The Many, Many Ways to Cut a Cake: 7 Puzzles To Try

Updated: Dec 16, 2020

Hello GLeaM members! Welcome back! It's been a little while since I've posted because I've had a lot of obligations lately, but don't worry—I'll be posting every week from now into the foreseeable future! I'm excited to introduce some new concepts and ideas to this site this summer, and I'm really hoping to grow our community even more, so stay tuned!

Today, I thought it would be fun to look at a couple more fascinating puzzles. I know a lot of you enjoyed the format of my prisoner-themed puzzle article, and this one will give you even more to try—this time themed around food! Specifically, we're going to be working our way through a series of problems about cake—and how many different ways we can cut our cake up. I thought it would only be fitting if we did a cake article after spending March 14 focusing on pie, and you'll be surprised by just how many cake-cutting problems there are! :)

I won't be giving you the answers to all the puzzles and questions I leave you today. I'd like to see you try them out yourselves, so leave the solutions you find in the comments. I'll be responding to correct answers!

For the puzzles I do leave solutions (or simply hints) to, they'll be at the bottom of the article. Make sure to avoid that section if you don't want to see them. They'll be leading you along the path to the answer!

Geometric puzzles are fun because they seem a little easier than they often reveal themselves to be. Don't be afraid to perservere!

Without further ado, let's begin!

Let's start with a simple question. What is the maximum number of pieces we can cut a circular cake into if we only slice it one time? What about twice? Thrice? Four times?

You can only cut the cake from the aerial view—imagine it's a 2D circle, so no cuts through the side.

Try drawing out the circles and see what you can come up with. Is a pattern emerging?

Here are some of the first few examples:

Photo courtesy of Fun with Num3ers

Notice we're not concerned at all with how big the pieces are! Later, we'll be looking at more equitable cake cutting—because, after all, people want to share a cake equally. But for now, we only care about number of pieces.

Here's the puzzle for you. Can you come up with a rule that gives the maximum number of pieces for a given number of slices? Perhaps even a formula?

Hint: Think about the triangular numbers.

Once you find the formula, see if you can take it a step further. Why do you think this formula emerges? Why does it make sense that the terms are growing at the rate they are, or that we're adding the number we are as we move along the sequence? Why do we seem to be avoiding intersections in making these diagrams?

This is a nice simple problem to come up with an idea for, but it's also a great problem to think about deeply—there are lots of relationships here. I'd love to see your thoughts in the comments!

What about a donut?

We've looked at cutting a circular cake into pieces but what about a donut? Does the hole in the middle of a donut change the number of pieces we can cut it into given a fixed number of slices?

Hint: It'll give us more pieces somehow, but how much more? Can you prove why?

This problem can easily turn into a fun mini investigation with the questions above, but start with this bite-sized puzzle: How many pieces can you make with two slices? What about three? This puzzle may be a little harder than it looks, but keep trying!

You can even extend this puzzle even further. What if you put two holes in your donut? What if you put three? See if you can write a formula based on your formula from the first puzzle in terms of n, the number of slices allowed, and h, the number of holes in your donut. Does your formula accurately describe the scenario or does it break down at some point?

Hints for the first part of the problem can be found below.

Maybe instead of a donut, we can think of this one as a large donut cake!

Triple Stack Donut Cake. Photo courtesy of Sprinkle Bakes.

We got to stick with the cake theme!

The L-shaped cake

It's time we try cutting a cake fairly, but we won't be looking at a simple-shaped cake.

Imagine you're hosting a surprise birthday party for your friend Alice and you invite two other people, Bob and Cara, to attend. You've bought a square cake, and you're excited for everyone to eat it. Cara's running late, so you start cutting the cake without her. You've just cut a fourth of the cake off for Alice, the birthday girl, and handed it to her, leaving the following shaped cake behind:

Photo courtesy of Numberphile "A Quick Cake Conundrum"

Then, Cara suddenly walks through the door, and she brings her friend Darla with her. You suddenly have to divide your L-shaped cake into four pieces instead of three. To make matters worse, your friends insist that not only will the pieces be the same exact size but also the same exact shape. How can you complete the task?

Who Took a Bite?

Imagine you've just bought a beautiful rectangular cake for your sister's birthday, but as soon as you take it home, your eager sister takes a small rectangular piece out of it when you turned your back. You let her get away with her theft, and you still want to divide the cake equally into two pieces—one for you and one for your sister. Can you complete the job in one slice?

Your cake is in the following shape:

Like always, you can't cut straight through the cake on the side. You must cut from the top in a 2D view! This time, it's totally fine to end up with two pieces of completely different shapes, but they must have the same area.

See if you can come up with the answer regardless of the size of the cake and the size of the piece removed from it. There is some property of the slice that you must cut that is universal for all setups! See if you can find it.

Cake for Five

This may be one of the hardest puzzles in this article, but it's a super interesting one. Imagine you're given a square cake that you must divide among five people. You don't have any ruler or other tools, but the cake, however, comes with a 5x5 grid. You can use the grid to measure your slices, but nothing else! How do you slice the cake into five equal pieces (not necessarily the same shape, but the same area)?

Each slice must cut vertically through the cake. Additionally, they must be "slice-like" with the tip of each slice at the center of the cake and the end of each slice at the perimeter of the square.