## Solution to Puzzle #43: The Triangular Duel

This is the best response to a puzzle in recent times – thanks everyone!

This was a slightly difficult question, the first correct answer for the first part came from Girish Tutakne again. Only one fully correct answer came from Anirudh Baddepudi, a 14 year old from Singapore. Well done Anirudh – truly brilliant!

I liked Anirudh’s answer and his articulation a lot and hence taking the liberty of reproducing his answer completely.

As the question said that each man is a logical thinker, I thought about what each man would do if it were his turn to shoot. As Brown and Smith each know that the other is their greatest threat, their motive will be to shoot each other if it is their turn, and if the other is still alive. As for Jones, he will not be targeted by Brown or Smith until one of them are dead, so if he has a shot and the other two are alive, he will shoot to the floor (not kill anyone). Therefore, as Jones will be at the playing field with 2 people left regardless, he has the best chance of survival. This is also proved in solving the second question:

I decided to systematically look at the chance of survival for all three people.

First, I looked at Brown:

In his duel with Smith, if he does not get the first shot he dies. So, he has a 1/2 chance of having the first shot. Then, he has a 4/5 chance of killing Smith, and if he does not, then he dies in Smith’s next shot. Thus, he has a (4/5 * 1/2) = 4/10 = 2/5 chance of surviving his duel with Smith. Then, if he kills Smith, Jones will have the next shot at him. However, this is where I got a little confused, as they can keep shooting each other infinitely with none of them dying. So, I looked at this as a never ending geometric series with terms (4/10, 4/100, 4/1000, 4/10000………..) I found these terms as I looked at each round of both of them shooting/missing as such:

1) 1/2 * 4/5 = 4/10

2) 1/2 * 1/5 * 1/2 * 4/5 = 4/100

3) 1/2 * 1/5 * 1/2 * 1/5 * 1/2 * 4/5 = 4/1000………

So then, the formula for the sum of an infinite geometric series is S= a/(1-r), with S being the sum, ‘a’ being the first term, and ‘r’ being the ratio between each of the terms. So if this formula is applied:

S=a/(1-r)

S= (4/10)/(1-(1/10))

S= (4/10) / (9/10)

S= 4/9

Therefore, Brown has a 4/9 chance of surviving his duel with Jones once Smith is dead. So, his overall probability of surviving is:

2/5 * 4/9 = 8/45

Next, I looked at Smith:

He has a 1/2 chance of getting the first shot against Brown, and if he gets the first shot, he kills Brown. However, he also has a 1/2 chance of not getting the first shot. In this scenario, he must hope that Brown misses his first shot, so he can kill him. This probability of surviving is (1/2 * 1/5) = 1/10. Therefore, his chance of surviving against Brown is 1/2 + 1/10 = 3/5.

If he kills Brown, Jones will have the next shot. Then, for Smith to survive, Jones must miss this shot, so the overall probability of Smith surviving is:

3/5 * 1/2 = 3/10.

Now this makes the rest fairly simple. We know that the sum of the chance of survival for each man must be 1, so Jones’ chance of survival is naturally:

1 – (3/10+8/45)

= 1 – (215/450)

= 235/450

= 47/90 = Jones’ probability of surviving.

So, as stated earlier, Jones has the greatest chance of survival out of the three men, not what one would assume when they first read the problem!

Hope you all enjoyed the puzzle!

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