# Collection of interesting, but hard to prove, theorems

Here's a list of theorems that are interesting, but beyond the ability of high-schoolers to prove. Some of the proofs will be accessible to undergrads, some only to grads, and some only to the world's most capable mathematicians. These are theorems we should be relaying to K-12 students, in order to show that math can be used to reason about anything, it is not limited to basic theorems for solving equations and basic theorems for proving geometric things. Math is precise reasoning in general. Most of these theorems are covered by Numberphile, and I would recommend showing their videos, or at least, getting ideas from those videos. They are top notch.

Fix a wobbly table theorem
Interesting because it says something about the real world.
Fold and cut theorem
Interesting because you wouldn't think it possible to cut any complex polygon out of paper. You would think that paper-cutting is limited in some way because the operations are so crude.
Abel–Ruffini theorem
Interesting because it shows there's a limit on what polynomials can be solved in general.
Four color theorem
Interesting because it's visual. Anyone can play with the idea, but yet it's extremely difficult to prove. Also interesting because it was a long-standing problem. Further, it's interesting because it was proven partially by computer-program. Further, it was proven recently.
God's number
Interesting because it tells us the maximum number of moves necessary to solve a well-known puzzle, the Rubik's cube. Also, it was proven partially by computer-program. Further, it was proven recently.
Minimum number of clues for a Sudoku to have a unique solution is 17
Interesting because it says something about a puzzle which is currently popular. Also interesting because it was proven recently.
Pólya's recurrence theorem
Interesting because it can be demonstrated visually, and also, you would think that a property which holds in 1D and 2D would carry over to 3D, but in this case it doesn't.
Fermat's last theorem
Interesting because Fermat's tongue-in-cheek comment caused such unrest, and also, because it was a long-standing problem, which was easy to describe, but very hard to prove. Further, it's interesting because it was proven recently.
Ham sandwich theorem, 3D variant
Interesting because it's visual, and because it's proof is non-constructive. This means we have a way of proving a fair cut exists, but we don't have a way of finding it. Numberphile video
Impossibility of squaring the square, or trisecting the angle
Interesting because they are visual. Also, both were long-standing problems, only proven impossible after the introduction of algebra. Further, they are interesting problems because both can be done with specialized tools, they just can't be done with compass and straightedge.
Mohr–Mascheroni theorem
Interesting because it says that one of the two basic tools of geometric construction is unnecessary. This is wildly unexpected. The proof is also unexpectedly easy. One instance of the proof can be found on Wikipedia, and it can be understood by an undergrad, or a dedicated high-school student. The proof would not be difficult for a high-schooler, but it is significantly longer than most proofs intended for high-schoolers.
Kepler conjecture
Interesting because it's somewhat visual. Also, it the proof was verified by a computer program. Further, this theorem was proven in 2017, which is somewhat recent. Numberphile video
Poincaré conjecture
Interesting because it seems intuitive in 2D and 3D, yet was very hard to prove for higher dimensions, especially 4D. Also interesting because it had so many wrong proof attempts. This, at least for me, raises the importance of developing computer-verifiable proofs, as some proofs are ultimately incorrect, and a huge waste of effort to determine them so. Numberphile video