Analysis of Pan-cake sorting
Let's suppose the input array is $A = [3,1,4,5,2]$.  When sorting this array using pancake sorting, only flips of prefixes of the array are allowed.  Flipping the first four elements, i.e. replacing $3145$ by $5413$, gives $A = [5,4,1,3,2]$. Then, flipping all five elements gives $[2,3,1,4,5]$, which brings the largest element to its correct position.  By repeating this process to bring the next largest element to the last-but-one position, and so on, the array can be sorted.  

The number of flips required to sort an array of length $n$ is at most $2n-3$. This is because bringing the largest $n-2$ elements to their correct positions can be done using at most $2n-4$ flips (because at most two flips are required for each element, as mentioned in previous paragraph).  Finally, the elements in the first two positions can be sorted using at most one flip.

The problem of finding the minimum number of flips to pancake-sort an array is still open.