If $A$ and $B$ are complex $n\times n$ matrices, the unital algebra ${\cal A}$ generated by $A$ and $B$ is the linear span of all possible (finite length) words in $A$ and $B$. It is not difficult to show that the words of length at most $n^2-1$ span this algebra. The smallest non-negative integer $l$ such that the words in $A$ and $B$ of length at most $l$ span ${\cal A}$ is called the length of the pair $\{A, B\}$.

It is conjectured that  the length of any pair is at most $2n-2$ and this has been verified for $n\le 6$. Also, provided ${\cal A}=M_{n}(\DC)$,  (i) the length of an arbitrary pair is at most $\sqrt{2}\,n^{3/2} + 3n$ and (ii) the length of the pair is at most $2n-2$ if one of the matrices is unicellular.

The problem, its history, some of its partial solutions and its low order exact solutions will be discussed.