Let $x_1, x_2,\dots, x_n$ be an arrangement of the numbers $1, 2,\dots, n$. Show that if n is odd, then $(x_1 − 1)(x_2 − 2)\dots(x_n − n)$ is even.

Question

For example, for $n = 6$ we could have $4, 3, 6, 1, 2, 5$.

I found this question in "How to Think Like a Mathematician" by Kevin Houston.

I know that $(x_1-1)(x_2-2)\dots(x_n-n)$ is even only if one of the factors is even. However, I do not know how to formally prove this whatsoever.

Answer 1

In order for there to be no even terms in the product, every odd number must be paired with an even number. However n is odd, so the number of odd numbers in the list is one more than the number of even numbers. Therefore at least one pair $x_k-k$ has to have two odd numbers, resulting in an even number difference.