A street magician has three cups labelled, in order, A, B, C that he has upside down on his table. He has a sequence of moves that he uses to scramble the three cups: he swaps the first and second, then he swaps the second and third, then he swaps the first and third. If he goes through this sequence of three moves a total of nine times, in what order will the cups be?
Starting with $\mathrm{ABC}$, the magician

- swaps the first and second cups to obtain BAC, then
- swaps the second and third cups to obtain BCA, then
- swaps the first and third cups to obtain ACB.

The net effect of these 3 moves is to swap the second and third cups.

When the magician goes through this sequence of moves 9 times, the net effect starting from $\mathrm{ABC}$ is that he swaps the second and third cups 9 times.

Since 9 is odd, then the final order is ACB.

ANswer: ACB