A question on parity from book "Mathematics Circles"

Question

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 too.He does this 25 times. Can he return the three pucks to their starting positions.(THIS IS A PART OF QUESTION ONLY)

Now, can you describe how this question is related to parity. I am finding some patterns in the position changed but not able to conclude anything. This question is from "Mathematics Circles" Chapter-1 Parity. The answer given behind is not satisfactory.

Answer 1

• Consider describing pucks' position by naming them clockwise. For example, you can get "ABC" or "BCA" or whatever.
• Observe that some positions are essentially the same and their naming depends on which puck you start, for example "ABC" = "BCA" = "CAB".
• (here comes the parity) Observe that there are two major classes of states: "ABC" = "BCA" = "CAB" and "ACB" = "CBA" = "BAC". You may call them "odd state" and "even state" or whatever. This is a very crucial step. I'm basically saying: "I'll neglect all the information on pucks' positions, except a tiny bit."
• (here comes the solution) Observe that hitting a puck you always transfer between states. For example "ABC" (odd) can be transformed to "ACB" (even) or "CBA" (even),but never to "BCA" (odd) or "CAB" (odd).
• Consider a sequence of hits that transfer odd -(first hit)-> even -(second hit)-> odd -...-(25-th hit)->even. Make a conclusion on impossibility.