There are three virtual piles of stones. In one operation one may add to, or remove from, one of
the piles the number of stones equivalent to the quantity in the other two piles combined, leaving the numbers in those
two piles unchanged.
Thus, e.g., (12,3,5) can become (12,20,5) by adding 12+5=17
stones to the second pile, or (12,3,5) can become (4,3,5) by removing 3+5=8 stones from
the first pile.
Assume a starting state (1111,111,11).
Is it possible, by a sequence of
such operations, reach a state where one of the piles is empty?
(In reply to re: Other cases
Charlie: It looks like our posts crossed. I realized as soon as I posted that (1110,110,10) was not the best example.
(1110,108,10) is an example of a triplet where 0 cannot be achieved, but which does not reduce to odd parities.
(1,2,2) is the smallest such triplet. For that matter, the legs of any proper triangle will do, or any triplet that can be achieved through addition from a proper triangle. For instance (1,2,32) can be reached by addition starting with (1,2,2), so it cannot be reduced to zero.