Problem
COM-B2-M05-P003 Three Negative Signs
#3
★★☆☆☆ Level 2 of 5
Three signs \(+,+,+\) are written on a board. In one move, choose two signs and replace each by the opposite sign. Prove that \(-,-,-\) cannot be obtained.
Consider the product of the three signs.
Interpret \(+\) as \(1\) and \(-\) as \(-1\). One move changes two signs, so the product of the three numbers is multiplied by \((-1)^2=1\).
Initially the product is \(1\). For the state \(-,-,-\), the product is \(-1\). The invariant differs, so the state is impossible.
Preparation for harder sign-changing problems on graphs.