The Prisoner's Balls
In far-off Puzzlania, a prisoner waits on death row. By custom, the night before a man is to be executed, he plays a game- either of chance or skill, it is the judge's discretion. This game will decide whether the prisoner will indeed die, or have his sentence commuted.
This particular prisoner was presented with a game that was perhaps a little of both. Before him are two large urns. One urn contains fifty black balls, the other fifty white balls. Tomorrow, the executioner will, while blindfolded, draw a ball randomly from one of the two urns. If it's black, it's curtains for the prisoner. If it's white his sentence will be commuted to life.
The prisoner wants very much to live, and is pleased that with the current state of affairs his chances of living are fifty-fifty. He is then presented with an option- he may change the contents of the urns. He can swap white balls for black, move balls from urn to urn, etc. There is a stipulation that when he is done, there must be fifty white and fifty black balls total between the two urns- he can't eat some of the black balls or paint them or anything.
It occurs to the prisoner he might be able to help his situation by moving the balls so that there were twenty-five of each color in each urn, then making sure the white balls were on top. But the executioner might have guessed this, and may shake up the urns. Worse yet, he might deliberately reach to the bottom of the urn he chooses.
Is there another way the prisoner can help himself?