Embedded Files
When Am I Ever Going to Use This?
When Am I Ever Going to Use This?
If you enjoyed solving this problem, you may enjoy looking for patterns and solving problems as a career!
In the video below, you'll hear from Li-An Chen, a current Boise State Lecturer in the Mathematics Department as she discusses the math she's used in her career and the advise she has for someone who is interested in a career in Mathematics or STEM
"Every major or job has difficult parts and you can only go past it if you have the passion"
"Every major or job has difficult parts and you can only go past it if you have the passion"
Fun Fact: Mathematics & Studying Law
Fun Fact: Mathematics & Studying Law
Another career you may be interested if you enjoy logic and looking for patterns is a career in law!
Students who major in mathematics or statistics have higher LSAT scores and acceptance rates.
"Lawyers with STEM backgrounds—particularly in computer science and electrical engineering—are in high demand, especially at law firms with intellectual property practices,"
"Seven of the 10 majors with the highest median LSAT scores in the fall 2023 applicant pool were STEM,"
What Do Mathematicians Do?
What Do Mathematicians Do?
Want to learn more about what Mathematicians do? The American Mathematical Society has put together several resources for you to learn more! Click here to check them out.
What We Learned
What We Learned
Visualizing the Situation
Visualizing the Situation
Using the free online tool, GeoGebra, I created a visualization of this situation, which you can check it out here. Using the slider on the left (labeled "size"), you can adjust the number of points on the circle to see visualizations of the regions created. The points are placed randomly.
Solution - Ruize Sun.pdfRuize's Solution
Ruize's Solution
The PDF shown here was sent in by Ruize Sun, a student here in the Treasure Valley.
Ruize analyzed this pattern by identifying what happens every time a new dot is added (how many new regions will be added). He described the relationship of number of regions from one figure to the next with clever summations, and then used summation identities to create an explicit formula.
This solution shows elegance in how he translated rules and patterns into a mathematical formula!
Solution - Tracen.pdfTracen's Solution
Tracen's Solution
The PDF shown here was sent in by Tracen Knowles, a student here in the Treasure Valley.
Tracen analyzed this pattern by creating tables of values to see how many new regions are added with each new point, and described this pattern by looking at how many regions are being added with each new line that is drawn from the new point.
After using this pattern to create an explicit formula, Tracen added a fun connection to Calculus by looking at the limit of the number of regions and analyzing the percent error that occurs if we use the 2^(n-1) pattern that seems to be present early on in the pattern!
Josie's Solution Proving the Function is Quartic
Josie's Solution Proving the Function is Quartic
Similar to Ruize and Tracen's solutions, when I tried to solve this problem, I approached it by looking iteratively. I asked myself: "What happens when I add a new point?" Once I began to think I found a pattern, I needed to convince myself that that patterns would always hold! Then, I used my pattern to find an equation.
I've described my solution in three videos:
Where did the pattern I notice come from and how did I know it would always hold for any number of points? Check out the first video to the left, or click here to watch on YouTube
How can I prove that my pattern can be described by a fourth degree polynomial and use technology to create a formula? Check out the second video to the left, or click here to watch on YouTube
Sum of Squares video referenced
How could you find the formula without technology ? (the video discusses conceptually how you could go about finding the formula, but because of laziness and lack of time I did not fully derive the formula by hand) Check out the third video to the left, or click here to watch on YouTube
A solution with Combinatorics
A solution with Combinatorics
I found this video after reading Ruize's solution and trying it myself as shown in the videos above. I was looking for other ways to solve the problem, and stumbled upon this one. 3Blue1Brown is an excellent YouTuber that always has amazing visualization for complex mathematical ideas!
This video does an excellent job articulating what is happening in this situation and uses the formula for Combinations to get the answer. There is also an interesting connection to Pascal's triangle that gets discussed!
I also learned that this problem is called Moser's Circle Problem!
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