A sphere is divided into regions by 9 planes that are passing through its center. What is the largest possible number of regions that are created on its surface?
a.
b.
c. 81
d. 76
e. 74
This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)
Reflection
I didnβt really need the projection into the plane for the solution, but my problem-solving self needed it to be able to count points and regions in slow motion. So, I should present a cleaned-up solution:
Solution
Since there are 9 planes, each plane must intersect with every other in a line, creating two points on the surface of the sphere. Thus, there are (9β8)/2 * 2 = 72 points of intersection, and for n planes, there are π(π β 1) points of intersection. With the first plane, there are zero points of intersection and two regions. Suppose we now have n planes and N regions. We add another plane, creating a circle on the sphere. For each segment that the circle intersects, it creates an additional intersection point as it enters, and it divides the region into two parts, adding one additional region. Hence, for each point added, a region is added as well. Since there are two
regions with zero points, there are thus 74 regions with 72 points of intersection.
A sphere is divided into regions by 9 planes that are passing through its center. What is the largest possible number of regions that are created on its surface?
a.
b.
c. 81
d. 76
e. 74
This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)
OK, so I wanted to prove that each region would be a triangle. So I decided to project the sphere onto a plane.
The projection of four planes:
Conjecture: The max number of regions is the number of intersection points plus 2. Proof (by induction)
If we have 1 plane, we have no intersection points and 2 regions. Suppose we have n planes with π(π β 1) intersection points and π(π β 1) + 2 regions. Now we add the next plane to our figure. The plane creates a circle on the sphere. To maximize the number of regions, we angle the plane so that our circle does not intersect any already-existing intersection points. So the circle goes through a number of segments. Each time it does, it cuts the region bounded by that segment into two. So for each new intersection point, we lose one region and gain two, for a net gain of one region. That is, however many intersection points are added, that will be the number of regions added as well. And since π + 1 planes have (π + 1)(π) intersection points, we will
have (π + 1)(π) + 2 max regions. DONE.
For the original competition problem, we have 9 planes and hence 9*8 + 2 = 74 regions, answer e.
A sphere is divided into regions by 9 planes that are passing through its center. What is the largest possible number of regions that are created on its surface?
a.
b.
c. 81
d. 76
e. 74
This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)
OK, so I wanted to prove that each region would be a triangle. So I decided to project the sphere onto a plane.
The projection of four planes:
After a while, I had a chart for max possible regions.
1 plane: Max regions = 2
2 planes: Max regions = 4
3 planes: Max regions = 8 (exponential?)
4 planes: Max regions = 14 (nope!)
5 planes: Max regions = 22 (huh?)
Then, really because I had no other ideas, I tried counting intersection points AND max regions
(remembering that one intersection point is βat infinityβ β that is, the north pole).
1 plane: Intersection Points = 0, Max regions = 2
2 planes: Intersection Points = 2, Max regions = 4
3 planes: Intersection Points = 6, Max regions = 8
4 planes: Intersection Points = 12, Max regions = 14
5 planes: Intersection PointsΒ 20, Max regions = 22
Oh. My. Goodness. The max regions are simply the number of intersection points plus 2. Could it really REALLY be that simple?
A sphere is divided into regions by 9 planes that are passing through its center. What is the largest possible number of regions that are created on its surface?
a.
b.
c. 81
d. 76
e. 74
This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)
OK, so I wanted to prove that each region would be a triangle. So I decided to project the sphere onto a plane.
The projection of four planes:
After a while, I had a chart for max possible regions.
A sphere is divided into regions by 9 planes that are passing through its center. What is the largest possible number of regions that are created on its surface?
a.
b.
c. 81
d. 76
e. 74
This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)
OK, so I wanted to prove that each region would be a triangle. So I decided to project the sphere onto a plane.
For a while, I toyed with the situation where we have
Plane 1 β equator (this always happens: Just make plane 1 the equator) π1(0π, 0πΈ).
Plane 2 β Prime Meridian π2(90π, 0πΈ)
Plane 3 β Intl Date Line π3(90π, 90πΈ)
Plane 4 β at an angle to all of those π4(45π, 45πΈ)
Here is our mapping with P1, P2, and P3 on it:
Now, how to represent P4? Aha! The inside of the unit circle is the southern hemisphere, and the outside is the northern. P4 must hit the equator a two points 180 degrees apart, go inside the southern hemisphere, and then outside to the northern. Thus:
The white region is a NONtriangular region created by the intersection of four planes. These are strange-looking regions, and I spent a long time β several days β vainly trying to count max regions created when I added P5, P6 etc. But one thing was clear: not all of the regions are triangular, nor can they be. For if a plane (say P4) cuts through a triangular region, it will create a new triangular region and a non-triangular βquadrilateralβ, as in the figure below. So counting triangles from points is NOT the solution here!
