Three hockey pucks, A, B and C, lie on a playing field. A hockey player hits one of them in such a way that it passes between the other two. He does this 25 times. Can he return to three pucks to their starting position?
This question from the book, "Mathematical Circles (Russian Experience)". They have given an answer, but it isn't satisfactory.
On
No he cannot. There are six total possible orders, and you can assign each a number based on how many flips it takes to return the order to $(A,B,C)$. You can then check that if an ordering returns to $(A,B,C)$ in an even number of steps, then every ordering you can change it into takes an odd number of steps and vice versa. It then follows that you cannot get back to $(A,B,C)$ in an odd number of steps.
Consider the clockwise vs. counter-clockwise orientation of the A-B-C on the plane.