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Problem of the Week

Problem 2 (due Monday, February 17)

Let $d(n)$ be the smallest number such that among any $d(n)$ points inside a regular $n$-gon with side of length 1 there are two points whose distance from each other is at most 1. Prove that

(a) $d(n)=n$ for $4\leq n\leq 6$.

(b) $\displaystyle \lim_{n\to \infty} \frac{d(n)}{n}=\infty $.

Overview

Every other Monday (starting 01/22/24), we will post a problem to engage our mathematical community in the problem solving activity and to enjoy mathematics outside of the classroom. Students (both undergraduate and graduate) are particularly encouraged to participate as there is no better way to practice math than working on challenging problems. If you have a solution and want to be a part of it, e-mail your solution to Marcin Mazur (mazur@math.binghamton.edu) by the due date. We will post our solutions as well as novel solutions from the participants and record the names of those who've got the most number of solutions throughout each semester.

When you submit your solutions, please provide a detailed reasoning rather than just an answer. Also, please include some short info about yourself for our records.