  There are (at least) two unsolved problems concerning the current state of
the thermal gas in clusters of galaxies. The first is identifying the source of
the heating which offsets cooling in the centers of clusters with short cooling
times (the ``cooling flow'' problem). The second is understanding the mechanism
which boosts the entropy in cluster and group gas. Since both of these problems
involve an unknown source of heating it is tempting to identify them with the
same process, particular since AGN heating is observed to be operating at some
level in a sample of well-observed ``cooling flow'' clusters. Here we show,
using numerical simulations of cluster formation, that much of the gas ending
up in clusters cools at high redshift and so the heating is also needed at
high-redshift, well before the cluster forms. This indicates that the same
process operating to solve the cooling flow problem may not also resolve the
cluster entropy problem.
