Olivia Morrison
Math 007
Prof. Winkler
04.11.2013
Three Men and a Duel: Strategy, Procedure, and Probability
The three-man duel puzzle requires a great deal of planning and patience to solve. One
cannot solve the puzzle without mapping out different strategic choices and following the chains
of probability through to the end of the line. The intricacies of this puzzle make it tricky, but the
challenge makes the solution worth attaining. Martin Gardner mentions the infamous puzzle in
his 2006 book, The Colossal Book of Short Puzzles and Problems. I have always found this
particular puzzle to be intriguing, frustrating, and, at times, extremely confusing. However, the
solution is straightforward once the person working on the puzzle determines the specific
strategies involved in the duel. An adaptation of the three-man duel puzzle, as well as its
solution, is below.
It is the year 1865. Three men – Mr. Crane, Mr. Fitz, and Mr. Ryan – lay claim to the
same plot of land. Each wants it for himself, but of course only one can have it. In order to settle
the dispute, they agree to a duel. Because there are three men, instead of the traditional two, this
pistol duel will be unique. Before it begins, the three men decide to draw straws to determine
who will shoot first, second, and third. Each will then take his place at one corner of an
equilateral triangle, so that he is the same distance from the other two. Then, the duel will
proceed. The three men will each get one shot per turn, and must take it, though they can shoot
anywhere they like. The turns progress in the order determined by the straws. If the three men
must shoot again, it will be in the same order. The duel will continue until two men are dead and
only one is left standing. Mr. Crane’s shots are always accurate, Mr. Fitz’s shots are accurate
80% of the time, and Mr. Ryan’s shots are accurate 50% of the time. Assuming that each man
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always utilizes the best strategy available to him, which man stands the best chance of survival,
and thus of claiming the plot of land? What is the exact probability that each man will survive?
In answer to the first question posed in this puzzle, Mr. Ryan is most likely to survive and
claim the plot of land, followed by Mr. Crane and, subsequently, Mr. Fitz. To reach this
conclusion, we must first look at each man’s overall strategy. In order to give themselves the
greatest chance of survival, Fitz and Crane will shoot at the man who poses the greatest threat. In
Fitz’s case, Crane poses the greatest threat: his shots are always accurate, so whomever he
chooses to shoot will die. Thus, we know that if Crane is still alive, Fitz will always shoot at him.
Crane’s biggest threat is Fitz, who has 80% accuracy (as compared with Ryan’s 50%). If Fitz is
alive, Crane will always shoot at him. Ryan, however, knows that both of his opponents will
always shoot at each other, provided both are still alive. So until one of them is dead, Ryan’s best
strategy is to shoot neither of the two men. Now that we have established the men’s initial
strategies, we must break up the problem into each of six possible scenarios based on the
drawing of straws. As we will see later on, these six scenarios can also be broken up into three
similar groups of two. The six scenarios (broken into groups) are as follows:
Crane – Fitz – Ryan (C-F-R)
Crane – Ryan – Fitz (C-R-F)
Ryan – Crane – Fitz (R-C-F)
Fitz – Ryan – Crane (F-R-C)
Ryan – Fitz – Crane (R-F-C)
Fitz – Crane – Ryan (F-C-R)
In the first case, C-F-R, Crane shoots first. He will shoot at Fitz, who poses the biggest
threat. Because Crane’s shots are always accurate, he will kill Fitz, and it will be Ryan’s turn to
shoot. Ryan will shoot at Crane, the only remaining opponent. He has a 50% chance of killing
Crane. If he does, he will be the lone survivor and thus will claim the plot of land. If Ryan
misses, Crane will shoot and kill him. Crane will then be the survivor and will claim the land. To
determine each man’s percent chance of survival in this case, let us first look at his percent
chance of death:
2
Crane 50%
Fitz
100%
Ryan 50%
The order in the second scenario is R-C-F. It proceeds almost identically to the first case.
