Skip to main content

Algorithm solves cake-cutting problem that has haunted mathematicians

cake cutting algorithm fruitcake
Image used with permission by copyright holder
We’ve all been there before: moving into a new apartment with housemates and trying to figure out who owes what each month. It’s easy enough if everyone gets the same identical room, but inevitably there is one person who gets the slightly larger bed, or the nice view with no road noise, or the ceiling that doesn’t leak, or the en-suite bathroom.

The same is true of a variety of other division tasks: from dividing up leftover pizza — does one slice of meat feast equal two slices of cheese? — to more serious examples like divorce settlements.

It is this type of problem that mathematicians have long investigated with the so-called ‘cake-cutting problem.’ Cake cutting is based around a scenario in which a varying amount of siblings or friends divide up a cake with multiple toppings, where everyone wants to be treated equally, but has different requirements and preferences.

“Fair division is something a lot of real world problems rely on,” Simon Mackenzie, a postdoctoral researcher at Carnegie Mellon University, told Digital Trends. “Cake cutting is a good metaphor to think about these fairness problems. It’s also an incredibly elegant problem from a mathematical perspective.”

As Quanta Magazine notes, in the 1960s an algorithm was created that could divide up a cake between three players, without sparking any envy. But for more than three players, people have been relying on a 1995 algorithm which could conceivably run for a virtually unlimited number of steps to come up with an answer.

In a new paper, due to be shown off at next week’s 57th annual IEEE Symposium on Foundations of Computer Science, Mackenzie and colleague Haris Aziz describe a more efficient, envy-free cake cutting algorithm, capable of solving the problem in a finite number of steps. This can be anywhere between three and 203 cuts of the cake. They have previously come up with solutions to both the tree and four-person variants of the puzzle.

“Even before I started my Ph.D. and got interested in fair division, this is a problem I’d been interested in out of pure curiosity,” Mackenzie said. “What got me working on it seriously was my co-author on the paper coming to me and saying we should collaborate. He thought there were some interesting communication complexity problems in cake cutting that we could look at. That’s what got us started.”

The algorithm the pair have come up with shows a drastically reduced running time is possible — and they have plans to make it even simpler and faster. It is already being hailed by computer scientists as an impressive breakthrough, partially on the complexity of the solution alone.

“This is a problem a lot of people are surprised is doable,” Mackenzie noted. “A lot of people thought it would be totally impossible.”

While part of this is certainly most exciting from a theoretical perspective — showing that a puzzle people once thought couldn’t be done, actually can be — it also has some promising implications for future solutions to real-world fair division problems and making these systems more efficient.

And who can really get upset about making the world a fairer and more efficient place?

Luke Dormehl
I'm a UK-based tech writer covering Cool Tech at Digital Trends. I've also written for Fast Company, Wired, the Guardian…
How to update your Gmail picture on desktop and mobile
A man holding a teacup staring at laptop screen.

There are lots of reasons why you'd want to change your Gmail profile picture. Maybe you have a great, new selfie you want to show off. Or you just want to update your work email with a photo that's recent and professional-looking. Whatever the reason, we can help you update your Gmail picture in just a few quick steps. We've also got you covered whether you choose to change your photo via Gmail's desktop website or through its mobile app.

Read more
How to build your own GPT chatbot without coding
Infinix Zero 30 5G Android phone in gold color with ChatGPT virtual assistant.

For $20 per month, a ChatGPT Plus subscription unlocks far more than just access to GPT-4. With a little know-how, you’ll actually be able to use some of OpenAI’s more advanced features to build a custom GPT chatbot all your own. We did it ourselves, and the results were simply astounding! And while this may sound like an intimidating task to undertake, you won’t even need to know any coding.

Read more
How to change your MAC address on Windows and Mac
The TP-Link Archer AXE7800 tri-band Wi-Fi 6E router in a room.

Each and every web-connected component in your household has a MAC address. One can think of these 12-digit identifiers as digital name tags that computers, smartphones, tablets, and smart TVs use to say “present” on our home Wi-Fi networks. MAC addresses are also integral in maintaining a safe and reliable gateway to the internet.

Read more