  Cellular automata (CA) are fully discrete alternatives to partial
differential equations (PDE). For PDEs, one often considers the Cauchy problem,
or initial value problem: find the solution of the PDE satisfying a given
initial condition. For many PDEs of the first order in time, it is possible to
find explicit formulae for the solution at the time $t>0$ if the solution is
known at $t=0$. Can something similar be achieved for CA? We demonstrate that
this is indeed possible in some cases, using elementary CA rule 172 as an
example. We derive an explicit expression for the state of a given cell after
$n$ iteration of the rule 172, assuming that states of all cells are known at
$n=0$. We then show that this expression ("solution of the CA") can be used to
obtain an expected value of a given cell after $n$ iterations, provided that
the initial condition is drawn from a Bernoulli distribution. This can be done
for both finite and infinite lattices, thus providing an interesting test case
for investigating finite size effects in CA.