Ryan has the first turn, but because Crane and Fitz are both still alive, he will shoot neither. This
makes it Crane’s turn. He will shoot at Fitz with 100% accuracy. It will be Ryan’s turn to shoot,
as Fitz will be dead. Ryan will shoot at Crane with a 50% chance of killing him. Ryan wins the
duel if he hits. If he misses, Crane will shoot and kill him, thus winning the duel and claiming
the land. Each man has the same percent chance of death as in the previous case:
Crane 50%
Fitz
100%
Ryan 50%
The third case, C-R-F, is similar to the first two. Crane shoots first, killing Fitz. It is then
Ryan’s turn. He will shoot Crane, with a 50% chance of killing him. If Ryan does not, Crane will
shoot and kill him. The same percent chances of death as in the previous two cases apply:
Crane 50%
Fitz
100%
Ryan 50%
Let’s move onto the second group of scenarios by looking at the fourth: F-R-C. Fitz
shoots first. He shoots at Crane, with an 80% chance of killing him. If he does, Ryan’s turn is
next. Ryan will shoot at Fitz – the only other man left standing – with 50% accuracy. If he kills
Fitz, the duel ends. If not, Fitz shoots at Ryan with 80% accuracy, and the two continue shooting
until one kills the other. Going back to Fitz’s initial shot, he only has an 80% chance of killing
Crane. If Fitz misses, it will be Ryan’s turn. Because neither Crane nor Fitz is dead yet, Ryan
will shoot neither. It becomes Crane’s turn. He will shoot at Fitz and kill him. It is now Ryan’s
turn again, and he must shoot Crane this time. Ryan has a 50% chance of killing Crane and
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winning. If not, Crane shoots and kills Ryan, thus winning the duel. The percent chances of death
are as follows:
Crane
Fitz
Ryan
80% + 10% = 𝟗𝟎%
̅%
40% + 20% + 4% + 0.4% + ⋯ = 𝟔𝟒. 𝟒
̅%
32% + 10% + 3.2% + 0.32 + ⋯ = 𝟒𝟓. 𝟓
The fifth case, which plays out similarly to the fourth, is ordered R-F-C. It is Ryan’s turn
to shoot first. Because both of his opponents are still alive, Ryan does not shoot either of them. It
is Fitz’s turn next. He shoots at Crane, with an 80% chance of killing him. If he kills Crane, it is
Ryan’s turn again. Now that Crane is dead, Ryan shoots at Fitz with 50% accuracy. If he kills
Fitz, the duel is over. If not, Fitz shoots back with 80% accuracy and the two men continue
shooting at each other until one is dead. Returning to Fitz’s first shot, there is a 20% chance that
he does not kill Crane. If this is the case, then it is Crane’s turn. He will shoot and kill Fitz.
Ryan’s turn is next, and he shoots at Crane with a 50% chance of killing him. The duel is over if
Ryan kills Crane. If Ryan misses, Crane will shoot and kill him. The percent chances of death in
this scenario are the same as in the fourth:
Crane
Fitz
Ryan
80% + 10% = 𝟗𝟎%
̅%
40% + 20% + 4% + 0.4% + ⋯ = 𝟔𝟒. 𝟒
̅%
32% + 10% + 3.2% + 0.32 + ⋯ = 𝟒𝟓. 𝟓
We now move on to the final scenario, ordered F-C-R. In this scenario, Fitz shoots first, aiming
at Crane. He has an 80% chance of killing Crane. If Fitz does so, then it is Ryan’s turn. He
shoots at Fitz with 50% accuracy. Ryan wins the duel if he kills Fitz. If he does not, Fitz will
shoot at Ryan with 80% accuracy and the two will trade shots until one dies. Let us look back to
the first shot of the duel. If Fitz does not kill Crane, it is Crane’s turn. He will shoot and kill Fitz.
Ryan’s turn is next. He will shoot at Crane with 50% accuracy. If Ryan kills Crane, he wins the
duel. If Ryan does not hit his target, Crane will shoot and kill him, thus winning the duel and the
land for himself. The percent chances of death are identical to the previous two scenarios:
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Crane
Fitz
Ryan
80% + 10% = 𝟗𝟎%
̅%
40% + 20% + 4% + 0.4% + ⋯ = 𝟔𝟒. 𝟒
̅%
32% + 10% + 3.2% + 0.32 + ⋯ = 𝟒𝟓. 𝟓
Now onto our final set of calculations. First, we will calculate the overall percent chance
of death for each participant in the duel. As there are two different sets of percentages, each of
which is equally likely, we simply average the two sets:
Crane
50% + 90%
= 𝟕𝟎%
2
Fitz
100% + 64. 4̅%
̅%
= 𝟖𝟐. 𝟐
2
Ryan
50% + 45. 5̅%
̅%
= 𝟒𝟕. 𝟕
2
We are now able to find the final chances of survival of each duelist by subtracting the percent
chance of death of each man from 100. Each man’s chance of survival is as follows:
Crane
Fitz
Ryan
̅
̅%
𝟑𝟎% 𝟏𝟕. 𝟕% 𝟓𝟐. 𝟐
Thus, we know that Mr. Ryan has the greatest chance of survival, followed by Mr. Crane, and
then Mr. Fitz, who is most likely to die in the duel.
In order to solve the puzzle of the three-man duel, one must first analyze the strategies of
each man, and then break the puzzle down into each of its six scenarios. This makes determining
the probabilities much more manageable, as each man’s choice of who to shoot, if anyone, is
dependent on the shooting order. From there, the strategy of each man governs his choice, and
his accuracy defines the outcome of the duel. If one is able to see the breakdown of the problem,
the puzzle of the three-man duel becomes a matter of step-by-step procedure, and the puzzle
solver can finally determine the chances of each man surviving the duel and thus claiming the
contested plot of land.
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