I often use a logic puzzle as an ice-breaker to get workshop participants thinking and working together. Working under time pressure often also brings out some interesting group dynamics.
Problem
Three logicians have all managed to sleight each other (don’t ask) and they meet for a three-way pistol duel at dawn. Alfred has a one-third chance of hitting whomever he aims at. Bertram is a better shot and has a two-thirds chance of hitting his target. Charles is a crack shot and always hits his target. Each will fire a single shot a shot in turn, first Alfred, then Bertram, then Charles, and repeating as necessary. Once hit, an individual is assumed to be unable to continue. What is Alfred’s best strategy for his first shot, to be the last person standing and hence to win the duel?
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Solution
Answers are usually evenly spread on this one, and occasionally, we get one we haven’t heard before! The best gambit, though, is for Alfred to fire into the ground (and we assume that he will not miss this “target”!). This way he has 40% chance of winning the duel!
For those who need it, here is a wordy explanation.
First let’s work out the optimal strategies for the possible two-player duels. Each player should shoot at the other, and the chance of winning depends on who has the first shot.
If Charles is facing Alfred or Bbertram, and has the next shot, he wins! If Charles does not have the next shot, he wins with probability 2/3 or 1/3 if he is facing Alfred or Bertram respectively. Alfred and Bertram’s chance of winning are therefore 1/3 and 2/3 respectively.
If Alfred is facing Bertram and has the next shot, he has a 1/3 chance of winning, a 4/9 chance of losing on the next shot by Bertram and 2/9 chance of two misses, getting us back to the same Alfred vs Bertram (Alfered to shoot) face-off. Alfred’s overall chance of winning against Bertram is therefore 3/7. If it is Bertram’s shot, he has a 6/7 chance of winning.
Now let’s look at Alfred’s options in the three-player duel.
If he shoots at Bertram and hits, he loses on the next shot by Charles. So shooting at Bertram looks like a bad strategy.
If he shoots at Charles and hits, he has only a 1/7 chance of surviving the Alfred vs Bertram (Bertram throws first) contest that follows.
If Alfred misses, deliberately or otherwise: Bertram’s best strategy is to shoot at Charles—shooting at Alfred is suicidal if he hits, while success against Charles gives him a 4/7 chance of winning. And Bertram knows (see below) that not hitting Charles means that Charles will hit him with the very next shot.
If nobody is hit before Charles’s turn: Charles’s best strategy is to shoot at Bertram—Charles then has a 2/3 chance of winning the contest with Alfred. This is better than the 1/3 chance of winning a contest against Bertram if he takes out Alfred.
Let’s put it all together. If Alfred misses: Bertram has a 2/3 chance of hitting Charles, leaving Alfred with a 3/7 chance of then winning. Bertram has 1/3 chance of missing Charles, whereupon Charles will hit Bertram leaving Alfred a single 1/3 chance of hitting Charles to win. So Alfred’s overall chance of winning is 2/7 + 1/9 = 0.397.
This is better than the chance of winning if he hits either potential target, so he should aim to miss!
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