  We are given a stack of pancakes of different sizes and the only allowed
operation is to take several pancakes from top and flip them. The unburnt
version requires the pancakes to be sorted by their sizes at the end, while in
the burnt version they additionally need to be oriented burnt-side down. We
present an algorithm with the average number of flips, needed to sort a stack
of n burnt pancakes, equal to 7n/4+O(1) and a randomized algorithm for the
unburnt version with at most 17n/12+O(1) flips on average.
  In addition, we show that in the burnt version, the average number of flips
of any algorithm is at least n+\Omega(n/log n) and conjecture that some
algorithm can reach n+\Theta(n/log n).
  We also slightly increase the lower bound on g(n), the minimum number of
flips needed to sort the worst stack of n burnt pancakes. This bound, together
with the upper bound found by Heydari and Sudborough in 1997, gives the exact
number of flips to sort the previously conjectured worst stack -I_n for n=3 mod
4 and n>=15. Finally we present exact values of f(n) up to n=19 and of g(n) up
to n=17 and disprove a conjecture of Cohen and Blum by showing that the burnt
stack -I_{15} is not the worst one for n=15.
