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There are four stones, positioned on the ground at the vertices of a square. At any time, you may pick up a stone and "hop" it over another one so that it lands an equal distance beyond the hopped stone. Can you find a series of hops which will make these stones form the vertices of a larger square? If so, how, if not, why?

To clarify what a "hop" is: if there is a stone a point $p$, you are allowed to move it to a point $p'$ provided there is another stone at the midpoint of $p$ and $p'$.

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    $\begingroup$ interesting. hmm time to use pencil and paper $\endgroup$ – JLee May 30 '15 at 1:10
  • $\begingroup$ can two stones be on the same spot at one time? $\endgroup$ – MisterEman22 May 30 '15 at 1:40
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    $\begingroup$ it really feels like it is impossible, but i hope it is not, so i will learn something cool. $\endgroup$ – JLee May 30 '15 at 1:58
  • $\begingroup$ @MisterEman22 Lets say no, though I think it is impossible for that situation to occur $\endgroup$ – Mike Earnest May 30 '15 at 2:11
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Answer:

It is impossible

Explanation:

Take these two rules:

  1. When you are making a move, the move is also able to be reversed, meaning that after a move, it can be undone and still follow the rule of how you are allowed to move.
  2. Assuming you have a grid where each square is 1x1, and you start with a square that surrounds 1 grid square (let's say points (0,0), (1,0), (0,1), (1,1)) then there is no way using the rule to get any of these points closer than 1 unit from each other. For this to happen, a point would have to be at an integer point while another is at a fraction point. Since they all started at integer points, they cannot be reflected across each other to end up at fractional points (e.g. (.5,1)). Thus it can be concluded that you would also not be able to create a smaller square from a square of any size.
Using these rules, we can determine that for you to be able to create a larger square from a square of size 1x1, then you would also be able to create a square of 1x1 from a larger square, since all the moves can be reversed. As shown in rule 2, you cannot create a 1x1 square from a larger square, so it can be concluded that you are unable to create a larger square from a 1x1 square using this hopping rule.

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  • $\begingroup$ This is exactly what I came here to post. Beat me to it! $\endgroup$ – Ian MacDonald May 30 '15 at 4:55
  • $\begingroup$ @IanMacDonald I spent half an hour trying to make a larger square and pretty much figured out it was impossible, but it took me a while to come up with an explanation for it xD $\endgroup$ – MisterEman22 May 30 '15 at 4:57
  • $\begingroup$ Do not try to make the square larger; that's impossible. Instead, only try to realise the truth: there is no square. $\endgroup$ – Ian MacDonald May 30 '15 at 5:05
  • $\begingroup$ @IanMacDonald 'tis what I did. Then I saw, that it's not the square that enlarges, it is myself. $\endgroup$ – MisterEman22 May 30 '15 at 5:09
  • $\begingroup$ What excludes (0,0) hopping over (1,1)? $\endgroup$ – Magnas May 28 '17 at 21:29

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