Three logic puzzles, with one thing in common: Gun fight!

**Puzzle #1**

A revolver is loaded with three bullets in three consecutive chambers. [Hence there are three chambers with bullets, and three empty ones.] Once the barrel is spun, each player points the gun at his own head and pulls the trigger. If he doesn’t die, the gun is passed to the other player. There are only two players. The game ends when one player dies. Would you prefer to be the first player to pull the trigger or the second one, or does it make any difference?

**Puzzle #2**

Imagine a gun fight between three people. Ina is the worst shot; she hits the target once every three trials. Mina is better; she hits the target twice every three shots. Dika is a dead shot; she never misses. Each gets one shot (i.e. only one round). As the worst shooter Ina goes first, then Mina, and then Dika. Whom should Ina aim for her one shot?

**Puzzle #3**

Three men are fighting in a truel. Andrew is the worst shot; he misses 2/3 of the time. Bob is better; he misses 1/3 of the time. Connor is the best shot; he always hits. Each of the three men have an infinite number of bullets. Each shot is either a kill or a miss. They have to shoot at each other in order until two of them are dead. To make it more fair they decide to start with Andrew, followed by Bob, and then Connor. We assume that they choose their strategies to maximize their probability of survival. At whom should Andrew aim for his first shot?

I will post the answers next week.

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The second and third puzzles are from Tanya Khovanova’s Math Blog.

Pic Courtesy: Wikipedia

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**Answer to the first puzzle:** see the comments section below.

**Answer to the second puzzle:** See the image below – click to embiggen.

Three scenarios are considered for Ina (who goes first): Ina shoots at Mina, Ina shoots at Dika, and Ina shoots in the air. As explained in the diagram below, the probability that Ina dies is lowest in the third scenario. Hence, she should fire the shot in the air (as Ramanand correctly inferred in the comment section below.)

Note that I’ve made a couple of assumptions: (1) These players are bloodthirsty – so if Ina kills Mina with her first (and only) shot, Dika would still kill Ina even if she poses no threat to her. (2) If it’s Dika’s turn, and both Ina and Mina are alive, then Dika will randomly kill either Ina or Mina. Changing this assumption (to “Dika would always kill Mina”, for instance) doesn’t change the conclusion though.

**Answer to the third puzzle:** I actually don’t have the correct answer for this puzzle. If anyone does have a solution, please do share!

Great puzzles as usual. My shots (pun intended :-))

1. You didn’t mention the number of chambers, so I assumed 6 (double). So I assume the P(getting shot) is 1/2, so I’d rather let the other person shoot first. In case he dies, I don’t have to do any shooting at all (the best case)

2. Ina should shoot in the air, thereby forcing Mina to shoot at Dika (Mina can’t do the same because in the next turn, Dika would prefer to take out the better shot among Ina and Mina).

3. Couldn’t tell how this was very different from #2, but assuming her that Andrew cannot fire in the air, I guess he would want to aim at Connor first. (Tried to work it out and that’s what I seem to get. Not so sure about this.)

Ramanand,

I’ve modified the first puzzle to mention the number of chambers (six).

Your answer to the first puzzle is correct, but your description is not.

For the second one, I didn’t really think about shooting in the air. I have to give it to you for thinking outside the box! This may change the original answer that I have, because one assumption was that the players can only shoot at each other (which was not explicitly mentioned in the puzzle).

The difference between 2nd puzzle and 3rd one is as follows: the game in the second puzzle is played only once (i.e. a single round), but in the third puzzle the shooters keep shooting until two of them are dead.

I will post the answers soon. Thanks for being a loyal reader to my puzzle posts. 🙂

Heh – the puzzles are really interesting.

I’ll try and work out an (a better one) explanation for (1). In (2), initially, I was trying to work out the probabilities for Ina, but then once I looked at the options Ina had, I had to include “not shooting at anyone” – hence the argument. Will try to work it out with the original assumption too.

In (3), I tried listing out the different event paths – there are one or two in which you could hit an infinite loop (potentially) – where A & B are left behind. Didn’t follow through with the exercise.

Hey Vishal, could you post the answers? thanks!

Below are the answers to puzzle #1 and #2. I am still working on #3, which is one of the two reasons behind the delay in posting the answers. (The second reason is: I got really busy at work!)

#1 There are total six chambers. Three consecutive chambers have bullets, and the other three are empty, which means that there are total six different arrangements possible:

1. {Bullet, Bullet, Bullet, Empty, Empty, Empty}

2. {Empty, Bullet, Bullet, Bullet,Empty, Empty}

3. {Empty, Empty, Bullet, Bullet, Bullet,Empty}

4. {Empty, Empty, Empty, Bullet, Bullet, Bullet}

5. {Bullet, Empty, Empty, Empty, Bullet, Bullet}

6. {Bullet, Bullet, Empty, Empty, Empty, Bullet}

Where the first position refers to the chamber that’s facing the muzzle.

Now, let’s consider the scenario in which you go first:

1. You die.

2. You live. (The first chamber is empty, so you survive when you fire the first shot. Then the second player takes his turn and dies because the second chamber has a bullet.)

3. You die.

4. You live.

5. You die.

6. You die.

Hence, if you go first, you would die 4 out of 6 times; your probability of dying is 4/6 = 2/3.

Your probability of dying when you go second is 1 minus your probability of dying when you go first = 1 – 2/3 = 1/3. (Basically, 2 out of 6.)

Hence, the chance of dying is less when you go second. So you should go second.

Ramanand, I have posted the answer to the second puzzle in the blog-post itself. (Couldn’t figure out how to embed an image within a comment.)

Thanks!

Pertaining to 2nd puzzle, is it necessary that Ina can point towards Mina only or Mina can target only Dika. Like in the 3rd case, Can Mina shoot at Ina back?

Deepak,

Not sure if I understand your question correctly. But if Mina survives the first shot (either Ina shoots in the air or Ina shoots at Mina and misses), she has to shoot at Dika. That maximizes her chance or survival – because if she kills Dika then she survives, and if she misses Dika then her chance of survival are 50/50 (Dika may choose to shoot at Mina or Ina). There’s no reason for her to shoot at Ina because whether Mina kills Ina or not, neither situation reduces her chance of survival (Dika will kill Mina with 50/50 probability).

Also, keep in mind that in puzzle #2, every shooter has only one chance to take a shot. There’s no second round.

when would the 3 answer be revealed its been agesssssssssssssssss?????

Angelo,

I actually don’t have the correct answer for the third puzzle. I thought I had figured out the probabilities, but upon further research realized that this Sequential Truel problem is “recognized as both a significant problem is game theory and an important paradigm of political behavior”, and there is an extensive bibliography available in various academic journals. The mathematics behind the solution is quite complicated and given my limited mathematical knowledge, I found it hard to follow. For a brief discussion, see the Wikipedia page and here’s a link to a white paper from The Journal of Game Theory Vol 4, Issue 3, Page 151-174: The Sequential Duel. Also, Truels, Survival of the weakest has some interesting discussion.

Vishal

In the air again.

1. If you shoot at Bob and kill him, you die for sure, so you are better off shooting in the air than at Bob. (Shooting at Bob and missing is same as shooting in the air.) So shooting at Bob is out.

2. If you shoot at Connor and kill him, then Bob takes the next shot at you. If you shoot in the air and Bob kills Connor, you get a shot at Connor before he shoots at you. This is better than shooting at Connor. (If you shoot at Connor and miss, it’s the same as shooting in the air.)