Pancake Numbers

Lesson Plan

Slides online

Some maths topics need a good story to capture the attention of students. This however, has your students engaged from the first slide. It also has a GREAT story. Bill Gates only published one scientific paper whilst studying, and it was all about pancakes. His research on this particular problem hasn’t really been bettered (should that be buttered?) since. Another problem which remains unsolved, it is a great lesson to introduce systematic thinking and setting out work in a way that is legible for others to understand.

As always there is a great numberphile video for this problem – but for ease let me try and explain how it works.

Basically, if a chef haphazardly stacked pancakes of different sizes (diameters) on top of each other, and then you wanted to use a spatula to flip them so that they ended up from smallest to biggest, how many flips would it take?

So if you had three pancakes piled in the order: medium, small, large then this would only need one flip, slide the spatula between small and large and flip the top two pancakes.

There are two questions to explore here: How many flips does it take to get back to the start, and what is the maximum number of flips for any stack of three/four/five pancakes? The first is a low entry, easy to start problem. The second is a deep investigation with no answer for when the amount of pancakes reach 20. Simpsons and Futurama writer David X. Cohen also wrote a paper on this topic, but made it even more difficult by suggesting that one side of each pancake was burnt, and so you have to try and order the pancakes so they all have burnt side down.

My Experience

I have tried this practically and without, and to be honest, the majority of my high ability students went straight to diagrams, but it was still fun to have a practice with some cut out circular card first (see slides).

We start with four pancakes – and this is much harder on purpose. I like to leave the option where possible for my students to simplify the problem, which is a proven maths problem solving strategy. I set out the pancakes as 3,1,4,2 and ask them to try and get it back to 1,2,3,4.

A key part of this is for the students to note down their method. Normally I’ll have someone claim to have done it, I’ll congratulate them and quickly move it back to the starting position. “Show me again” I’ll say and they go about it as if they’ve never done it before, in a similar way to the jumping frogs problem that is one of my favourites.

Once we discover the best path to solving that configuration of pancakes, I ask if it matters what order they start in. We then discuss the worst way to start with four pancakes and eventually how many different ways to stack four pancakes. Then I share about Bill Gates and his research and how no-one has an answer for 20 pancakes.

We then if possible move onto a great Scratch project, where they can have a go at a digital version and see how quick they can turn 13 pancakes. At this point students now spot the strategy for getting the order correct. This strategy always works (flip under the biggest pancake that is not in order so that it goes to the top, then flip the stack so that the top one is in the right place, and repeat) but it isn’t the fastest way often!

Have flipping fun!