The Mathematical Magic Behind Undergraduate Divisibility Problems
This blog post explores the origin of common problems in undergraduate mathematics courses, such as proving that a polynomial is always a multiple of a certain integer. The author points out that these problems stem from combinatorial counting, specifically Pólya-Redfield counting. This method uses the orbit-counting formula under group action to connect the value of a polynomial to the counting of a certain combinatorial structure, ensuring the polynomial is always a multiple of a specific integer. The article uses two examples, bracelet counting and tic-tac-toe board counting, to explain how Pólya-Redfield counting is used to construct these problems. It also proposes a conjecture about whether all such polynomials originate from Pólya-Redfield counting.
Read more