The cleanest solution has the priest flip one coin and the pirate flip one coin. If their coins match, the priest goes, if not the pirate goes. It's straightforward to check that each outcome occurs exactly 50% of the time.
Another favorite came from past Problem of the Week Champ Kevin Bourrillion who offered the following: Each flips both the coins. Whoever gets more heads wins. If they tie, they flip again. Neither coin needs to be fair.
One other "solution" deserve to be mentioned, for its entertainment value if for no other reason. Have the priest put both coins in one hand and put both hands behind his back. If the pirate guesses which hand holds the coins, she goes.
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Page last updated 23 April 1998.