Harary–Hill conjecture for the crossing number of complete graphs

Let $K_n$ be the complete graph on $n$ vertices, and define

$$H(n):=\frac{1}{4}\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{n-1}{2}\right\rfloor\left\lfloor\frac{n-2}{2}\right\rfloor\left\lfloor\frac{n-3}{2}\right\rfloor.$$

Here $\operatorname{cr}(G)$ denotes the minimum number of crossings in a plane drawing of a graph $G$. Harary–Hill conjecture.

$$\operatorname{cr}(K_n)=H(n).$$

The formula is achieved by Hill's drawing of $K_n$ and is conjectured to be optimal; determining the crossing number of complete graphs remains a central open problem in topological graph theory.