The warden is at it again. The entire prison population will be set free if the inmates can achieve a simple result. They must stack three cards, an Ace, King and Queen, on a table, in that order, with the Ace on top.

Alone and in a closed room, the warden begins the process by placing the three cards face up on a desk in some or all of three bins, appropriately marked Left, Middle, and Right. If they all occupy a single bin, only the top card is visible. If they occupy only two of the bins, then only two cards are visible, and it is impossible to tell which of the two visible cards conceals the third. Of course if they are all in separate bins, all three are visible. How the cards are initially laid out is totally up to the warden, but for the purposes of this puzzle we may assume the placement is random.

At 8:00am on the fateful day, a prisoner chosen at random enters the room and moves one of the visible cards from its bin to a different (possibly empty) bin. That is, from the top of one stack to the top of another (possibly empty) stack. The prisoner then leaves the room and is led back to his cell. He does not communicate in any way with the other inmates. Then at 9:00am, and at one-hour intervals thereafter, a second, third, etc., randomly chosen prisoner enters the room and again moves a single card from the top of one pile to the top of another. A prison guard inspects the cards after each move and informs the warden if at any time the three cards become stacked in a single bin in the desired order: Ace, King, Queen, with Ace on top.

The cards must be correctly stacked by the time the 5:00pm prisoner leaves the room, or before, for the prisoners to be released. They are all executed otherwise.

The prisoners are not permitted to work out a strategy beforehand. In fact, the prisoners do not know what they are expected to do until they enter the room. We could say that prisoners enter the room, read the above description of the problem, make their move, and then leave.

What are the prisoners' chances?

We can assume they are smart. Smart enough to be Brain Denizens.

The warden is at it again. The entire prison population will be set free if the inmates can achieve a simple result. They must stack three cards, an Ace, King and Queen, on a table, in that order, with the Ace on top.

Alone and in a closed room, the warden begins the process by placing the three cards face up on a desk in some or all of three bins, appropriately marked Left, Middle, and Right. If they all occupy a single bin, only the top card is visible. If they occupy only two of the bins, then only two cards are visible, and it is impossible to tell which of the two visible cards conceals the third. Of course if they are all in separate bins, all three are visible. How the cards are initially laid out is totally up to the warden, but for the purposes of this puzzle we may assume the placement is random.

At 8:00am on the fateful day, a prisoner chosen at random enters the room and moves one of the visible cards from its bin to a different (possibly empty) bin. That is, from the top of one stack to the top of another (possibly empty) stack. The prisoner then leaves the room and is led back to his cell. He does not communicate in any way with the other inmates. Then at 9:00am, and at one-hour intervals thereafter, a second, third, etc., randomly chosen prisoner enters the room and again moves a single card from the top of one pile to the top of another. A prison guard inspects the cards after each move and informs the warden if at any time the three cards become stacked in a single bin in the desired order: Ace, King, Queen, with Ace on top.

The cards must be correctly stacked by the time the 5:00pm prisoner leaves the room, or before, for the prisoners to be released. They are all executed otherwise.

The prisoners are not permitted to work out a strategy beforehand. In fact, the prisoners do not know what they are expected to do until they enter the room. We could say that prisoners enter the room, read the above description of the problem, make their move, and then leave.

What are the prisoners' chances?

We can assume they are smart. Smart enough to be Brain Denizens.

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