vertex and every permutation is connected by an edge. Each edge has a weight associated with it; the weight is calculated by seeing how many characters can be added to the end of one permutation (dropping the same number of characters from the start) to result in the other permutation. For instance, the edge from 123 to 312 has weight 2 because 123 + 12 = 12312 = 312. Any hamiltonian path through the created graph is a superpermutation, and the problem of finding the path with the smallest weight becomes a form of the traveling salesman problem. The first instance of a superpermutation smaller than length was found using a computer search on this method by Robin Houston. Lower bounds, or the Haruhi problem In September 2011, an anonymous poster on the Science & Math ("/sci/") board of 4chan proved that the smallest superpermutation on n symbols (n ≥ 2) has at least length n! + (n−1)! + (n−2)! + n − 3. In reference to the Japanese anime series The Melancholy of Haruhi Suzumiya, the problem was presented on the imageboard as "The Haruhi Problem": if you wanted to watch the 14 episodes of the first season of the series in every possible order, what would be the shortest string of episodes you would need to watch? The proof for this lower bound came to the general public interest in October 2018, after mathematician and computer scientist Robin Houston tweeted about it. On 25 October 2018, Robin Houston, Jay Pantone, and Vince Vatter posted a refined version of this proof in the On-Line Encyclopedia of Integer Sequences (OEIS). A published version of this proof, credited to "Anonymous 4chan poster", appears in . For "The Haruhi Problem" specifically (the case for 14 symbols), the current lower and upper bound are 93,884,313,611 and 93,924,230,411, respectively. This means that watching the series in every possible order would require about 4 million years. Upper bounds On 20 October 2018, by adapting a construction by Aaron Williams for constructing Hamiltonian paths through the Cayley graph of the symmetric group, Science Fiction author and mathematician Greg Egan devised an algorithm to produce superpermutations of length n! + (n−1)! + (n−2)! + (n−3)! + n − 3. Up to 2018, these were the smallest superpermutations known for n ≥ 7. However, on 1 February 2019, Bogdan Coanda announced that he had found a superpermutation for