Odd and Even

Put the integers 1, 2, 3, … n in any order and call them a₁, a₂, a₃, … a_n. Then form the product

P = (a₁ – 1) × (a₂ – 2) × (a₃ – 3) … × (a_n – n).

Now: If n is odd, prove that P is even.

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If n is odd then the sequence (1, 2, 3, … n) begins and ends with an odd number. This means it contains one more odd number than even. Likewise (a₁, a₂, … a_n), which are only the same numbers in a different order. This means that two odd numbers must end up in the same bracket — there are not enough even numbers available to pair each with an odd. And if two odd numbers appear in the same bracket, they’ll produce an even difference and hence an even product. From Ross Honsberger, Ingenuity in Mathematics, 1970.