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:
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!
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:
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.
Can you generalize? Can you cut 7 slices given a 7x7 grid, 8 slices given an 8x8 grid or 9 slices given a 9x9 grid?
How do you share any cake among three people, so everyone is satisfied?
This is a little more of an open-ended puzzle. You know the typical way of dividing a cake when there's no "right way" to divide it, when there's random amounts of icing and random amounts of sprinkles, and so on?
You use the "I Cut, You Choose" Method. If for example, you and your friend Alice are dividing a cake, and she cuts it into two pieces she believes are equal, she will be satisfied with either one, and then you can choose which piece you want, and since you chose it, everything seems fair and you are satisfied.
Is there a way to extend this method to three people? Say if you're now trying to share a cake so everyone is satisfied (meaning, they all believe their piece is better than or equal to everyone else's) with two friends: Alice and Bob. Can you devise a system that works?
Cut your cake—scientifically!
My final puzzle is as much scientific as mathematical. Alex Bellos, the mathematician that actually collected many of the puzzles I used above, once came across a hundred-year-old method of cutting cake. Come to find out, the standard way of cutting a cake into slices that meet at the center is bad for the cake if you don't eat it all at once—it allows the moist part of the cake to sit out and get stale. Instead, this is a method of cutting a cake where only the outside perimeter of the cake is exposed once you remove the slices. This is actually one of the first Numberphile videos I watched as a kid, and it's a fun idea to play with. See if you can come up with a method, before you actually watch the video:
The video actually managed to get over 16 million views, and the news even published a couple pieces on it, such as this one:
Hey, they might be hating on it, but the mathematical way is still the best!
SOLUTIONS AND HINTS
What about a donut? HINT
Here's an example of a donut with two cuts, creating five pieces.
Here's an example of a donut with three cuts, creating nine pieces.
There are actually a number of different possibilities for the diagrams above—ways to get two cuts on the donut to create 5 pieces and to get three cuts on the donut to create 9 pieces. See if you can draw one where at least one of the cuts goes through the donut hole!
Notice that with two cuts, we created 4 pieces for the cake and 5 for the donut, and 4+1=5.
Notice that with three cuts, we created 7 pieces for the cake and 9 for the donut, and 7+2=9.
See where you can take that idea!
The L-shaped Cake HINT
Hint: The pieces look just like the overall cake!
I'll leave you the Numberphile video I found this puzzle off of to explain the whole solution if you need it: https://www.youtube.com/watch?v=ezdeBrPnzyc&t=153s
Who Took a Bite? SOLUTION
The cut is the one that goes through the center of the cake and the center of the missing slice:
Let's think about why this is true.
First off, any line through the center of the overall rectangle (before the piece was removed) divides it into two equal pieces. (See if you convince yourself that's true!) Similarly, any line through the center of the missing slice in the cake divides the hole into two equal pieces as well. Therefore this line divides the overall cake into two equal pieces—both are an equal piece of cake minus an equal piece of hole.
Sometimes, depending on the size of the rectangle and removed slice, the line will cut into the corner of the piece on the left. (Imagine the line above rotating slightly counterclockwise.) This will result in one person getting two pieces and the other getting one, but what's important here is they end up with the same amount of cake!
Cake For Five SOLUTION
Cut five slices that all contain the same length of the squares' perimeter. The perimeter is 20 units, so simply make sure the slices each have 4 units along the perimeter. Start at a corner, perhaps, and mark every point 4 units along. That's it.
Let's see if we can explain why this works. We're guaranteeing perimeter is the same, but how does that correspond to area being the same as well? Think of it this way:
Here, I labeled the sections 1 to 5 to make it easier for us to refer to them, and I also added in some red lines from the center to the corners. You'll see why in a minute.
First, let's note that there are 8 triangles now drawn in the diagram and every single one has the same height (if we consider the base to be the side along the perimeter of the square.) The height of every triangle is 5/2 = 2.5 units.
The areas of section 1 and 2, which both contain just one triangle are both simply Base*Height/2 = 4*2.5/2 = 5.
The areas of section 3 and 5 are the sum of two triangles that have the same base and heights: Base #1 *Height/2 + Base #2 *Height/2 = (Base #1 + Base #2)*Height/2 = (3+1)*2.5/2 = 5.
The area of section 4 is also the sum of two triangles: Base #1*Height/2 + Base #2*Height/2 = (Base #1 + Base #2)*Height/2 = (2+2)*2.5/2 = 5.
Notice what we did here. Instead of considering these sections as weird quadrilaterals, we broke them down into two triangles and to find their total area, we SUMMED THEIR BASES, and since the sum of the bases is the same for every section (we said they all contain the same number of units of the perimeter), every section has the same area.
This is the key to this problem. This proves that when every section contains the same perimeter of the overall square, every section also has the same area.
This can also be easily generalized to other grids: grids with any number of units you can think of, divided into the same number of pieces as that length. In fact, the numbers you consider easy to divide a square cake into, like 2, 4 or 8, also use this property: the perimeter of each of the slices cut into halves again and again in the typical way are also the same.
In fact, each of the pieces not only have the same area, they also have the same amount of icing, EVEN if that icing is on the sides of the cake.
Extending the "I Cut, You Choose" Method SOLUTION
Here's a very creative way of solving the puzzle, from Numberphile courtesy of the amazing mathematician Hannah Fry:
"What about a donut?", "Who took a bite?" and "Cake for Five" are courtesy of Alex Bello's fantastic book, So You Think You've Got Problems?, a book I highly recommend reading this summer break. They're listed under slightly different names but you can find them as Puzzles 52-54 in his compilation.
Check out the videos I referenced from Numberphile on cake-cutting:
https://www.youtube.com/watch?v=ezdeBrPnzyc&t=153s "A Quick Cake Conundrum"
https://www.youtube.com/watch?v=kaMKInkV7Vs "Equally Sharing a Cake Between Three People"
https://www.youtube.com/watch?v=wBU9N35ZHIw "A Scientific Way to Cut a Cake"
And here's one more crazy question for you: What if we look at the maximum number of pieces we can cut a cake into if we do allow 3D cuts, meaning cuts through the side of the cake? This channel tried it:
See if you can take it any further!
As I wrap this up, I just want to say that I'm excited for all of you to read this article! I'm really proud of all the puzzles I gave you this time, and I'd love to answer any comments you have. Ask me in the comments if you want any solutions or hints that aren't given here!
Also, let me know if you like this article format, where it's more about puzzles and solutions than simply an article-length exploration of a topic or a story from the math world. I'll certainly be writing more of the latter as well, but let me know what your preference is!
Two more things before I wrap up: First, check out one of my recent articles here. It's a list of all the ideas I have of math-related things you can try over quarantine, and I have a lot—for all kinds of math lovers!
Second, there's a story I read in the news lately that is very fascinating. A graduate student named Lisa Piccirillo only needed less than a week to solve an outstanding math problem that's lasted for decades, proving an essential property of a knot originally discovered by the late John Conway. Read the article from Quanta Magazine here if you're interested!
I hope you enjoyed this article, and that the puzzles were challenging enough for you! I'm pretty sure they all weren't a piece of cake. :) Have a great week!
Cover Image Courtesy of Quanta Magazine and Scientific American