Scientifically and Mathematically Speaking – What’s the Best Way to Cut a Cake?

Is there a perfect way to cut a cake? I explore. Image courtesy jeremyfoo, Flickr via CC license.

We all know how to cut a cake, right? Even Wilton gives diagrams on how to cut a cake in every conceivable shape there is, including the most common round shape (below).

Cake Cutting Diagram for Round Cakes – via Wilton.

But, is there a best way to cut a cake, for example to so that everyone has the ‘best’ piece or that each slice is fresh, not ‘dry’? For these questions, leave it to mathematicians to come up with these answers.

Cutting a Cake to Prevent Dry Surfaces

First, we look to Sir Francis Galton. A few years before he was knighted he had his humorous answer for the dilemma of cutting a cake in a way “so as to leave a minimum of exposed surface to become dry” in Nature, 1906. He wanted the best way to take a small cake for two and have it ‘fresh’ for 3 days, meaning no dry exposed areas. Each day, a third of the cake was removed and served to two people. I posted Galton’s Cutting a Round Cake on Scientific Principles on Old School Pastry for you to read.

Today, we use plastic wrap. Galton’ answer in 1906:

• Take a cake 5 inches across, and make two cuts down the middle to remove the center slice. Cut that center slice in half, and serve two people. Press the remaining halves together, secure with a rubber band (I’m assuming he only consumed fondant-covered or stiff royal icing-coverd cakes? Rubber bands don’t work too well with buttercream.).
• Next day, make two cuts to remove the center piece again, and serve two people. Push the remainder halves together, securing like the day before.
• Final day, you are left with 4 small pieces. Divide in half and serve the same two people two portions each, and there you have 3 servings of cake for 2 people, each having a slice that has no dry edges.

Not very practical in today’s world, with an endless supply of plastic containers large enough to hold a cake, or cellophane or foil doing the job of keeping the cake fresh nicely. Scroll to the very bottom to see how Numberphile.com has demonstrated this technique. But what if you wanted the best way to cut a cake, based on equability, ‘envy-freeness’, or efficiency?

Cutting a Cake so Everyone Gets a “Perfect” Slice

Steven J. Brams, Michael A. Jones, and Christian Klamer from the American Mathematical Society, together published Better Ways to Cut a Cake in the December 2006 issue of Notices, Volume 53, Issue 11. It is described as:

• A mathematical cake, as viewed by n persons participating in its division, is modeled by their n (covert) value functions on the unit interval. Each participant can cut the cake at a point by a vertical line at that point, and each is assumed to make their cut so as to maximize the value of the minimum size piece they might receive. The authors explore algorithms that allow this division to be fair to all.

I loved this, and the paper perfectly illustrates why I love algebra – you can find any answer with it. They based their cake cutting algorithms on the assumption that every person is “risk-adverse: He or she will never choose a strategy that may yield a more valuable piece of cake if it entails the possibility of getting less than a maximum piece.” This is especially true in the case of kids cutting their own pieces. I would know; I have three.

Their research led them to search the answer to satisfy these three key features.

1. The first is envy-freeness – where every person thinks they are getting the best piece so they don’t envy the next person’s slice.
2. The second is efficiency – where there is no other cut that is better.
3. And the last is equability – where every person thinks that each cut is the same as, or just as good as, the next person’s.

It will always remain a challenge to cut a cake so that everyone has that perceived ‘perfect slice of cake,’ especially when multiple people are vying for it. But even the authors admit “there may be no envy-free division that is also equitable.” I guess even in cake cutting, tough choices have to be made. The paper discusses proportional equability, and in some parts base their assumptions that the people involved are being truthful in the equitable procedure, and is a pleasing pastry/mathematical read all around.

Below: Galton’s Example via Numberphile.