  Time-reversible dynamical simulations of nonequilibrium systems exemplify
both Loschmidt's and Zerm\'elo's paradoxes. That is, computational
time-reversible simulations invariably produce solutions consistent with the
{\it irreversible} Second Law of Thermodynamics (Loschmidt's) as well as {\it
periodic} in the time (Zerm\'elo's, illustrating Poincar\'e recurrence).
Understanding these paradoxical aspects of time-reversible systems is enhanced
here by studying the simplest pair of such model systems.
  The first is time-reversible, but nevertheless dissipative and periodic, the
piecewise-linear compressible Baker Map. The fractal properties of that
two-dimensional map are mirrored by an even simpler example, the
one-dimensional random walk, confined to the unit interval. As a further puzzle
the two models yield ambiguities in determining the fractals' information
dimensions. These puzzles, including the classical paradoxes, are reviewed and
explored here. We review our investigations presented in Budapest in 1997 and
end with presentday questions posed as the Snook Prize Problems in 2020 and
2021.
