Problem
Simple but Fun
A permutation of an odd-sized set produces an even product. Can you see why?
Suppose \(n\) is an odd number and \(a_1, a_2, \dots, a_n\) is a permutation of \(1, 2, \dots, n\). Prove that the product
\[(a_1-1)\cdot(a_2-2)\dots (a_n-n)\]is always an even number.
Explore the solution
One only needs to observe that
\[(a_1-1) + (a_2-2) + \dots + (a_n-n) = 0.\]As the sum of an odd number of integers results in zero, at least one of them should be even.