  A nonautonomous version of the ultradiscrete hungry Toda lattice with a
finite lattice boundary condition is derived by applying reduction and
ultradiscretization to a nonautonomous two-dimensional discrete Toda lattice.
It is shown that the derived ultradiscrete system has a direct connection to
the box-ball system with many kinds of balls and finite carrier capacity.
Particular solutions to the ultradiscrete system are constructed by using the
theory of some sort of discrete biorthogonal polynomials.
