Consider the equation:
P₁* P₂ + P₂* P₃+…..+ P_n-1* P_n + P_n* P₁ = 0, where n is a positive integer, and each P_i= 1 or -1

Derive the general form of n such that the above equation has at least one valid solution.

(In reply to seems to be a solution by Charlie)

For any multiple of 4, say n=4*k, repeat the string {1,1,-1,-1} k times.  Then the products in the sum alternate 1,-1,1,-1... with an even number of terms, 2*k each of 1 and -1.  The sum of these must be 0.  This proves all multiples of 4 have at least one valid equaiton.

Posted by Brian Smith on 2016-09-05 12:56:31