Problem of the Week 1223

The Evil Warden

Alice and Bob are prisoners of warden Charlie. Alice will be brought into Charlie's room on Sunday and shown 5 cards, numbered 1, 2, 3, 4, 5, face-up in a row in a random order. Alice can, if she wishes, interchange two cards. She will then leave the room and Charlie will turn all cards face-down in their places.

Bob will then enter the room. Charlie will call out a random target card T. Bob will be allowed to turn over ONE CARD ONLY; and if, and only if, that card is T, the two prisoners are freed.

As always, Alice and Bob can plan a strategy before Sunday, and have no means of communication on Sunday. Note that Charlie's two choices — the initial shuffle and the choice of target — are assumed to be purely random.

What is the prisoners' best strategy? Express the probability of success as n%, where n is the nearest integer to the actual probability of your strategy.

The odds of success seem poor ...

Source: Larry Carter, Mark Rickert, and Stan Wagon, who have been working on many variations of this problem.

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