MathDB

Consider

m+1

horizontal and

n+1

vertical lines (

m,n\ge 4

) in the plane forming an

m\times n

table. Cosider a closed path on the segments of this table such that it does not intersect itself and also it passes through all

(m-1)(n-1)

interior vertices (each vertex is an intersection point of two lines) and it doesn't pass through any of outer vertices. Suppose

A

is the number of vertices such that the path passes through them straight forward,

B

number of the table squares that only their two opposite sides are used in the path, and

C

number of the table squares that none of their sides is used in the path. Prove that

A=B-C+m+n-1.

Proposed by Ali Khezeli

Problem Statement