Greedy flipping of pancakes and burnt pancakes

We prove that a stack of n pancakes is rearranged in all n! ways by repeatedly applying the following rule: Flip the maximum number of pancakes that gives a new stack. This complements the previously known pancake flipping Gray code (S. Zaks, A New Algorithm for Generation of Permutations BIT 24 (1984), 196–204) which we also describe as a greedy algorithm: Flip the minimum number of pancakes that gives a new stack. Surprisingly, these maximum and minimum flip algorithms also rearrange stacks of n ‘burnt’ pancakes in all 2n! ways. We conjecture that these four algorithms are essentially the only greedy algorithms for rearranging pancakes and burnt pancakes in all possible ways using flips.