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lostkiwi
Icarian Member

 Posted: Mon Oct 19, 2009 1:30 am    Post subject: 1 This is a modification of the original three duel puzzle http://en.wikipedia.org/wiki/Three_way_duel_(puzzle) The puzzle is this: Three men are in a pistol duel. Each man will shoot in turn. The three men are identified as A, B, and C: A is a poor shot, and only hits 40% of the time B is a reasonable shot, and hits 60% of the time C is a good shot, and hits 80% of the time. Each man knows each others shooting capabilities, and therefore in order to maximise their survival chances, they will always target the best shooter still standing. They will always aim for a legitimate shot, contenstants are unable to forfeit their shots by firing into the ground. Once someone is hit, they are no longer able to fire any shots. (on account of being dead and all) Each man shoots in turn, and the shooting order is determined before the fight, and that shooting order cycles until there is only one contestant standing. Question 1) Assume that you are one of these contestants, and that you are able to specify the shooting order. Which contestant would you be, and what shooting order would you choose, in order to maximise your chances of survival? Question 2) Alternatively, assume you are one of these contestants, however you do not know the shooting order until the duel starts (it is determined randomly from a third party source). Which contestant would you be in order to maximise your chances of survival?Last edited by lostkiwi on Tue Oct 20, 2009 6:54 am; edited 2 times in total
lostdummy
Daedalian Member

Posted: Mon Oct 19, 2009 3:06 pm    Post subject: 2

 Quote: in order to maximise their survival chances, they will always target the person that is the highest threat during their shot

That part of problem setup can have two issues:
- it is not always true that person who is highest threat is one who should be targeted to maximize survival
- that condition prevent anyone to "chose shooting order" as asked in Q1, since target order would be fixed by requiring to shoot "highest threat" person

Above probably hold even if "threat" is redefined to mean something other than "whoever is best shot".
lostkiwi
Icarian Member

Posted: Mon Oct 19, 2009 11:34 pm    Post subject: 3

lostdummy wrote:
 Quote: in order to maximise their survival chances, they will always target the person that is the highest threat during their shot

I have changed the wording for clarification.

 lostdummy wrote: That part of problem setup can have two issues: - it is not always true that person who is highest threat is one who should be targeted to maximize survival

When there are only 3 people, statistically, each contestant will always have a higher chance to survive if they take out the person with the highest shooting capability.

Consider shooting order A->B->C. A's best chance is to aim at C. If they hit, then they only have a 60% chance of being hit by B, where as if they aim at B and hit, they stand a 80% chance of being hit by C.

This quickly becomes a Geometric Series problem, but needless to say, I am confident that shooting the highest capable shooter will always provide the highest survivlal rates for that individual (ie B will always aim for C, C will always aim for B)

 lostdummy wrote: - that condition prevent anyone to "chose shooting order" as asked in Q1, since target order would be fixed by requiring to shoot "highest threat" person

The shooting order is not the order in which the contestants get shot, it is the order that contestants take their shots. There are 6 different possible combinations, which obviously make a dramatic difference in a contestants survivlal chance. For instance, C has a much better chance of winning if the shooting order is C->A->B compared to A->B->C

I hope that clarify's things.

Last edited by lostkiwi on Tue Oct 20, 2009 12:00 am; edited 1 time in total
ralphmerridew
Daedalian Member

 Posted: Mon Oct 19, 2009 11:57 pm    Post subject: 4 Given that lostdummy joined in June 2004 and that lostkiwi joined in January 2005, what is the probability lostdummy's name is mocking lostkiwi's?
lostkiwi
Icarian Member

Posted: Tue Oct 20, 2009 12:03 am    Post subject: 5

 ralphmerridew wrote: Given that lostdummy joined in June 2004 and that lostkiwi joined in January 2005, what is the probability lostdummy's name is mocking lostkiwi's?

Fair point. Although I joined in 2005, I rarely come to these boards, I am not familiar how to identify a members join date, so I couldn't be sure if it was a new account or not.

It did seem a bit too coincidential, however it's now obvious that it is just that - coincidental. Anyway, I've retracted the comment.
lostdummy
Daedalian Member

 Posted: Tue Oct 20, 2009 9:29 am    Post subject: 6 Well, if I correctly disabled intentional misses from my solution to last "truel" problem, it results in: P[ 0.40 ; 0.60 ; 0.80 ] = (3,3,2) 37.916% 51.084% 11.001% P[ 0.40 ; 0.80 ; 0.60 ] = (2,3,2) 35.744% 27.502% 36.754% P[ 0.60 ; 0.40 ; 0.80 ] = (3,3,1) 43.123% 45.877% 11.001% P[ 0.60 ; 0.80 ; 0.40 ] = (2,1,2) 32.508% 18.335% 49.158% P[ 0.80 ; 0.40 ; 0.60 ] = (3,1,1) 45.837% 43.947% 10.217% P[ 0.80 ; 0.60 ; 0.40 ] = (2,1,1) 45.837% 8.625% 45.539% where P[order of shooting] = (their target order) their survival chances% Also, average survival chances for players regardless of order are A:43.03% B:30.39% C:26.59% From that, it follows: Q1) Shooting order A,B,C and best chance for survival for B (51%) Q2) Best chance for survival regardless of shooting order has A (43%)
lostdummy
Daedalian Member

 Posted: Tue Oct 20, 2009 9:50 am    Post subject: 7 Also, related to "it is always best to target best shot", it may hold true for 3 players, but it is not true for 4 players or more, even when missing is not allowed. Consider this same problem with 4 players: A=80%, B=60%, C=40% and D=20% What would be targeting order (if shooting order is ABCD) that would maximize survival chance? Would it be BAAA (ie, target best shoot)?
lostkiwi
Icarian Member

Posted: Tue Oct 20, 2009 10:56 am    Post subject: 8

 lostdummy wrote: Well, if I correctly disabled intentional misses from my solution to last "truel" problem, it results in: P[ 0.40 ; 0.60 ; 0.80 ] = (3,3,2) 37.916% 51.084% 11.001% P[ 0.40 ; 0.80 ; 0.60 ] = (2,3,2) 35.744% 27.502% 36.754% P[ 0.60 ; 0.40 ; 0.80 ] = (3,3,1) 43.123% 45.877% 11.001% P[ 0.60 ; 0.80 ; 0.40 ] = (2,1,2) 32.508% 18.335% 49.158% P[ 0.80 ; 0.40 ; 0.60 ] = (3,1,1) 45.837% 43.947% 10.217% P[ 0.80 ; 0.60 ; 0.40 ] = (2,1,1) 45.837% 8.625% 45.539% where P[order of shooting] = (their target order) their survival chances% Also, average survival chances for players regardless of order are A:43.03% B:30.39% C:26.59% From that, it follows: Q1) Shooting order A,B,C and best chance for survival for B (51%) Q2) Best chance for survival regardless of shooting order has A (43%)

Nice, that's the exact same answer I had calculated as well. I declare you the winner!
lostkiwi
Icarian Member

Posted: Tue Oct 20, 2009 10:58 am    Post subject: 9

 lostdummy wrote: Also, related to "it is always best to target best shot", it may hold true for 3 players, but it is not true for 4 players or more, even when missing is not allowed. Consider this same problem with 4 players: A=80%, B=60%, C=40% and D=20% What would be targeting order (if shooting order is ABCD) that would maximize survival chance? Would it be BAAA (ie, target best shoot)?

I'm going to give this a crack tomorrow, but I'm going to need a bigger excel spreadsheet...
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