recreational mathematics - There are 314 coins in 21 open boxes. In each move you can take 1 coin from each of any two boxes and put them into a third box and...

There are 314 coins in 21 open boxes. In each move you can take 1 coin from each of any two boxes and put them into a third box and in the final move you take all the coins from one box. What is the maximum number of coins you can get?

The answer is 314 and I am struggling to prove it is possible to get 314 coins at last for every possible distribution of coins among the 21 boxes.

Answer

$$(10,1,0,0)\to(9,0,2,0)\to(8,2,1,0)\to(7,1,1,2)\to (9,0,1,1)\to(11,0,0,0)$$