The necessary condition that is proved above—a voting paradox can happen only if all three preference lists remaining after cancellation have the same spin—is not also sufficient.

(a) Continuing the positive cycle case considered in the proof, use the two inequalities 0 < a - b + c and 0 < -a + b + c to show that |a – b| < c.

(b) Also show that c < a + b, and hence that |a – b| < c < a + b.

(c) Give an example of a vote where there is a majority cycle, and addition of one more voter with the same spin causes the cycle to go away.

(d) Can the opposite happen; can addition of one voter with a “wrong” spin cause a cycle to appear?