Here is the general question:
Start with a square, with a natural number at each corner. Now write, at the
midpoint of each edge, the absolute value |a-b| of the difference of numbers
(a, b) at the endpoints of that edge. Draw a smaller square connecting those
midpoints, and repeat. The game stops if you get to (0; 0; 0; 0). For instance,
(1; 3; 9; 7) ! (2; 6; 2; 6) ! (4; 4; 4; 4) ! (0; 0; 0; 0):
For a longer game, try (44; 24; 13; 7).
Are there games of infinite length, or of arbitrarily long finite length?
Given a bound on the size of the initial numbers, what can one say about
the maximum length of the game?
Now, based on examination of several cases, I have found that all odd (2N+1) sided games have at least one base case that never ends [often it's the case of (2*N) 1s and (1) zero...(1,1,1,1,1...1,1,0)]
I've also found an upper bound on the steps it takes for the game to "recycle," meaning the game returns to its base stage (or at least a rotation of the base stage...for example 1,1,1,1,0 = 1,1,0,1,1 because the sides are connected), and this upper bound is [(2^N)-1]...in fact, all games with this "binary identity" (all 1s and a 0) will take exactly that many steps to circulate...the only exceptions are when N is a power of two...in this case, it takes (2*N)-1 steps to "recycle."
I was wondering if anyone can prove why this is true...since it was only presumed based upon the collected data.