20+ Math Research Topics for High School Students

If you are a high school student who is passionate about mathematics, pursuing research in mathematics can be a unique way to dive deep into the field. 

Math research can be a methodical, structured approach to exploring the field, finding answers to unsolved questions, analyzing patterns, and applying mathematical thinking to complex issues. The experience can also help you build skills like data collection, analysis, interpretation, and research paper writing. You can use this experience to boost your future college and job applications, as well as submit your research to competitions, journals, and conferences!

Here, we have compiled a list of over 20 math research topics for high school students. Here, you’ll find ideas across key categories such as statistics, number theory, geometry, and applied mathematics. The topics are both interesting and approachable at the high school level.

Probability and statistics

Probability and statistics focus on analyzing patterns in data and predicting outcomes. Many project topics in this research area are accessible if you’re familiar with basic probability rules and statistical tools like mean, standard deviation, and chi-square tests.

1. Modeling card game probabilities: Analyze the probability of getting certain hands in poker, bridge, or other card games.

2. Lottery systems and odds: Evaluate how different lottery systems work and calculate your chances of winning.

3. Sports analytics: Use data from sports, like basketball or soccer, to calculate player efficiency ratings or predict match outcomes.

4. Polling and bias: Design and test different types of surveys to explore how sampling bias affects results.

5. Weather and probability: Study local weather data to calculate the probability of specific weather events over time.

Number theory

Number theory, the study of integers and their properties, is a great research area to explore if you are interested in mathematical puzzles and patterns. Additionally, number theory has practical applications in sought-after disciplines like cryptography and coding. So, a research project in the area can be a chance to build a solid math foundation that can help you explore other interests in the future. You can work on a realistic and independent research project in this broad research area with just high school math as your base.

6. Exploring prime gaps: Investigate how primes are distributed and whether there are patterns in the gaps between them.

7. Perfect and abundant numbers: Research special types of numbers and try to identify rules or generate new examples.

8. Modular arithmetic in cryptography: Study how concepts like congruences are used in secure digital communication.

9. Patterns in Pascal’s triangle: Explore interesting patterns in Pascal’s triangle, such as Fibonacci numbers or binomial identities.

10. Fermat’s Little Theorem: Investigate how this theorem is applied in number theory or used in cryptography.

Geometry and topology

Geometry is an extremely important and practical field with connections to physics, architecture, and art. It deals with the quantitative properties of space, involving measurements, angles, and volumes of shapes. Topology is the study of qualitative properties, focusing on the spaces preserved under continuous deformations like stretching, bending, and twisting. The broad research area may require some reading in addition to school math, but it is still accessible to high schoolers.

11. Tiling patterns and symmetry: Analyze the mathematics behind tessellations and the classification of symmetrical tiling patterns.

12. Topology of everyday objects: Explore how basic concepts of topology (like connectedness or genus) apply to objects like coffee cups and donuts.

13. Geometric constructions with compass and straightedge: Study which figures can and cannot be constructed using classical tools.

14. Origami and geometry: Investigate the geometric principles that allow complex shapes to be created from simple folds.

15. Fractals in nature: Research how fractal geometry appears in natural structures like trees, coastlines, or snowflakes.

Algebra and functions

This area focuses on relationships between variables, algebraic structures, and patterns in functions. Algebra has critical real-world applications in population growth, financial models, and the spread of diseases, which you can study through structured research, especially if you want to work on an interdisciplinary study. Additionally, if you are interested in equations, sequences, or mathematical modeling, the research topics below are worth considering.

16. Exploring polynomial roots: Investigate how changing the coefficients of a polynomial affects the number and type of roots.

17. Fibonacci sequences and their generalizations: Study how these famous sequences appear in nature and art, and explore how they relate to the golden ratio.

18. Game theory in decision making: Model real-life strategic situations using basic principles from game theory.

19. Linear programming and optimization: Study how to formulate and solve optimization problems using constraints and objectives.

20. Patterns in arithmetic and geometric sequences: Analyze how sequences behave over time and where they apply in the real world.

Bonus research topics in math for high school students - 

Applied mathematics and interdisciplinary research

If you are interested in using math to address real challenges in various disciplines, consider choosing a topic in applied math. This area connects mathematical thinking to other disciplines, such as economics, computer science, biology, and social science. These topics often involve real-world data and modeling.

1. Epidemic modeling with SIR models: Study how diseases spread using compartmental models and real-world data.

2. Traffic flow and queuing theory: Work on a model that explains how traffic patterns work and how congestion can be predicted or minimized.

3. Math in music: Explore how musical scales, rhythms, and harmonics can be described using mathematical concepts.

4. Voting theory and fairness: Analyze different voting systems to evaluate how fair or manipulable they are.

5. Environmental modeling: Use differential equations or basic simulations to model things like pollution spread or the impact of climate change.

One other option—Lumiere Research Scholar Program

If you want to pursue independent math or interdisciplinary research under the guidance of a mentor, consider applying to Lumiere Research Scholar Programs, which are selective high school programs founded in partnership with Harvard and Oxford researchers. In the past, 4,000+ students applied for just 500 spots in the research program. You can learn about the application and available programs on the application form here.

Also, you can check out the Lumiere Research Inclusion Foundation—a non-profit research program for talented, low-income students. Last year, it offered 150 students full need-based financial assistance.

Stephen is one of the founders of Lumiere and a Harvard College graduate. He founded Lumiere as a Ph.D. student at Harvard Business School. Lumiere is a selective research program in which students work one-on-one with a research mentor to develop a research paper.

Image Source - Lumiere Research Scholar Program logo