Solution to Riddle of the Week

Solution to Riddle of the Week: The Truel

Difficulty level: Moderate

You need to think outside the box a little to solve this one. See the original question here.

SOLUTION

Mr. Black should shoot into the air.

To secure the best odds of survival, Mr. Black wants to miss on his first shot. We assume that Mr. Gray and Mr. White will make the logical choice and shoot at each other, rather than at Mr. Black, to dispatch the greater threat. Shooting into the air and intentionally missing gives Mr. Black a very important advantage: he will get to shoot first when there are only two people left.

Consider if Mr. Black were to try to kill Mr. White or Gray with his first shot. If Mr. Black shoots at Mr. White, who hits every time, and he happens to kill him, then Mr. Gray gets the first shot at Mr. Black, with a 2/3 chance of taking him out. If Mr. Black shoots at Mr. Gray, and he happens to hit him, then Mr. Black is dead, because Mr. White and his 100 percent accuracy will take him out with the next shot.

But suppose, instead, that Mr. Black intentionally misses. Mr. Gray, up next, will take aim at Mr. White, who is the greatest threat because of his accuracy. Gray has a 2/3 chance of hitting. If he hits, then Mr. White is dead and Mr. Black gets the next shot, now in a two-man duel with Mr. Gray. Even if Mr. Gray misses Mr. White, then Mr. White will shoot Mr. Gray dead with the next shot, because Mr. Gray is a bigger threat than Mr. Black. After that, it's down to Mr. Black and Mr. White, but at least Mr. Black gets to take one shot with a 1/3 chance.

So what is the probability of survival?

If Mr. Black misses intentionally, and Mr. Gray misses Mr. White, then there is a 1/6 chance of survival for Mr. Black. Here's the math: 1 (B intentional miss) • 1/3 (odds G misses W) • 1 (W shoots G) • 1/3 (B last chance to take out W) = 1/6.

If Mr. Gray successfully shoots Mr. White, then the duel could possibly continue for much longer. The math is a little more complicated: 1 (B intentional miss) • 2/3 (odds G hits W), then Mr. Black gets a shot with 1/3 to end it there. But even if he misses, he is not guaranteed dead, as Mr. Gray could miss on the next shot, and it could potentially continue back and forth for many shots.

Nigel Coldwell built a simulation for this problem that runs through hundreds of thousands of possibilities, which you can see here. The bottom line: if Mr. Black shoots into the air initially, he has just shy of a 40 percent chance of survival. If he shoots at Mr. White, he has closer to a 31 percent chance of survival. And if he shoots at Mr. Gray, his odds are the worst, with about a 26.5 percent chance of survival.