That's a very beautiful problem. I've appended a short solution to the end of this e-mail.
----- Original Message ----- From: Eric Angelini Sent: 01/29/14 10:30 AM To: math-fun Subject: [math-fun] Jumping square points
Hello Math-Fun, Four points shape a square of size "s". Every minute a point randomly jumps over another point and lands symmetrically behind it. Is it possible that, at some stage, the 4 points shape a new square having size "t" > "s"? Best, É.
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***** spoiler alert ***** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . No. Operate in the ring Z[i] of Gaussian integers, and let {a,b,c,d} be the Gaussian integers corresponding to the vertices of the square. When you apply the process to obtain {a,2a-b,c,d}, the greatest common divisor (again a Gaussian integer, up to multiplication by the ring units) remains invariant (similar to Euclid's algorithm). Hence, if they form a square at any time, it must be the same size and orientation as the original square. Sincerely, Adam P. Goucher