Pancake problem The pancake graph is described on < Is there any stack of pancakes possible where the bottom pancake is already on its proper place, but where every shortest solution involves moving the bottom pancake? It seems almost evident that the answer is "no", but how can we prove it? The same question in another form: the pancake graph can be recursively built from smaller pancake graphs. My question is, whether or not these smaller graphs are isometric subgraphs.

If you have a solution for a scramble that started with the largest pancake on the bottom, then you can edit the solution such that you do the same flips, except you never interact with the bottom pancake. This way everything apart from the bottom pancake still gets sorted, and the bottom pancake is still in place.