  We confirm, for the primes up to 3000, the conjecture of Bourgain, Gamburd,
and Sarnak on strong approximation for the Markoff surface $x^2+y^2+z^2 = 3xyz$
modulo primes. For primes congruent to 3 modulo 4, we find data suggesting that
some natural graphs constructed from this equation are asymptotically
Ramanujan. For primes congruent to 1 modulo 4, the data suggest a weaker
spectral gap. In both cases, there is close agreement with the Kesten-McKay law
for the density of states for random 3-regular graphs. We also study the
connectedness of other level sets $x^2+y^2+z^2-3xyz = k$. In the degenerate
case of the Cayley cubic, we give a complete description of the orbits.
