1

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.

Sophi
  • 25
Coherent Sheaf
  • 1,333
  • Do you mean 'can he do it in such way that after the 25-th turn the three pucks are in their starting positions'...? – CiaPan Apr 18 '16 at 05:55
  • Well its given exactly as I have typed. i presume they are trying to say at the 25th turn, else the question would be ridiculous as it would be possible to bring it back in the second turn. – Coherent Sheaf Apr 18 '16 at 06:07
  • What is their answer, and why do you think it's unsatisfactory? – JRN Apr 18 '16 at 06:31
  • Well, after reading @CiaPan 's answer to my question, i think that they were trying to intend the same think, but it ended up confusing me. Now when I go back to the answer it makes more sense. – Coherent Sheaf Apr 18 '16 at 07:26

2 Answers2

1

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.

Stella Biderman
  • 31,155
1

Consider the clockwise vs. counter-clockwise orientation of the A-B-C on the plane.

CiaPan
  • 13,049
  • Thanks ! Each valid movement would cause the orientation to change from clockwise to counter-clockwise or vice versa. So we must need even no of moves. – Debashish May 23 '17 at 05:37