Consider a rectangular array of dots with an even number of rows and an even number of columns. Color the dots, each one red or blue, in such a way so that in each row half the
dots are red and half are blue, and also in each column half are red and half are blue. Now,whenever two points of the same color are adjacent (in a row or column), join them by an edge
of that color. Show that the number of red edges is the same as the number of blue edges.
To prove that the number of red edges is equal to the number of blue edges, we need to first establish a way to count the number of red and blue edges separately.
Let's consider a rectangular array of dots with m rows and n columns. Since both m and n are even, we can divide the dots into pairs in each row and column.
To count the number of red edges, we can focus on each pair of adjacent red dots in a row or column. We can count the number of horizontal red edges by counting the number of adjacent pairs of red dots in each row, and we can count the number of vertical red edges by counting the number of adjacent pairs of red dots in each column.
Similarly, we can count the number of blue edges by focusing on each pair of adjacent blue dots in a row or column. We can count the number of horizontal blue edges by counting the number of adjacent pairs of blue dots in each row, and we can count the number of vertical blue edges by counting the number of adjacent pairs of blue dots in each column.
Now, to prove that the number of red edges is equal to the number of blue edges, we can proceed by considering the number of pairs of adjacent dots in both rows and columns. Since each dot is adjacent to exactly one dot in its row and one dot in its column, the total number of pairs of adjacent dots is the same for both rows and columns.
Since we are dividing the dots into pairs, there are m*n/2 pairs in total. Since m and n are even, m*n/2 is an integer. Therefore, the number of red edges (counted by pairs of adjacent red dots) and the number of blue edges (counted by pairs of adjacent blue dots) is the same.
Hence, we have proven that the number of red edges is equal to the number of blue edges in the given rectangular array.