The Truel - Answer

Your best strategy is to fire into the air! Doing so will make you the most likely to survive of any of the truelists!

Let's consider the alternatives. Obviously, you don't want to fire at Sandy. If you fire at Sandy and kill him, then Pat is certain to kill you with the next shot. So shooting at Sandy is the worst strategy.

If you fire at Pat and kill her, then it's just down to you and Sandy, and Sandy gets the first shot. So Sandy has a 50% chance of killing you right away. If Sandy misses, you have a 30% chance of killing him, after which he has another 50% chance of killing you, if, as is 70% likely, you miss. So Sandy's chance of killing you on the second round is .5 * .7 * .5. Similarly, the chance that Sandy will kill you on the third round is .5 * .7 * .5 * .7 * .5. Continuing, we see that the total chance that Sandy will eventually kill you is the sum of a geometric progression with first term .5 and constant ratio .7 * .5. This sum is .5 / (1 - (.7 * .5)) = . 5 / (1 - .35) = .5 / .65 = .769 (approximately). So your chance of survival (if you kill Pat on the first round) is just .231.

But now suppose you fire into the air on the first round. Sandy, who shoots next, will, in his own self-interest, fire at his strongest opponent, Pat. (If he doesn't, Pat will kill him with the next shot and retain a 70% chance of survival for herself, or 100% if Sandy killed you with the first shot.) If Sandy kills Pat (50% chance), then it's down to you and Sandy again, but you get the first shot. An analysis similar to that in the previous paragraph shows that your chance of survival is now .3 / (1 - (.7 * .5)) = .3 / (1 - .35) = .462.

If Sandy misses Pat (50% chance), then Pat will, logically, choose to kill Sandy rather than you (since Sandy is more dangerous to her), leaving you with one shot at Pat before Pat kills you. Your chance of survival in this case is clearly 30%.

So overall, if you fire into the air, your chance of survival is (.5 * .462) + (.5 * .3) = .381. Not that great, but better than your chances if you fire at Sandy or Pat on the first round. (If you fire at Sandy on the first round, you have a 70% chance of missing, which leads to the .381 probability of survival just calculated, plus a 30% chance of hitting him and being immediately killed by Pat, so your chance of survival is .7 * .381 + .3 * 0 = .267. If you fire at Pat on the first round, your chance of survival, similarly, is .7 * .381 + .3 * .231 = .336.)

Your chances of survival after deloping are also better than those of the other truelists. After you delope, Sandy will survive if he kills Pat (50% chance) and survives the remaining duel with you (we previously calculated your chances in this duel as .462, so Sandy's is 1 - .462 = .538). So Sandy's overall chance of survival is .5 * .538 = .269.

Pat will survive if, after you delope, Sandy misses her (50% chance), and if, after she then kills Pat (100% chance), you miss her on your next shot (70% chance). So Pat's chance of survival is .5 * .7 = .35.

So you, the worst shot, are the most likely to survive the truel!

[Note: some people suggest that, based on reasoning similar to the above, each truelist will fire into the air on every shot, ensuring that all will survive. But the conditions of the problem specify that the truel goes on until there is only one survivor. The above solution assumes that any truelist prefers a strategy that leaves him or her with a better-than-even chance of survival to a strategy that condemns them all to a lifetime of continuing the truel. That ensures that if Pat gets a turn, she will definitely break the impasse by killing Sandy, which means that Sandy's best strategy must be to try to kill Pat.]

Important Disclaimer

No one was actually killed or even injured in the creation of this problem. No animals were harmed either. It's just a puzzle. Don't fight any real truels at home.