How would more experienced puzzlers go about this?
We - or at least I - would use a solving program.
I think you're right: there's likely no "nice" human-understandable path to solving this. The board is a 5×13 grid with 5 cells removed, which is very regular. That would make me suspect that the puzzle was computer-generated, even if you hadn't included what appears to be code output in your screenshot. And more importantly, there's no reason to assume there would be a "nice" solution. Packing puzzles are hard - if they weren't designed with a particular logical path in mind, it's very unlikely that they have one at all.
I agree with your deduction about the cells taken up by the five rightmost pentominoes, but that's the only real in-road. If we take "there is exactly one way to do this" as a given, we can rule out the 5×5 grid (because if there was a solution involving it, we could rotate only that grid to produce a different solution). But then there's nothing else you can do.
So, brute force is likely the only option. But I imagine you're looking for something better than "use a computer". If I was, say, locked in a room with only pencil and paper until I solved this by hand, and told that there is exactly one solution, here's what I'd do:
- Use the logic above to show that the "dented" shape must be correct.
- Calculate 12 choose 5, and get 792.
- Enumerate all 792 5-subsets of the pentominoes.
- Try filling the dented square with each 5-subset; cross out all the ones that don't work. (If I'm unsure of whether I've missed anything, I can pick a certain square to branch on - look at all pieces that can fill that square, and then try out each individual case there.)
- When I can successfully fill the dented-square with a 5-subset, attempt to pack the remaining pieces into the left side of the board. It would likely be easiest to bruteforce by branching first on whatever goes into the dent (which must be one of
LINVY), and then I'd probably pick a nearby square to further branch on if necessary.
It would still be a slow, laborious process... but it would at least be guaranteed to end eventually.