Question

Consider a rectangular array of dots with an even number of rows and
an even number of columns. Suppose the dots are colored red or blue
in such a way that every row has the same number of red and blue
dots, and likewise every column. Whenever two dots of the same color
are adjacent in a row or column, connect them with a line segment of
that color. Show that the total number of blue segments must equal
the total number of red segments.