A definitive solution to the most visited point problem in both the plane and space

Abstract

In this article, we will solve the problem of the most visited point within rectangles and parallelepipeds, with the problem already solved for squares in Santos and Castilho (2013). The problem is as follows: considering a rectangle in the first quadrant of the Cartesian plane with the lower-left vertex at the origin (0,0), we seek the integer coordinates through which the most paths pass. These paths are determined by integer steps either upwards or to the right, starting from the origin of the Cartesian system and reaching the upper-right vertex (M,N) of the rectangle. The conclusions we have reached are that the most visited point within the M by N rectangle, with M>N, is the point (1,0); in parallelepipeds of dimensions M by N by P, with M>N>=P, the most visited point is the point (1,0,0); in regular parallelepipeds of dimensions M by M by M, the most visited point is (1,1,1) for M=2, for M>2 the points will be (1,0,0), (0,1,0), and (0,0,1). We used basic tools of Combinatorial Analysis and the Principle of Induction for the calculations.