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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.