Monday Maths: Pancake Numbers

You may think that I’m making this up.  I promise I’m not.

Pancake Numbers are real.

Imagine you have a stack of pancakes, of different sizes.  You want to arrange them with the largest at the bottom and the smallest at the top.  You have a spatula, which you are allowed to insert anywhere in the stack.  When you do, you can flip the stack of pancakes on top of your spatula upside down.

Yum. Pancakes.

For example, imagine you had pancakes of sizes:


You insert the spatula between 7 and 9, and flip.  You now have:


What is the maximum number of flips you need to get them in size order?

That’s a pancake number.

Crepes.  They're like pancakes, only French.
Crepes. They’re like pancakes, only French.

Some stacks of pancakes can be sorted in fewer flips than the pancake number, but none of them will need more than the pancake number.  The pancake numbers for stacks of size 1, 2, 3, etc are: 0, 1, 3, 4, 5, 7, 8, 9, 10, 11, 13…

Mathematicians have only been able to calculate the pancake number of stacks up to 19 high (it’s 22).  Apparently 20 is just too complicated, although they do know that for any stack with n pancakes, the pancake number is less than or equal to (5n+5) / 3, so the number for 20 must be less than or equal to 35.

Why not try this on Pancake Day this year?  Create a stack of pancakes and challenge the girls to get them into the right order using the pancake sorting method.


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