Today is National Pretzel Day. Why? I have no earthly idea, but it is, and so here we are. Everyone (some, anyway) loves a good puzzle, and so to honor this day we’ve set aside to celebrate dough twisted a funny way, we at *Inverse* have compiled eight (8 being the most pretzel-looking numeral available) of the most frustrating, mystifying, mind-bending conundrums from the realms of math and science. Hopefully you get to enjoy this serving of brain-pretzels alongside an actual, less infuriating pretzel.

**1.** P versus NP

MIT calls this the most notorious of all theoretical computer science conundrums: Can all problems whose solutions can be verified by a computer also be quickly solved by a computer? So, does P = NP? Most scientists apparently believe P does not equal NP, but no one can prove it one way or the other. This is one of the six remaining “Millennium Prize Problems,” which is a fancy way of saying the Clay Mathematics Institute will literally pay you $1 million to solve this.

Twenty years ago, *Scientific American*’s Martin Gardner offered the tidy sum of $100 to anyone who could solve this iteration of the centuries-old magic square puzzle. No one has, so we can only hope your $100 prize would include interest.

This pertains to model theory. Vaught’s conjecture holds as follows: the number of countable models of a first-order complete theory in a countable language is finite or ℵ0 or 20. Counterexamples to solve it in full are periodically proposed, but nothing’s made the cut so far. The University of California Berkeley had a whole conference on this last year.

**4.** Lithium

When the universe was born, there was a pretty immediate reaction creating hydrogen, helium, and lithium. Scientists can account for the first two gases — but a huge percentage of the lithium is gone. No one knows where it went or even how it’s possible that it went anywhere to begin with. Research turns up only about one third of the gas, according to *Science News*. The rest is just gone, somehow.

This is one of those ones that seems really easy to solve, and so the fact that it remains a mystery is kind of stressful. Devil’s Kettle is a waterfall in Minnesota, which sounds straightforward enough except for one thing – no one knows where the water goes. That is dumb, I can hear you saying. Can’t scientists send, like, waterproof robotic cameras to follow it down? For that matter, can’t you just drop stuff and see where it eventually pops out? Not so far, no.

Since its introduction in 1939, mathematicians continue trying and failing to create a successful proof for this thing. No one’s even gotten close.

**7.** Whale sharks

First things first, whale sharks are the absolute coolest. But what makes these beasts even more intriguing than they already are is that no one knows where they give birth. Scientists have attempted to track the females for years, only to watch them just sort of fall off the map. Which is a nice thought, really – there are white spaces on the map, corners of the world we can’t find.

Technically this was proved in the ‘90s, but it’s too central to this kind of list to not include, the platonic ideal of tantalizing unsolved problems (despite the fact that it’s since been, you know, solved). Even if you’re not a mathematician, there’s a decent chance you’ve heard of Fermat’s Last Theorem. The unsolved proof has wormed its way into popular culture and was finally proved by Andrew Wiles in 1994. Simply put, the theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n strictly greater than two. Before Wiles, mathematicians had been wrestling with it for more than 350 years.