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In the Wild West, you challenge two cowboys, Hunter Jack and Sharp Shooter Leo, to a death match. They, being better shooters than you, readily accept your challenge. But they don't want to waste bullets and thus lay down certain rules that are accepted by you as well. Here are the rules:

1) Everyone shoots in a given order till only one is left.
2) Everyone shoots only once when his turn arrives.
3) If any one of you is injured, the other two will finish him off together.
4) The worst shooter (which is you) gets to shoot first and the best one shoots last.

Now, what tactics will you use if you know that you hit every third shot, Jack hits every second shot and Leo hits every shot?

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    $\begingroup$ You need to be more clear regarding the problem. if you hit, then is it an instant death? What counts as "injured" then? What happens if you miss? What does "finish off" mean? $\endgroup$ – votbear Jan 18 '18 at 10:31
  • $\begingroup$ Also, do Jack and Leo know that "you hit every third shot of yours, Jack hits every second shot and Leo hits every shot" as well ? And do they know you know ? $\endgroup$ – Keelhaul Jan 18 '18 at 10:42
  • $\begingroup$ Anyone can shoot at anyone or they have to shoot one that come next in the order ?? Be more specific please. $\endgroup$ – Aarsh Jan 18 '18 at 10:45
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    $\begingroup$ @narasimha could you rewrite your sentence then ? Something like "Your shots have a probability of 1/3 to hit", because as it is phrased, we can assume you miss twice, then hit once, then miss twice again, then hit once etc... $\endgroup$ – Keelhaul Jan 18 '18 at 12:20
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    $\begingroup$ Is this riddle original? I've seen this in many other places, and attribution is required on puzzling. $\endgroup$ – Quintec Jan 18 '18 at 14:13
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I've made some guesses on the rules as it was not really clear. I will suppose injuries is impossible: a hit = a kill. I will also suppose that when someone is killed we don't restart with the worst shooter. (ex: If I kill Jack then it is Leo's turn).
That being clear my strategy is:

Not shooting at anybody (or shooting in the air)!!

Explanation:

(1) I kill Leo first
That would make no sense to try killing jack, as if I kill him, Leo will kill me instantly and it is game over. If I kill Leo, I will have a 1/4 chance to win against Jack.
(2) Jack kills Leo first
If Jack decide to shoot, it won't be on me (because killing me means certain death for him). If he succeed in killing Leo, I have a good 1/2 chance to win the fight against him.
(3) Leo kills Jack first
If finally Jack decided not to shoot, Leo could shoot at his turn, as I am a bad shooter he will prefer to shoot Jack and kills him, from there I have 1/3 to kill him instantly or die next turn.
(2) OR (3) > (1) So whoever shoot first between Jack and Leo it gives me better odd than being the first to kill ==> I don't shoot or I shoot in the air.
To conclude: Sadly I guess every other guy will do the same (don't shoot) and the fight will end in a draw (out of bullet if we fire on the air) or dying because of age.

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  • $\begingroup$ This is the traditional answer, but I'm not so sure it's the best for this question which specifically uses the wording "every third shot" rather than probabilities $\endgroup$ – theonlygusti Jan 18 '18 at 14:55
  • $\begingroup$ @theonlygusti comments suggest that was just badly worded. $\endgroup$ – Brent Hackers Jan 18 '18 at 14:57
  • $\begingroup$ Nah, the other players do shoot. Your strategy maximises your chances of winning, but for Leo, he wants to fire first, because he's the better shot; that being the case, Jack knows that if he doesn't kill Leo, Leo will kill him, so he wants to try to get Leo first. $\endgroup$ – BlueHairedMeerkat Jan 18 '18 at 15:00
  • $\begingroup$ "If Jack decide to shoot, it won't be on me" Should be "If Jack decides to shoot, it won't be at me" $\endgroup$ – Acccumulation Jan 18 '18 at 20:24
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Despite the lack of precisions, I'll give it a try with some assumptions

Before the encounter, shoot twice to miss. Then you know your next shot will be a hit.
You're the first to shoot, and you shoot at Leo, but only to injure him! This way, you and Jack shoot at him a first time to finish him off, and you both miss. You both shoot a second time and while you miss again, Jack hits and kills him.
Now it's Jack's turn, and since his last shot was a hit, he misses you.
Your turn again. You missed your last two shots at Leo, so your next shot is a hit on Jack. Congratulation, you won!

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  • $\begingroup$ Your every third shoot can hit so how can you injured him ?? $\endgroup$ – Aarsh Jan 18 '18 at 11:00
  • $\begingroup$ @Aarsh hit is not necessarily instant death. If I aim at his hand, he won't die right away. Since it is not precisey stated what a "hit" is, I suppose it's hitting whatever you aim at, and I chose not too aim at Leo's heart or head. $\endgroup$ – Keelhaul Jan 18 '18 at 11:10
  • $\begingroup$ But it can consider as a hit ! If you want to injured him then you have to hit him. and you can hit at every third shot ! $\endgroup$ – Aarsh Jan 18 '18 at 11:12
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    $\begingroup$ @Aarsh Sorry for assuming that, just be aware that adding multiple exclamation marks is often considered "shouting" ;). And, again, I used a loophole to make his "first" shot a hit by firing twice (and miss) before the encounter. It's literally the first two sentences I wrote. However, as OP clarified his meaning of "once every three shots" as juste a probability of 1/3, I'm afraid my trick is moot now. $\endgroup$ – Keelhaul Jan 18 '18 at 12:31
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    $\begingroup$ @psmears Let's just hope they're not logicians! $\endgroup$ – Keelhaul Jan 18 '18 at 14:31
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My guess (and it really is a guess)

Your first shot is fired into the air. The next shooter, 'Jack', will then (because you're clearly very bad at this) attempt to shoot the best shooter 'Leo' who has a 50/50 chance of being hit (and therefore killed due to the finishing off rule). If Leo survives than Jack will be killed on the next shot. So by the time we reach your next turn, you have a 50/50 chance of being left with either Jack or Leo. Either way, you then take a shot at the remaining cowboy giving you either, a 1/3 chance of killing them on the next turn and a 75% chance of them killing you before you get another 1/3 shot. I don't think there's actually any way for you to improve your odds of winning beyond that.

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(1). Your first shot is missed. Then Jack also miss the shot. And Leo took shot at jack and that shot injured him. (2). Now You and Leo both take shot. But you missed and Leo killed him off. (3). Now you hit every third shot so now you shoot at Leo and kill him.

So this way you can win.

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    $\begingroup$ This assumes the actions of Leo, which could as well shoot you in the first place or directly kill Jack. $\endgroup$ – Keelhaul Jan 18 '18 at 11:13
  • $\begingroup$ That's why ask him that anyone can shoot at anyone. $\endgroup$ – Aarsh Jan 18 '18 at 11:14
  • $\begingroup$ Which still doesn't mean you can assume how the other cowboys will act. $\endgroup$ – Keelhaul Jan 18 '18 at 12:18
  • $\begingroup$ @Keelhaul. I think this is basically a game theory question in that you need to play the game in such a way as to make the other players more attractive targets than you which this question seems to do correctly albeit without a full explanation. The logic in your own answer appears sound however if the shots that you fire before the challenge are taken into account. $\endgroup$ – Brent Hackers Jan 18 '18 at 12:20
  • $\begingroup$ @Keelhaul you can; surely each person wants to make sure they themselves survive? Normally in puzzles like this other people are assumed to be PLBs $\endgroup$ – theonlygusti Jan 18 '18 at 14:56

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