The resulting cluster extension leaves nothing but clues, bv and a single cell with three candidates, a multiple solution situation the Sudoku community labels BUG + 1. BUG stands for Bi-Value Universal Grave, the reduction to clues and bv alone, which means either two solutions, or no solutions. If uniqueness is trustfully guaranteed, and we have a BUG +1, we can confirm the candidate whose removal would reduce the 3-candidate cell to a bv, because we know the puzzle has a single solution.
That isn’t appropriate here, of course, because the extended cluster has made multiple solutions obvious. We have three solutions, two green ones that interchange 7 and 9, and the blue one. In the BUG +1, each choice for confirmed candidate gives us a different valid solution!
With uniqueness guaranteed, we could also invoke a unique rectangle uniqueness method, insisting that the blue 8 remain in NWr1 and removing the green 8 in r3c1. This would identify a solution, but deny the obvious, and miss what follows.
In the next challenge, following up the alternative that
SdC Cc4 = 248 and SdC Sc4 =937, the blue/green cluster first traps 3 and 9 in r1c3, 9 in r4c2 and 3 in r8c4. Extensions trap 9 in r9c4 and r8c9.
The next extension forces two green 7’s in c1, confirming blue. Two clusters are imposed, to limit solutions. Purple and orange conflict, so pink or red must be in any solution. To enumerate solutions, we can examine :
pink & red, pink & orange, and red & purple
Each combination gives a solution, but 7 and 9 interchange in pink & orange, giving two.
Here are the four solutions: pink and red in upper left, red & purple in upper right, pink & orange in lower left and right. The pink & orange solutions match the green solutions found earlier, so we have now found five solutions.
So now to follow up on the remaining alternative,
SdC Sr9 = 2(3+9)(4+7).
The removal of 4-candidates from r9 leaves a finned 4-wing. On the 4-panel below, three victims appear. The victim of r5c9 is in the fin box, but the other two two imply the fin. Did you remember Kraken analysis?
The marking on the wing’s E9 goes:
E9=>(W9, E4m=>C3=>(C7,S3=>(S2=>Nc6np29=>NW4=>(NW5, SW5), SW9=>(NW9, NW2=>(N2, N9)))).
This leaves a 3-chain toxic set elimination in r5c1, and coloring produces a color trap in r1c2. The extended cluster traps 9 in r8c4 and 3 in r8c7, then forces two blue candidate in r3c6, proving green.
Surprisingly, the green solution below matches none of the previously found solutions, bringing the total number to 6!
Our encounter with the loosey goosey and difficult USAPE at least shows us the ability of Sue de Coq and Medusa coloring to enumerate multiple solutions.
With all of the advanced tools we have covered over the past year, it is hard to imagine we would ever need to rely on uniqueness methods to solve a puzzle, unless it is deliberately composed for this purpose. Nevertheless, I will spend the next few posts describing and analyzing what Sudoku writers have to say about uniqueness methods. Remember to read the label. They must be used with caution.
I’ll start with BUG+1, so perhaps you would like to bring to that point a classic example presented by ruud in www.sudocue.net