My guess is that the process terminates, although I have not been able to
prove it.  It is akin to solving the heat equation numerically on a
circle.

Basic idea: Over time, the numbers become closer.  Since their sum is
constant and positive, they end up nonnegative.  It suffices to show that
the sum of squares eventually decreases.  It does on one step if x+y+z>0,
but I cannot yet do the rest.

Best wishes,

John

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