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
Other cases by Steve Herman)
Clearly if the numbers of stones, each divided by the GCD of the three, are all odd, my argument holds as well. Such is the case with (1110,110,10), where the GCD is 10.
I wonder if, when the numbers are divided by their GCD, that the results are not all odd, whether in all those cases, a zero pile is possible.

Posted by Charlie
on 20170319 14:33:33 