4c2=8

we deduce that '''r4c2=8'''.

Actually, for this example, a much simpler deduction can be made. Because '''r7c2''' and '''r8c3''' see each other, we can immediately conclude that '''r7c2<>8''' and '''r8c3<>7''' (by a simple proof by cases).

== Practice puzzles ==
The following two puzzles are practice puzzles for BUG+1.
007000200000850010060002005900030140208001000004000000000500700402000000000017008
920000000080000400000100853008000000073040060090000070040500001009004630000260000

== External Links ==
* [http://forum.enjoysudoku.com/the-bug-bivalue-universal-grave-principle-t2352-15.html The BUG (Bivalue Universal Grave) principle] (''Sudoku Players' forum'')

== See also ==
* [[BUG Lite]]
* [[Reverse BUG]]
* [[Reverse BUG Lite]]
* [[Uniqueness Test]]
* [[Unique Rectangle]]
* [[Deadly pattern]]
* [[Uniqueness Controversy]]

[[Category:Uniqueness]]
[[Category:Solving Techniques]]

BUG - Revision history

Rooted: Created page with "A '''Bivalue Universal Grave''' or '''BUG''' is a state of the grid in which every unsolved cell has 2 candidates, without the presence of Hidden Singles. Thi..."