I haven't made a puzzles yet, but I kind of have the same problem when I cut BBP. I use the good blue painters tape to attach my patterns to when cutting. I found tho, that when I use it on BBP, when I peel the tape off it leaves holes in the wood. It pulls alot of the wood off and leaves holes. So i bought some of the no name, or cheap BPT to try and it woorks great. It doesn't pull any wood off with it. So now I use the cheap tape for all BBP projects. It might work for the picture problem too. It's not as sticky as the good tape. Just a thought.