A mathematician is killed and Sherlock discovers some mysterious math equations written in hidden ink on his walls. As he and Watson investigate, they discover that the slain mathematician and an unknown partner were working on the P vs NP problem. The show posits that someone has solved it, and the solution is the driving MacGuffin of the episode. I have to give the show credit for getting the essential details of the P vs NP problem and its implications correct. That said, I do have two quibbles with the plot.
First, no mathematician writes their work in paint on their walls. It's not remotely secure, even if you do need a UV light source to see it, and mathematicians are just like regular folk; they don't always get everything perfect the first time, so writing your work on your walls with paint would be madness. But I get that really the producers just liked the effect of hidden writing appearing on the wall and it's a minor detail so no big deal.
A much bigger deal is the solution they posit. Without getting too into the messy details, the P vs NP problem has two potential solutions. One solution would prove that P = NP while the other would prove P != NP. The show goes with someone finding a solution that is P = NP, which would essentially upend life as we know it on earth, and that's not an exaggeration. If P = NP, the solution proving it would necessarily involve showing how to solve a massive collection of related problems that would effect everything from cryptography to artificial intelligence to your weather forecast. Virtually every mathematician and computer scientist I have ever met, myself included, believes that P != NP, and that if the P vs NP problem does have a solution, it would be one proving the opposite of what they depict in the episode. I get that it facilitated what they wanted for the plot but it still feels a little sloppy, especially how they hand wave it away at the end by having he NSA "seize" everything so that they don't have to deal with the implications down the road.
To me it's a bit like they had an episode where Sherlock and Joan discover that there is real alchemy that can turn lead into gold, and at the end of the episode having found the secret recipe before the bad guys Sherlock says something like "I've put the secret recipe in my bank deposit box and we shall never discuss making more gold again, Watson." I mean, yah, you can write whatever, but who's not going to believe that people would level cities to get their hands on that recipe, right? A solution to P vs NP where P = NP would be like that, but worth much, much more. So a decent episode, with an asterisk that most people will probably not notice or care about; like an excellent meal at a restaurant with one tiny flaw that you can pick at all night. Just the way I like it.
