Slices of Pie Riddle
Write-up by Craig Corsi and Steven J Miller
Below we discuss the Slices of Pie riddle. While this problem may not have the same mathematical prerequisites as some of the other riddles, it exposes a few preconceptions that people often have when solving riddles. We examine these preconceptions and show how they make the solution more difficult to reach. After solving the riddle, we will look at how to explore the riddle in a more general setting using a website called the Online Encyclopedia of Integer Sequences: http://oeis.org/ .
Here's the original question:
The first step in solving a problem is to make sure we know what all the words mean. We need to clarify that the ďcircleĒ in this riddle refers to the entire disk, and not just the boundary. As a nice additional exercise, show you canít divide the perimeter of a circle into seven pieces with just three lines.
Letís look at what happens with fewer lines and try to sniff out a pattern. If we have zero lines then thereís just one region. If there is one line then there are two regions. If there are two lines then we can get four regions (and no more, as the best a line can do is divide each existing region in half). We see a pattern: 1, 2, 4; it seems natural to conjecture that the next term is 8. Unfortunately, this pattern doesnít continue, and the problem says that the next term is 7, not 8.
So what kind of preconceptions make our task seem impossible? That is, what assumptions might be made which don't need to (and shouldn't) be made? First, we might think that all pieces of the circle have to be the same size and shape, as if we were cutting a pie and giving the pieces to our friends. One way to accomplish this is to make seven cuts, each of which cuts halfway into the pie, touching the center. This gives the right number of pieces but with too many cuts. While this isnít the solution, it does help us get on the right track. If a problem is hard, try doing a simpler one first. Is there some number of cuts that will divide our circle into seven pieces? Perhaps we can do it with more than three, and then see ways to simplify or remove some of the cuts.
Alternatively, we could make three cuts through the center, dividing the circle into six pieces, each whose angle equals sixty degrees. This is the right number of cuts, but we don't have enough pieces. We would need one more cut. Thus, weíve shown the following fact:
Fact : At least one of the cuts does not pass through the center of the circle.
This is a major step forward. There are so many different ways to cut the circle, we need to find a way to test all the possibilities. This gives us a wonderful start. Weíll make one more observation before giving a solution:
Fact : Each line starts at a point on the circleís perimeter and ends at a point on the circleíperimeter.
Proof: If not, we can only create additional regions by extending a line that is either entirely within the circle, or going from the perimeter to some interior point.
The purpose of this observation is to help us navigate all the possible configurations. We know at least one line doesnít pass through the center of the circle, and the problem is equivalent to choosing 6 points on the circle (in three pairs of 2) so that the resulting lines divide the circle into 7 pieces.
Itís good to try lots of different sets of three cuts; this is a great way to build intuition and get a feel for what goes wrong (and maybe point out what weíll need to do to make things go right). Other attempts might include having two vertical lines and one horizontal line, or one vertical and two horizontal. These both give only six pieces. Three parallel lines is worse, as only four pieces are created. Do we need to make even weirder pieces to accomplish our task? All of our examples so far have had all of the pieces include some of the border of the circle. Is it possible to find an example where this does not happen?
Let's put these observations aside and instead try to search for a solution more methodically. A good way to think about the problem is to add lines on the circle one by one and see how many pieces we can get from the circle after each step. Let's be greedy: place each line in the best possible place (giving us the most number of pieces) before adding the next line. This may not be the way to go, but itís worth a try.
The first line can divide the circle into at most two pieces. Adding another line can make at most four pieces total. (We can't move pieces of the circle between cuts!) There are lots of ways to do two cuts dividing the circle into four pieces. Letís start with the simplest possible: take two lines going through the center at right angles to each other. There are lots of other configurations we could try, but this has the advantage of being simple. Letís see what this gives us.
So, where should we place the third line? For each of the four pieces of the circle determined by the first two lines, either the third line divides that piece into two, or it does not. So how many of the four pieces can we divide at once? It doesn't look like four is possible, but three is very doable. In the figure below, we start with four pieces. A thin third line divides the blue, green, and red pieces into two, giving a total of seven pieces, as desired.
One of my favorite features about mathematics is that frequently there is more than one way to solve a problem, with different solutions highlighting different aspects. This problem is a terrific example, as there are at least two solutions which appear fundamentally different. Letís talk about how to find another one.
Here's another way to attack the problem. Pick three points on the circle and connect the points with three lines. This divides the circle into three caps and a triangle, for four regions total. What we want to do now is perturb our initial three line segments, and see how that adds more regions as we move them.
Specifically, we slide each line just a little bit toward the middle of the circle, and we see that the lines overlap enough to divide the circle into seven pieces. So this gives us our answer as well!
GENERALIZATIONS and the OEIS:
A huge part of mathematics is taking a solution to a specific problem and trying to generalize. Are there related questions to ask? How does a particular problem fit into a more general framework? What are the key features? What properties should the solutions of the general case have?
There are lots of ways we could generalize this problem. Perhaps the most natural (but by no means the only) possibility is to ask:
How many pieces can we get with four lines? With five lines? With n lines?
As the number of lines increases and the number of possible configurations of lines skyrockets, we start to question whether the methods we used to solve the original riddle will work in the general case. Also, no one is challenging us to reach a particular number of pieces anymore, which means that we have to guess what is and is not possible and then prove our claim. Proving that a certain number of pieces is impossible can be difficult (maybe not for a small number of lines on the circle, but certainly as that number gets larger).
At this point we may be thinking that someone else with a lot more mathematical background has already solved this problem using advanced mathematics and it may instead be worth our while to try another problem. In this case, the Online Encyclopedia of Integer Sequences (OEIS) may be able to help us research what is known about the riddle: http://oeis.org/.
This encyclopedia is amazing. If we know the first few terms of some sequence of integers, this website can often actually identify our sequence and continue it for us, based on what others have already discovered. Try inputting (i) 1, 1, 2, 3, 5; ; (iii) 1, 3, 6, 10, 15, 21; (iii) 1, 1, 2, 5, 14, 42, 132; (iv) 27, 82, 41, 124, 62, 31, 94, 47, 142; (v) 4, 11, 31, 83, 227, 616, 1674. While some of these series are hopefully old friends, Iím hoping at least the last two are new.
Returning to our problem, we discovered earlier that with 0, 1, 2, and 3 cuts we can make up to 1, 2, 4, and 7 pieces.
If we input 1, 2, 4, 7, and scroll down a bit, we find:
Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pancake with n cuts.
So not only has the general problem been solved, but there is a really nice formula for the number of slices possible with n cuts. The first few terms are 1, 2, 4, 7, 11, 16, 22, 29, 37, 46.
If we take a look at the sequence, we see that the jump between terms increases by one each time. This makes sense. Going back to our discussion of how new cuts took existing pieces and cut them into two, we saw that the first slice affected one piece, the second slice affected two existing pieces, and the third slice affected three pieces, and so, as we would expect, the number of pieces that we can cut at a time increases by one for every new slice.
Another way of phrasing this is that if we look at the difference between adjacent terms, we get a new sequence: 1, 2, 3, 4, 5, 6, 7, 8, 9Ö. This sequence is much easier to understand, and once we understand it we can go back to our original sequence. This is a great lesson: when youíre given something to study, acquire data points if possible, and look for relations. It may be the case that a related problem is clearer than the original.
What else is cool about OEIS? If we look below a sequence we get a comments section that describes the sequence in many other ways. We also get a number of formulas and references for further reading which may help place the topic in a larger mathematical background.
Finally, we end with a few additional generalizations. If youíre teaching this in a class, stop here and donít read further. Try and come up with your own questions. See what you students or classmates create, and then come back.