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The three-cornered duel

An intense dispute about the Axiom of Choice has led three of our favorite mathematicians, Archimedes,
Bernoulli and Cauchy into a duel. The contestants are to take turns in alphabetical order.

  • Archimedes fires first, then Bernoulli (if he is still alive), then Cauchy (if he is still alive) and then back to Archimedes (if he is still alive), and so on.
  •  If a shot is successful, it is assumed to be fatal. The shooting proceeds until there is only one mam left standing.

From experience we know that Archimedes is not a very good shot (he lived 2000 years before gunpowder after all!) He is only successful 30% of the time. Bernoulli however is an excellent shot. He hits his target 100% of the time. Cauchy hits his target 70% of the time. After contemplating his predicament for a full night, Archimedes suddenly exclaims “Eureka!” Assuming every character will act in his own self-interest, what did Archimedes realize he should do with his first shot?

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