A sphere is divided into regions by 9 planes that are passing through its center. What is the largest possible number of regions that are created on its surface?
a.
b.
c. 81
d. 76
e. 74
This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)
OK, so I wanted to prove that each region would be a triangle. So I decided to project the sphere onto a plane. Thereβs a standard way of doing that, used both by map-makers and mathematicians. Place the sphere with the south pole on the plane at the origin. Then for each point on the sphere, run a line from the north pole through that point to the plane. This gives a 1-1 mapping of sphere to plane. The diagram below shows this mapping, with the points A and B on the sphere mapping to the points Aβ and Bβ on the plane respectively.
Notice that in the mapping above, the south pole is mapped to the origin (βstraight downβ), while the north pole itself cannot be mapped. We call the north point the βpoint at infinity.β Also notice that the equator gets mapped to a circle. And, any circle around the sphere that goes through the north pole will also go through the south pole, and so becomes a line in the plane.
A sphere is divided into regions by 9 planes that are passing through its center. What is the largest possible number of regions that are created on its surface?
a.
b.
c. 81
d. 76
e. 74
This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)
AHH! Insight! Each plane must intersect the others because they all pass through the center. And two planes intersect in a line. And the line must intersect the sphere at two points. SO, we can count intersection points: There are 9 planes, and each plane will intersect the other 8, so there are 9 β 8 = 72 intersection points IF we arrange the planes for maximum regions. More generally, if we have n planes arranged for max intersection points, we will have π(π β 1) intersection points.
Wait, letβs do this carefully. There are 9 planes, and they can each intersect 8 different planes; but that counts the intersections of plane A and plane B twice, so there are (9*8)/2 = 36 lines of intersection, but 36 β 2 = 72 points of intersection with the sphere. So our problem just got narrower: Given 72 intersection points defining various regions on the sphere, how many regions do we get?
And thatβs where the problem stands as of this writing. My preliminary conjecture is that each region will be a βtriangleβ (officially, spherical triangle) on the surface of the sphere, especially if we are maximizing regions. I need to prove that conjecture and then count triangles, which shouldnβt be too hard.
A sphere is divided into regions by 9 planes that are passing through its center. What is the largest possible number of regions that are created on its surface?
a.
b.
c. 81
d. 76
e. 74
This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)
At this point, various methods suggested themselves. Perhaps we could use recursion: let be the regions created by planes, and then we could examine the number of additional regions formed by planes?
Or, related to this, perhaps we needed to find the number of intersection points of each of the planes, and then relate the number of intersection points to the number of regions. But how to describe the intersection points?
It did occur to me that if we have n planes situated for maximal regions, they will divide the equator up into subintervals, and adding another plane will divide up two of those subintervals into 4. Did that help? Well, it could help count the number of regions touching the equator: two for each subinterval (one north of equator, one south). But what about the regions not touching the equator? Hmph.
One possible way to visualize this problem is to project the plane onto a sphere. I know how to
do that, but counting the regions still seems hard.
For a while, I toyed with the situation where we have
Plane 1 β equator (this always happens: Just make plane 1 the equator) π1(0π, 0πΈ).
Plane 2 β Prime Meridian π2(90π, 0πΈ)
Plane 3 β Intl Date Line π3(90π, 90πΈ)
Plane 4 β at an angle to all of those π4(45π, 45πΈ)
I looked at my daughterβs wall map of the world: P4 goes through Tblisi Georgia and south of French Polynesia.
Where does P4 intersect the others? Could I make a formula to find the intersection points?
A sphere is divided into regions by 9 planes that are passing through its center. What is the largest possible number of regions that are created on its surface?
a.
b.
c. 81
d. 76
e. 74
This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)
On my first pass, all I could do was to visualize the first three planes, one at the equator, one passing through the prime meridian in Greenwich England, and one passing through the International Date Line. That gave me regions, so my preliminary conjecture was βb. β. But I couldnβt prove it. And when I tried to mentally add a fourth plane to my diagram β one starting in Ukraine or something and hitting the equator halfway between the others β I found that I couldnβt clearly see that plane and count the regions formed. That vexed me for a while, and I put it away for the day.
The next day, I realized that I wasnβt going to be able to picture these planes, and I needed to find a way to describe their directions mathematically. The picture I had was of the equatorial plane and a second plane passing through it in the center. That second plane could be rotated any amount around the equator β described by one angle β and then elevated by tilting to a different angle. So I conjectured that two angles uniquely describe each plane: π to describe angle around and π to describe angle of elevation.
