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 the same as the number of blue edges in the given scenario, we need to establish a few concepts and observations.
Observation 1: Each dot has exactly two adjacent dots in the same row, as well as exactly two adjacent dots in the same column.
Now, let's start by counting the total number of edges in the given rectangular array of dots. Each dot contributes towards two edges - one in the row and one in the column. Since there are an even number of rows and an even number of columns, the total number of edges is even.
For simplicity, let's assume that the array has m rows and n columns, where both m and n are even numbers.
The total number of red edges can be obtained by counting the number of edges connecting two adjacent red dots. Similarly, the total number of blue edges can be obtained by counting the number of edges connecting two adjacent blue dots.
To establish a connection between the number of red edges and the number of blue edges, we will consider two cases:
Case 1: The first dot in the first row is red (denoted as R).
In this case, we have two possibilities for the second dot in the first row:
a) Blue (denoted as B): This will result in a red edge connecting the first and second dots in the first row and a blue edge connecting the first dots in the first and second columns.
b) Red (denoted as R): This will result in a red edge connecting the first and second dots in the first row and a red edge connecting the first dots in the first and second columns.
Observation 2: In Case 1, if the second dot in the first row is blue, it will force all the dots in the first row to alternate between red and blue.
From Observation 2, it follows that the number of red edges will be equal to the number of blue edges if the second dot in the first row is blue.
Case 2: The first dot in the first row is blue (denoted as B).
In this case, we have two possibilities for the second dot in the first row:
a) Red (denoted as R): This will result in a blue edge connecting the first and second dots in the first row and a red edge connecting the first dots in the first and second columns.
b) Blue (denoted as B): This will result in a blue edge connecting the first and second dots in the first row and a blue edge connecting the first dots in the first and second columns.
Observation 3: In Case 2, if the second dot in the first row is red, it will force all the dots in the first row to alternate between red and blue.
From Observation 3, it follows that the number of red edges will be equal to the number of blue edges if the second dot in the first row is red.
Therefore, in both cases, the number of red edges is equal to the number of blue edges.
To summarize, by observing the patterns in the arrangement of dots and considering the possibilities for the coloring of the second dot in the first row, we can establish that the number of red edges is always equal to the number of blue edges in the given scenario.