TITLE: What is the shortest string that contains all permutations of an alphabet?
QUESTION [50 upvotes]: What is the shortest string $S$ over an alphabet of size $n$, such that every permutation of the alphabet is a substring of $S$?
Edit 2019-10-17: 
This is called a superpermutation, and there is a wikipedia page that keeps track of the best results. Turns out the problem is still open.

REPLY [9 votes]: I researched this question 20 years ago and found that the length of the shortest string containing all the permutations of n objects to be as stated in http://www.notatt.com/permutations.pdf. We created a computer algorithm to generate all possible strings containing all permutations of n objects and proved this minimal length through brute force for alphabets up to 11 objects. We never could find a proof that our algorithm generated the shortest strings for any n and I would love for someone to pick this subject up. I've found that most mathematicians disregard this topic as already done when in fact, upon close examination, it has not been proven. If anyone knows of such a proof, please pass it along. You can find our paper at Minimal Superpermutations, Ashlock D., and J. Tillotson, Congressus Numerantium 93(1993), 91-98.