riddle of the day

In a duel, it seems natural to aim at the most dangerous enemy. But if you are the worst shooter, your best defense may be not looking like a threat.

The Duel of the Three Gunslingers

Strategist
Master plays

Riddle statement

Three gunslingers are in a duel. They shoot in turns, always in this order:

  • Abel
  • Bruno
  • Cesar

Their accuracy is different:

  • Abel hits 1 out of every 3 shots.
  • Bruno hits 2 out of every 3 shots.
  • Cesar always hits.

They all want to survive. Abel shoots first.

Whom should he aim at?

Show solution

Solution

Abel should miss on purpose.

It is not enough to say that this makes his rivals attack each other: we can compare his survival probabilities.

When only two gunslingers remain and Abel shoots first:

  • against Bruno, Abel survives with probability $3/7$;
  • against Cesar, who never misses, Abel survives only if he hits his first shot: probability $1/3$.

If Abel fires into the air, Bruno will attack Cesar, his main threat.

  • With probability $2/3$, Bruno kills Cesar. Then Abel faces Bruno with Abel shooting first: survival probability $3/7$.
  • With probability $1/3$, Bruno misses. Cesar kills Bruno, and Abel shoots first against Cesar: survival probability $1/3$.

Therefore:

$ P(\text{survive by missing})=\frac23\cdot\frac37+\frac13\cdot\frac13=\frac{25}{63}. $

If Abel shoots at Cesar, his probability drops to:

$ \frac{59}{189}. $

If he shoots at Bruno, it drops even further:

$ \frac{50}{189}. $

Since:

$ \frac{25}{63}=\frac{75}{189}>\frac{59}{189}>\frac{50}{189}, $

missing on purpose really is his best strategy.

Answer: Abel should fire into the air.