You are in the desert and you have 3 buckets of water containing a,b,c liters respectively (a,b,c - positive integers).
You need an empty bucket for an unspecified purpose. Being in the desert you need the water and cannot just pour it away.
You have to pour the contents of one bucket into another one. But in any pouring, you must double the contents of the bucket which receives the water.
For example the sequence of bucket contents could be:
3 2 1
1 4 1
0 4 2
Now show that no matter what a,b,c are, you can always manage to empty a bucket under this constraint.
You may assume:
(capacity of each bucket)>(a+b)
Well, three hints were not enough for me. I await the solution with anticipation.
I did examine the behavior of two buckets, one odd and one even. What I see that there is only one available move (pouring the larger one into the smaller). The process only stops if the two are equal, but they never are equal because their sum is odd, so there is always a next move. There are only a finite number of divisions possible, so the process must eventually repeat (cycle). That cycle seems to include the initial state, but it is not clear to me that the initial state is repeated in all cases.
And I do not see what this gets me. Perhaps I am missing something about the two bucket case. I wish somebody else was working on this also.