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.

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Vanilla Beans + Sea Salt = Vanilla Sea Salt


Vanilla sea salt. Just the two ingredients together bring out the best in foods: the vanilla beans giving a slight perfumy aroma and the sea salt heightening flavors. (The Independent has a great article on the science of salt.)

Sea salt is courser than table salt, and while it is less processed, both have the same nutritional value. In pastry, sea salt's courser grind is often reserved for special purposes like a black treacle and stone-ground whole wheat flour bread (Saveur), or sprinkling it on sweets like Double Chocolate Pudding (James Beard Foundation) and Pecan Shortbread with Fennel (Bon Appetit). So why not give the sea salt a little vanilla kick?

How to make it? Here is a basic recipe for vanilla sea salt, and it has just two ingredients. Take about a cup of sea salt and place it in a jar with a couple of inches of head space for stirring and shaking, and make sure the lid is tight fitting. Choose 2 full, plump fresh vanilla beans - the best that you have as the more fresh they are, the more fragrant the salt will be. Cut them up into 1 to 2 inch size pieces. Add them to the salt and mix so they are completely covered by the sea salt.

Place the jar on the counter, spaced away from the sun. Shaking every couple of days until the salt is fragrant from the vanilla beans, at least 3 weeks for the full flavor to begin to develop.

The vanilla bean will keep infidelity in the salt so there is no need to remove it. Use the vanilla sea salt as desired.


Fred's Bakery and Deli - For Sale


The latest comb through on commercial bakery commercial real estate for sale - Fred's Bakery and Deli.
  • 2500 square feet.
  • $105,000. - at $42/sq ft
  • Robertson Ave., off the 10 freeway in Los Angeles.

From the LoopNet real estate site:
This Bakery has been in business for more than 60 years. Great location and loyal costumers. Fully operated by employees.
See the site listing for more details and for contact info.

Behind the Scenes in Arnaud Larher's Kitchens

Inside Arnaud Larher's Kitchen -
 Image via YouTube.


Ever wondered what goes on behind the scenes at a French pastry shop? This video shows what a typical day is inside the kitchens of Arnaud Larher, a French pastry chef. I wish I could taste one of everything that is made here.

A link to his shop is here: Arnaud Larher. How can anyone resist a seemingly Californian flip flop tart created by a French pastry chef, MOF 2007 at that? (Find it on his website.)

Enjoy.





 

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