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Old 2021-08-02, 12:30   #2
Dr Sardonicus
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Feb 2017

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This reminds me of a problem in The 2nd Scientific American Book of Mathematical Puzzles & Diversions, a collection of Martin Garner's "Mathematical Games" columns from Scientific American, of which a PDF may be found here. It is the ninth of the first set of "Nine Problems."

Different versions may be found online, generally without any attribution or reference to earlier versions. Here is Gardner's version:


SMITH, Brown and Jones agree to fight a pistol duel under the following unusual conditions. After drawing lots to determine who fires first, second and third, they take their places at the corners of an equilateral triangle. It is agreed that they will fire single shots in turn and continue in the same cyclic order until two of them are dead. At each turn the man who is firing may aim wherever he pleases. All three duelists know that Smith always hits his target, Brown is 80 per cent accurate and Jones is 50 per cent accurate.

Assuming that all three adopt the best strategy, and that no one is killed by a wild shot not intended for him, who has the best chance to survive? A more difficult question: What are the exact survival probabilities of the three men?
Gardner also traces the problem further back:
The problem, in variant forms, appears in several puzzle books. The earliest reference known to me is Hubert Phillip's Question Time, 1938, Problem 223. A different version of the problem can be found in Clark Kinnaird's Encyclopedia of Puzzles and Pastimes, 1946, but the answer is incorrect. Correct probability figures for Kinnaird's version are given in The American Mathematical Monthly, December 1948, page 640.
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