The Reverse-BUG requires some knowledge about unavoidable sets, more specifically two digit unavoidable sets. An unavoidable set is a set of cellvalues that cannot be solved uniquely if none of the values are given. Two digit unavoidable sets always take up exactly two cells in each unit (row, column and box) they occupy. Theres several such sets in every grid, minimum amount is 36 and the theoretical maximum is 144. They come in various shapes and sizes, some are marked here:
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*--------------------------------*
|#8 7 9 |+6 5 #4 | 2 +3 1 |
|+6 5 #4 |+3 2 1 | 9 #8 7 |
|+3 2 1 | 9 #8 7 | 5 +6 #4 |
|----------+----------+----------|
| 5 #4 3 | 7 1 2 |#8 9 6 |
| 2 1 #8 | 5 9 6 | 7 #4 3 |
| 7 9 6 |#4 3 #8 | 1 2 5 |
|----------+----------+----------|
|#4 6 2 |#8 7 5 | 3 1 9 |
| 1 3 *7 | 2 #4 9 | 6 *5 #8 |
| 9 #8 *5 | 1 6 3 |#4 *7 2 |
*--------------------------------*
A special feature for 2-digit unavoidable sets is that unless all 18 instances of the two digits are included, the remaining instances of the digits will also form an unavoidable set:
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*--------------------------------*
| 8 #7 9 | 6 #5 4 | 2 3 1 |
| 6 #5 4 | 3 2 1 | 9 8 #7 |
| 3 2 1 | 9 8 #7 |#5 6 4 |
|----------+----------+----------|
|#5 4 3 |#7 1 2 | 8 9 6 |
| 2 1 8 |#5 9 6 |#7 4 3 |
|#7 9 6 | 4 3 8 | 1 2 #5 |
|----------+----------+----------|
| 4 6 2 | 8 #7 #5 | 3 1 9 |
| 1 3 *7 | 2 4 9 | 6 *5 8 |
| 9 8 *5 | 1 6 3 | 4 *7 2 |
*--------------------------------*
When making a puzzle from the grid above, at least one of the cells marked * AND one of the cells marked # must be among the given values, or the puzzle has multiple solutions.
The Reverse-BUG makes use of this special feature. If all given values of digits A and B were included in one 2-digit unavoidable set, smaller than 18 cells, then the puzzle would have multiple solutions. Therefore we cannot complete such a set in an unique solution puzzle.
If the given instances of digits A and B are placed like this:
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A . . | B . .
B . . | * . .
then the cell marked * cannot have the value A in an unique puzzle.
An example:
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.7.2..3..6.21..45.....7...8....6...4...8.7...5...3....7...5.....69..87.5..5..3.6.
*-----------------------------------------------------------*
| 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 |
*-----------------------------------------------------------*
r1c3<>1
A type 1 UR and one XY-wing then solves the puzzle. Some other examples can be found here and here.
The reverse-BUG is a very rare creature, but when it appears it is very easy to spot as you only need to consider already solved cells. The smaller the creature is, the easier it is to spot. When it gets big enough, it is usually easier to spot the potential deadly pattern from the unsolved cells:
(UR-puzzle posted by Mike Barker here.)
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...1.49..7..9..42....7..6...6........2...5...8...3...4.8.....7....5.....231.6...5
*-----------------------------------------------------------*
| 3 *5 6 | 1 2 *4 | 9 8 7 |
| 7 1 8 | 9 *5 6 |*4 2 3 |
| 9 *4 2 | 7 8 3 | 6 *5 1 |
|-------------------+-------------------+-------------------|
|#45 6 -3459 | 28 7 19 | 25 13 89 |
| 1 2 39 | 68 *4 *5 | 7 36 89 |
| 8 79 579 | 26 3 19 |-25 16 *4 |
|-------------------+-------------------+-------------------|
|#45 8 #45 | 3 9 2 | 1 7 6 |
| 6 79 79 |*5 1 8 | 3 *4 2 |
| 2 3 1 |*4 6 7 | 8 9 *5 |
*-----------------------------------------------------------*
Reverse-BUG in r1c26, r2c57, r3c28, r5c56, r6c79, r8c48, r9c49: r6c7<>5
UR in r47c13: r4c3<>45
RW