Question 57
Jesse and Tjeerd are playing a game. Jesse has $n \ge 2$ stones. There are two boxes: in the black box there is space for half of the stones (rounded down) and in the white box there is space for half of the stones (rounded up). Jesse and Tjeerd alternate turns, with Jesse as first player. In his turn, Jesse takes one new stone, writes a positive real number on the stone and puts it in one of the boxes which is not full yet. Tjeerd can see all the numbers on the stones in each of the boxes and is allowed to move one stone of his choice to the other box, if that other box is not full yet, but he is also allowed to choose to do nothing. The game stops when both boxes are full. If the total value of the stones in the black box is greater than the total value of the stones in the white box, Jesse wins; otherwise Tjeerd wins. Determine for each $n \ge 2$ who can always win this game (and give a winning strategy).