  We investigate the boundedness of positive solutions of the reciprocal
max-type difference equation
  \[ x_{n}=\max\left\{\frac{A_{n-1}^{1}}{x_{n-1}}, \frac{A_{n-1}^{2}}{x_{n-2}},
  \ldots, \frac{A_{n-1}^{t}}{x_{n-t}}\right\}, \ \ n=1, 2, \ldots, \]
  where, for each value of $i$, the sequence $\{A_{n}^{i}\}_{n=0}^{\infty}$ of
positive numbers is periodic with period $p_{i}$. We give both sufficient
conditions on the $p_{i}$'s for the boundedness of all solutions and sufficient
conditions for all solutions to be unbounded. This work essentially complements
the work by Biddell and Franke, who showed that as long as every positive
solution of our equation is \emph{bounded}, then every positive solution is
eventually periodic, thereby leaving open the question as to when solutions are
bounded.
