The following problem is open in the sense that no solution is currently known. Progress on the problem may be awarded points. An example of progress on the problem is a non-trivial bound on the sequence defined below.

For each integer $n \geq 2$, consider a regular polygon with $2 n$ sides, all of length 1 . Let $C(n)$ denote the number of ways to tile this polygon using quadrilaterals whose sides all have length 1 .

Determine the limit inferior and the limit superior of the sequence defined by
$$
\frac{1}{n^{2}} \log _{2} C(n)
$$
The first terms of the sequence $C(1), C(2), C(3), \ldots$ are
$$
\begin{array}{l}
1,1,2,8,62,908,24698,1232944,112018190,18410581880,5449192389984,2894710651370536, \\
2752596959306389652,4675651520558571537540,14163808995580022218786390, \ldots
\end{array}
$$

This is sequence A006245 in The On-Line Encyclopedia of Integer Sequences. It appears that no further terms are known.

The problem of studying this sequence was first proposed by Donald E. Knuth in his book Axioms and Hulls, published in 1992. The number $C(n)$ counts various other objects of mathematical interest, such as
- commutation classes of the longest element in the symmetric group $S_{n}$;
- heaps for the longest element in the symmetric group $S_{n}$;
- primitive sorting networks on $n$ elements;
- uniform oriented matroids of rank three on $n$ elements; and
- arrangements of $n$ pseudolines.

For further information on these problems and their equivalence, we point the interested reader to the paper On the number of commutation classes of the longest element in the symmetric group by Hugh Denoncourt, Dana C. Ernst and Dustin Story, as well as to the references therein.

Perhaps the best bounds for $C(n)$ were obtained by Stefan Felsner and Pavel Valtr in their paper Coding and counting arrangements of pseudolines. They prove that
$$
2^{0.1887 n^{2}} \leq C(n) \leq 2^{0.6571 n^{2}}
$$
for sufficiently large $n$.