  We investigate mushroom billiards, a class of dynamical systems with sharply
divided phase space. For typical values of the control parameter of the system
$\rho$, an infinite number of marginally unstable periodic orbits (MUPOs) exist
making the system sticky in the sense that unstable orbits approach regular
regions in phase space and thus exhibit regular behaviour for long periods of
time. The problem of finding these MUPOs is expressed as the well known problem
of finding optimal rational approximations of a real number, subject to some
system-specific constraints. By introducing a generalized mushroom and using
properties of continued fractions, we describe a zero measure set of control
parameter values $\rho\in(0,1)$ for which all MUPOs are destroyed and therefore
the system is less sticky. The open mushroom (billiard with a hole) is then
considered in order to quantify the stickiness exhibited and exact leading
order expressions for the algebraic decay of the survival probability function
$P(t)$ are calculated for mushrooms with triangular and rectangular stems.
