  An open problem of interest, first infused into the applied probability
community in the work of Bingham and Doney in 1988, (see \cite{Bingham}) is
stated as follows: find the distribution of the quadrant occupation time of
planar Brownian motion. In this short communication, we study an alternate
formulation of this longstanding open problem: let $X(t), Y(t), t \geq 0$ be
standard Brownian motions starting at $x,y$ respectively. Find the distribution
of the total time $T=Leb\{t \in [0,1]: X(t) \times Y(t) >0\}$, when $x=y=0$,
i.e., the occupation time of the union of the first and third quadrants. If two
adjacent quadrants are used, the problem becomes much easier and the
distribution of $T$ follows the arcsine law.
