Blog Post 8

\(1\) \(\underline{\text{Parity}}\)

Parity, in terms of mathematics, describes the classification of an integer as either even or odd. An even number is defined as an integer that is divisible by \(2\) while an odd number is one that is not. A more formal definition states that an even number is an integer \(n\) of the form \(n=2k\) where \(k\) is an integer. On the other hand, an odd number is an integer of the form \(n=2k+1\). In set notation we see:

\[\text{Even} \hspace{0.5mm} = \hspace{0.5mm} {2k : k \in \mathbb{Z}}\]

\[\text{Odd} \hspace{0.5mm} = \hspace{0.5mm} {2k+1 : k \in \mathbb{Z}}\]

In number theory, the idea of parity allows us to solve some mathematical problems simply by making note of odd and even numbers. In the same way, the impossibility of some mathematical constructions can be proven. For example, consider the following question:

\(\hspace{1cm}\) Is it possible to fill in the blanks below with \(+\) and \(-\) signs to create a total of \(20\)?

\(\hspace{1cm}\) \(1 \)_\( 2 \)_\( 3 \)_\( 4 \)_\( 5 \)_\( 6 \)_\( 7 \)_\( 8 \)_\( 9\)

In this case, we find that such a construction is not possible. Here, we have \(5\) odd numbers and therefore the total must be odd. Since \(20\) is even, this proves it cannot be done.

This example makes reference to some properties of the parity of numbers that should be noted. In particular, notice that:

\(\underline{Addition}\) \(\hspace{7.5cm}\) \(\underline{Multiplication:}\)

  • even \(\pm\) even \( =\) even \(\hspace{5cm}\) even \(\cdot\) even \(=\) even

  • even \(\pm\) odd \(=\) odd \(\hspace{5.4cm}\) even \(\cdot\) odd \(=\) even

  • odd \(\pm\) odd \(=\) even \(\hspace{5.5cm}\) odd \(\cdot\) odd \(=\) odd

Parity is manifested within the game of chess in multiple ways. Considering a chess board, the parity can be indicated by the color of the square. Thus, say dark squares are odd and light squares are even. According to the rules of chess, the bishop, for example, can only move diagonally. In a sense, it is moving one over, one up/down (and vice versa). Therefore, it is moving two spaces, or an even amount of spaces. Parity principles tell us, then, that if the bishop started on an even (light) square, it can only ever move onto other even squares. The same is true for an odd (dark) square.

This same idea is critical in solving the mutilated chess board problem which states: “Suppose a standard 8x8 chessboard has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoes of size 2x1 so as to cover all of these squares?”

It has been proven that because each domino covers one square of each parity and there are two more squares of one parity than of the other, it is not possible to cover all the squares.