Cake cutting

A special treat for you all today - a dissertation of my thoughts on how best to cut cakes up. This was prompted by my recent celebration of Willie Nelson's birthday and subsequent devouring of a large and delicious mudcake over the course of several days. I'd been thinking back on the various strategies of cutting up cakes and the principles underlying those strategies. For the most part they are principles of economy of action or aesthetic considerations, but some could also be considered political principles or even ethical principles.

First, let's note that this post concerns itself mainly with the circular cake. Rectangular cakes and square cakes are also interesting, but I think the round cake offers a greater challenge to the cake cutter due to its infinite rotational symmetry. An elliptical cake would be a greater challenge still, lacking both the convenient parallel edges of the oblong cake as well as the circular cake's advantage of equi-angular cuts from the centre having equal areas .

However, I can't recall ever having to cut up an elliptical cake so I'll not address it further here except to note that it wouldn't surprise me if someday someone discovered that Johannes Kepler actually came up with his laws of planetary motion while trying to cut an elliptical cake into pieces of equal size. It would be an ingenious party trick to perform but you'd have to come up with some way to calculate a velocity around the perimeter of the cake that was appropriate to an orbiting body. Perhaps if you served the cake on a hyperbolic plate and rolled marbles around its edge whilst observing closely with an accurate timepiece on hand.

Anyway, that's enough of that. On with the cake-cutting fun!

In my youth, I always endevoured to cut a cake into equal-sized pieces, just like the good Mr. Kepler. What's more, I always cut the cake into exactly as many pieces as there were persons present to eat it. I call the Birthday Strategy, because if you're at a kids birthday party this is the only way to cut up a cake without being lynched by angry children. Kids are quite unlike adults in that no child has ever, in the history of the world, said, "I'll just have a very thin piece thanks, I'm already pretty full". Children will eat just as much cake as is made available to them. In addition, children will closely monitor both the relative sizes of all the pieces cut and any available leftover cake, and will view any shortfall of their own piece most unkindly.

The Birthday Strategy works well for these circumstances but it does have some practical difficulties. The worst case situation is when the number of people present is prime. It's very difficult to divide a cake into a prime number of equal-sized portions without measuring instruments. Even having a composite but odd number of people makes it tricky.

For this reason, a meta-strategy that can be effective is to employ the Birthday Strategy for cutting cakes but avoid difficulty by always ensuring that the number of people present is a power of two. I call this the Turing Strategy. The Turing Strategy is very convenient from the point of view of cake cutting because it involves only having to bisect angles, something that is very easy to perform sufficiently accurately by eye, particularly if you have a rotating cake stand available.

For some time in my early twenties I employed a pseudo-Turing Strategy, whereby I always cut the cake into a power of two pieces but made no attempt to control the number of people present. This means that there will usually be pieces left over. This is acceptable when most of those present are adults, as some adults will want only one piece and others will want more. I left it up to the hungry eaters to squabble over the leftovers amongst themselves. The downfall of the pseudo-Turing strategy is threefold. First, if the number of people present is one greater than a power of two, you will leave almost half of the cake as leftovers. Second, once you start cutting a cake into 32 pieces or more, each piece becomes impractically thin and difficult to serve intact. You might scoff at having 32 people there to eat a cake but remember: even if you only have 17 people there you'll need 32 pieces of cake. Finally, people look at you funny if you cut a cake up like this.

Eventually I realized that the pseudo-Turing strategy doesn't really make sense, as it employs a solution to the problem of children but only in the context of adults, for whom there is no real requirement for equal sized pieces. At this point I began to employ a new consultative style, where I moved the knife slowly around the cake like the hands of a clock and asked the person for whom that piece was intended to indicate when the piece was large enough. This is called the Nnnnnow! Strategy.

The Nnnnnow! Strategy works well for small groups of people but not for large groups. It requires the sequential close proximity and attention of the eaters, which can be difficult to orchestrate in a large gathering. It also runs the risk of running out of cake before all people have been served if the early eaters do not moderate their appetites in response to population pressures. The only way of controlling this is to limit the size of the pieces served, which not only requires the implementation of the Birthday Strategy anyway to find such a limit, but requires this to be done iteratively in response to the remaining amount of cake and the remaining number of eaters, thus imposing a prohibitive computational demand.

I also have been known to get annoyed at the propensity of some young ladies to insist on wafer-thin slices of cake which, as noted above, are impractical to serve, for reasons which I believe to be more related to diet propaganda and self-denial than an accurate reflection of their desire for cake. This would then lead me to serve them more cake than they wanted, which in turn led to them becoming annoyed with me in return.

As a result of these difficulties I then adopted a new strategy for cake cutting, which I call the Amalric Strategy. This involves me deliberately cutting the cake into an assortment of differently sized pieces and letting the pieces fall where they may. Just make sure that some are big, some are small, and most are in-between. Then hand them out and let people swap amongst themselves until everybody is equally miserable.

But recently I have begun to turn against even the Amalric strategy. I've been chafing under the yoke of expectation - people always expect round cakes to be cut into wedge-shaped pieces from the centre using radial cuts. It's much more interesting to just start from the edge and cut out whatever shape you please. I call this the anti-Penn Strategy. The fun thing about doing this is you end up with some wonderfully irregular shapes and jagged edges. There's no need to restrict yourself to any kind of grid or regularity at all.

The anti-Penn works best when the cake is to be cut into pieces but left assembled for people to select their own piece, as otherwise the gestalt effect is lost. I like it because it's a way of really surprising people when they are least expecting to be surprised. Furthermore, it can be a real conversation starter! In fact, you could print out a copy of this article and carry it around in your wallet for just such an occasion and help to popularize the anti-Penn!

So now you are well-versed in the theory of cake-cutting and how some of these ingenious methods arose to address the seemingly intractable difficulties of the day. Why not vote now in the new poll to the right on how you like up your cakes? And if you have a different strategy, I'd love to hear about it!