Maybe I'm just a glutton for punishment, but for me, intractable and even undecidable problems are, if you pardon the pun, endless fun. Routing of PCBs and ICs

Maybe I'm just a glutton for punishment, but for me, intractable and even undecidable problems are, if you pardon the pun, endless fun. Routing of PCBs and ICs is one of my favorites, and decompliation/disassembly is another. I'm just totally content with knowing that I'll never be "that guy" who gets famous for finding "the" perfect solution (especially if we can already prove a perfect solution doesn't exist). But with enough effort, I might be one of the many people who come up with one of the many increasingly better incomplete solutions. A small step in the right direction, like the approximations of Schrödinger's equation you mentioned, can be very useful as well as an amazingly fun challenge. It's a bit like those video games that keep getting tougher but never end; all the fun is in seeing how far you can get.