A problem of transfers and invariants disguised as buckets and marbles: the move looks local, but the decisive idea is global.
The Last Bucket Full
Riddle statement
There are three buckets with marbles.
In each move, you may choose one bucket and double its number of marbles, taking exactly that many marbles from the other two buckets combined.
For example, if the buckets contain 5, 4 and 3 marbles, one way to double the bucket with 3 marbles is to take 2 from the bucket with 5 and 1 from the bucket with 4. You would get 3, 3 and 6.
Now compare these two cases:
- Case A: 12, 7 and 3 marbles.
- Case B: 10, 8 and 5 marbles.
In each case, can you make two buckets empty?