Problem of the Week

Problem 7 (due Monday, December 1 )

Consider an $m\times n$ rectangle divided into $mn$ unit squares. Let $T$ be the set of all vertices of the unit squares. At each point of $T$ we draw a short arrow (say of length $1/2$) pointing up, down, left, or right with the condition that no arrow sticks outside the rectangle. Prove that regardless of how the arrows are chosen, there always must exist two vertices of the same unit square at which the arrows point in opposite directions.

Overview

Every other Monday (starting 08/25/25), 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.