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Ponder This

October 1998

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Ponder This Challenge:

You must help Pete the pancake chef! He's desperately trying to sort his pancakes by size. How many flips will it take?

There is a stack of N pancakes -- each one a different size -- on top of the griddle. Your goal is to get the pancakes in a stack arranged by size, with the largest pancake on the bottom, smallest one on top. You must accomplish this by flipping the pancakes a limited number of times.

For each flip, you must (on Pete's behalf):
Choose a number k between 1 and N. Pick a pancake in the kth position (counting from the topmost pancake) and insert your spatula under it (between the pancakes in position k and (k+1). Then flip the pancakes on your spatula, reversing the order.

N=5 (Numbers on the pancakes represent the size of the pancake, not its position.) The smallest pancake is labeled "1":

Initial position: Top 1 3 5 2 4 Griddle

First move (k=3)

Yields: Top 5 3 1 2 4 Griddle

Second move (k=5) Yields: Top 4 2 1 3 5 Griddle
Third move (k=4) Yields: Top 3 1 2 4 5 Griddle
Fourth move (k=3) Yields: Top 2 1 3 4 5 Griddle
Fifth move (k=2) Yields: Top 1 2 3 4 5 Griddle
We used 5 moves to sort the stack.

Here's what we want you to tell us:
  • For a given N, what is the least number of flips with which you can guarantee that you will correctly sort the stack?
  • For every N, for every initial position, can it be done in N flips?
  • How many flips could be required for a given N and the worst possible initial position?

In other words, if we consider N to be fixed, for each initial permutation, there is some minimum number of flips that works. Find the maximum of these minima as you vary the permutations, i.e. find a hardest permutation.

E.g. if N=5, the permutation (Top 2,1,3,4,5 Griddle) would require only one flip, although you could take a roundabout path and take 13 flips. The minimum for this permutation is 1. The minimum for the permutation (Top 1 3 5 2 4 Griddle) is 5 flips as shown. It turns out that any other permutation can also be done in 5 flips. So 5 is the answer we're looking for here.

Or, to put this in simple mathematical terms: For N=number of pancakes, p=permutation Let f(N,p) be the minimum number of flips when faced with initial permutation p; let g(N) be the maximum of f(N,p) over choices of p. Find g (N).

FYI: Although this problem may appear simple at first, the solution is actually fairly complex.

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Challenge: 10/01/98 @ 12:00 AM EST
Solution: 11/01/98 @ 12:00 AM EST
List Updated: 11/15/01 @ 10:00 AM EST

People who answered correctly:

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