  The aim of this essay is to better understand the Grasshopper Problem on the
surface of the unit sphere. The problem is motivated by analysing Bell
inequalities, but can be formulated as a geometric puzzle as follows. Given a
white sphere and a bucket of black paint, one is asked to paint half of the
sphere, such that antipodal pairs of points are oppositely coloured. A
grasshopper lands on the sphere, and jumps a fixed distance in a random
direction. How should the sphere be coloured such that the probability of the
grasshopper landing on the same colour is maximized? Goulko and Kent have
explored this problem on the plane without an antipodality constraint. This
essay gives clear indication that the spherical problem with the antipodality
constraint yields colourings with similar shapes as the planar problem does.
  This research has discretised the problem and used a simulated annealing
algorithm to search for the optimal solution. Results are consistent with the
planar results of \cite{goulkokent}. For $0.10\pi\leq\theta\leq0.44\pi$
cogwheel solutions are found to be optimal, with odd integer of cogs $n_o$ such
that $n_o$ is close to $\frac{2\pi}{\theta}$. For $0.45\pi\leq\theta\leq
0.55\pi$ critical solutions are found, in which domains of identical colour
decrease in size towards $0.5\pi$ (moving from either side). For $\theta\geq
0.55$ colourings are found consisting of stripes with cogs. Towards
$\theta=\pi$ colourings are generated that display just stripes that scale in
width with $\pi-\theta$.
