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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.

Rishabh Shrivastava
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  • It is hard to interpret the intended question from your posting. Please edit your posting to [1] provide the exact wording of the original question and [2] provide the answer given that you regard as unsatisfactory. This will help mathSE reviewers to [A] reverse engineer the problem composer's intent and [B] either correct the unsatisfactory given solution or explain it to you in a way that makes sense to you. – user2661923 Jan 18 '22 at 16:28
  • When you edit the post for clarity, I suggest leaving off all of the hockey stuff...it's just a distraction. I think, but am not at all sure, that you are saying that an ordered triple $(x,y,z)$ can be switched to $(y,x,z)$ or to $(x,z,y)$. Is that correct? – lulu Jan 18 '22 at 16:31
  • Your are welcome to flag me with a comment when the editing is done. However, please do not provide the requested info in comments. Instead, please edit your question to provide this info. – user2661923 Jan 18 '22 at 16:31
  • @lulu Normally, I would agree with you. However, here, there is the danger of something being misinterpreted by the OP (i.e. original poster) when he translates the question into math terms. Therefore, I suggest that the OP provide the exact wording of the original question. – user2661923 Jan 18 '22 at 16:33
  • @user2661923 Fair enough. Though it is awfully difficult to picture what is being done with these hockey pucks. – lulu Jan 18 '22 at 16:36
  • @lulu No, I don't think ordered triples describe this. We have three objects (pucks) on a plane, and might as well consider them point-like. Each move changes the position of just one object. A move changing from position $A$ to $A'$ must have segments $AA'$ and $BC$ intersect, and not at an endpoint. – aschepler Jan 18 '22 at 16:55
  • To get to the original configuration, what does the parity have to be? You have $25$ reflections. What can you conclude? – John Douma Jan 18 '22 at 17:00
  • @aschepler Well...but doesn't that simply take $(A,B,C)$ into $(B,A',C)$?. I can certainly identify the position of the pucks with ordered triples up to cyclic order. I'd have thought that was enough. – lulu Jan 18 '22 at 17:03
  • @lulu Probably, after you appropriately apply the lynchpin idea that clockwise/counterclockwise orientation of the triangle is important and define what those triples describe. I think to state the problem in that way is skipping the interesting part. – aschepler Jan 18 '22 at 17:10
  • Guys, you can see the body of my question. All the details there. – Rishabh Shrivastava Jan 19 '22 at 17:22

1 Answers1

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  • 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.
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