We have 16 tanks in a 4x4 grid.

+--+--+--+--+
|1 |2 |3 |4 |
+--+--+--+--+
|5 |6 |7 |8 |
+--+--+--+--+
|9 |10|11|12|
+--+--+--+--+
|13|14|15|16|
+--+--+--+--+

All the tanks are empty, except the first one, which is filled. We'd like to drain it. If you drain the water outside of the tank system, certain baby animals will be very sad, but draining the water from one tank to another is alright.

There is a door between each pair of adjacent tanks (but no diagonal connections). If you open the door between two tanks, the water levels will instantaneously equalize. What's the most you can drain out of the first tank?

Initially, while this was still in the queue, I thought the answer would be 15/16, if all the tanks were allowed to equalize.

It took a while for me to realize that, just to take one simple method, half the water could be let out into tank 2 and then that tank sealed. Then 14/15 of the remaining half could be drained into the remaining 14 tanks, leaving 1/30 of the original water in tank 1, meaning that 29/30 had been drained.

Similarly 2/3 of the water could go into tanks 2 and 3, etc. Or some more complicated scheme might be devised to allow more water to drain out of tank 1.

Posted by Charlie on 2013-10-14 17:29:43