In the shower, I realized that I had just rediscovered latitude and longitude! That made me feel much better about my mathematical description as likely correct.
But now, how to turn the mathematical description into a solution? If I have one plane at (π1,π1), how do I count the regions it creates with the other planes?
In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Lisa Sun. Her topic, from Precalculus: using Pascal’s triangle.
How could you as a teacher create activity or project that involves your topic?
To introduce Pascalβs Triangle, I would create an activity where it involves coin tossing. I want to introduce them with coin tossing first before bringing in binomial expansions (or any other uses) because coin tossing are much more familiar to majority, if not all, students. Pascalβs Triangle can show you the probability of any combination of coin tossing (aka binomial distribution). Below are a few of the results and how they compare with Pascalβs Triangle:
Afterwards, I would ask the students guiding questions if they see anything interesting about the numbers that we gathered. I want them to notice that each number is the numbers directly above it added together (Ex: 1 + 2 = 3) and how those three numbers form a triangle hence, Pascalβs Triangle.
B2: How does this topic extend what your students should have learned in previous courses?
In previous courses, students should have already learned about binomial expansions. (Ex: (a+b)2 = a2+ 2ab + b2). This topic extends their prior knowledge even further because Pascalβs Triangle displays the coefficients in binomial expansions. Below are a few examples in comparison with Pascalβs Triangle:
If any of the students are having difficulties expanding any of the binomials or remembering the formula, they can remember Pascalβs Triangle. Using the Pascalβs Triangle for solving binomial expansions can aid the students when it comes to being in a stressful environment (ex: taking a test). Making a connection between their prior knowledge on binomial expansion and Pascalβs Triangle, I believe it would give the students a deeper understanding as to how Pascalβs Triangle was formed.
C2: How has this topic appeared in high culture?
Thereβs a computer scientist, John Biles, at Rochester University in New York State who used the series of Fibonacci numbers to make a piece of music. How do the Fibonacci numbers relate to Pascalβs Triangle you ask? Well, observe the following:
As you can see, the sum of the numbers diagonally gives you the Fibonacci numbers (a series of numbers in which each number is the sum of the two preceding numbers).
John Biles composed a piece called PGA -1 which is based on a Fibonacci sequence. Note that on a piano, from middle C to a one octave C, there are a total of eight white keys (a Fibonacci number). Also, when you do a chromatic C scale which includes all the black keys, there are a total of five black keys (another Fibonacci number) which are also separated in a group of two and three black keys (see the pattern?). When youβre creating chords, letβs take the C chord for example, it consists of the notes C, E, and G. Notice that harmonizing notes are coming from the third note and the fifth note of the whole C scale. So following similar ideas on the use of these numbers/sequences, John Biles was able to compose music.
The following may be a bit extra, but I also want to include this youtube link of this blogger who was very precise and compared the sequences to current pop music:
[I found this to be super interesting!]
How have different cultures throughout time used this topic in their society?
Hundreds of years before Blaise Pascal (mathematician whom Pascalβs Triangle was named after), many mathematicians in different societies applied their knowledge of the Triangle.
Indian mathematicians used the array of numbers to represent short and long sounds in poetic meters in their chants and conversations. A Chinese mathematician, Chu Shih Chieh, used the triangle for binomial expansions. Music composers, like Mozart and Debussy, used the sequence to compose their music to guide them what notes to play that would be pleasing to the audience. In the past, arithmetic composing was frowned upon however contemporary music to this day is now filled with them. When Pascalβs work on the triangle was published, society began to apply the knowledge of the Triangle towards gambling with dice. In the end, all cultures began to use Pascalβs Triangle similarly in their daily lives.
How can technology be used to effectively engage students with this topic?
The Youtube video above is a great tool for students who are visual learners. This video is to the point and clear with the message as to what Pascalβs Triangle is, the uses of it, and who aided in the discovery of it. I also believe the characters that were being used in this video would be appealing to students. This video was filled with facts that I want my students to know therefore, I would like them to follow along and write down important facts about Pascalβs Triangle. I would like to conclude that technology can be a βforce multiplierβ for all teachers in their classroom. Instead of having the teacher being the only source of help in a classroom, students can access web site, online tutorials, and more to assist them. Whatβs great is that students can access this at any time. Therefore, they can re-watch this video again once theyβre home when they need a refresher or didnβt understand something the first time.
be the regions created by 
planes, and then we could examine the number of additional regions formed by 
planes?
subintervals, and adding another plane will divide up two of those subintervals into 4. Did that help? Well, it could help count the number of regions touching the equator: two for each subinterval (one north of equator, one south). But what about the regions not touching the equator? Hmph.
regions, so my preliminary conjecture was βb. 
Mean Green Math 