How many Latin squares are there of a fixed size under certain given restrictions?What is Pascal’s tetrahedron?

Math 202 – Writing project

Each final project will have the following:

• 12 point font, Times New Roman
• 1” margins top, bottom, left, right
• Cover page with title, my name, and student name
• 6 – 12 page typed, double spaced paper
• APA 7 formatted paper and reference page (at least 5 sources that are scientific based)
• 5 – 10-minute presentation for the class on the research
  • What did you do?
  • What did you find?
  • What did you learn?

You may choose from the following topics:

1. The Evolution of Cryptography through Number Theory
   1. Who was influential?
   2. What is used and why?
   3. When did cryptography start? Was number theory before, during, or after?
   4. Other questions to consider…..
2. Scheduling problems: Look for explicit scheduling examples in your life. Here are some ideas to consider.
   1. When organizing a conference or event, how do you schedule the talks according to participant and room restrictions?
   2. How are final exams, sports competitions, and community events scheduled at NCC? How do you make sure that all or participants must be able to attend their events or exams?
   3. How many teachers’ committees are there in your school and how are they formed? When are regular faculty meetings scheduled? How are classes assigned to you? How are classes scheduled?
   4. Use different techniques and levels of difficulty: weighted graphs, SDRs, matchings, chromatic polynomials.
3. Power in games : Look for any kind of real-life examples where some kind of vote takes place. Model and determine the power that each involved party has using the Shapley-Shubik power index. Here are some possibilities.
   1. Find out how much power each state has in the US presidential elections according to the Shapley-Shubik power index.
   2. Are there student elections at your school? Do you vote on certain school decisions? How much power do you have in an election where each department has one vote, but some departments are larger than others? What kind of majority is needed?
4. Pascal’s Triangle and Binomial Coefficients:
   1. Study Pascal’s triangle and look for any kind of patterns and binomial coefficient identities.
   2. Do the same for Pascal’s triangle mod n.
   3. What is Pascal’s tetrahedron? How can you use it in algebraic expansions, that is, is there something like the binomial theorem for Pascal’s tetrahedron?
5. Latin squares Study Latin squares and their relationship to Sudoku.
   1. How many Latin squares are there of a fixed size under certain given restrictions? Come up with easier but still interesting Sudoku variations.
6. Algebraic representations of graphs
   1. Study the adjacency matrix of a graph. How can you find the number of edges, the degrees, the number of triangles, etc., without drawing the graph? Just by using the entries of the matrix.
7. Number Theory
   1. Survey of arithmetic functions σ, τ, µ, ϕ and how to compute them. Combinatorial proofs
   2. Understand divisibility criteria. Develop divisibility criteria for “nontraditional” primes, like 7 or 13. Can you explain why these are not usually mentioned in the regular literature?
8. Other ideas?
   1. They must be approved by Dr. Counterman prior to starting. Be sure to get your idea to her by the first check in assignment of the project.