Answer to Puzzle #54: Two Bullet Russian Roulette

Before reading the answer can I interest you in a clue?

If you have never thought about the analogous question with 1 bullet, (and why would you?) we can quickly cover that now. One bullet can be in 6 positions as shown.

1 B-----
2 -B----
3 --B---
4 ---B--
5 ----B-
6 -----B

Obviously you would, with a randomised gun, have a 1 in 6 chance of dying. However, if someone has just fired and survived then we have eliminated the first scenario, B-----, it definitely wasn't that. And we are on the second chamber. We have essentially reduced the game to the following (removing the first row, because it wasn't that, and the first column because that chamber has been fired.):

2 B----
3 -B---
4 --B--
5 ---B-
6 ----B

Which is just the same as a standard game of Russian Roulette if there were only 5 chambers in the gun. A 1 in 5 chance of dying is worse than a 1 in 6. So in the case of one bullet it would be in your interest to spin and re-randomise the gun.

Back to Two Bullets

We will apply the same technique as above but first I would like to explain it intuitively. There's more than a hint of the game battleships here. As the second player we can only be hit by the second bullet in the chamber, remember they are consecutive, if the first bullet has already been fired, and we know it has not. By not spinning we have essentially removed an entire bullet from our problem. That is why we should fire the gun as is. Now lets model it.

If we randomise the gun we have a 2 in 6 or 1 in 3 chance of dying. The possible set ups at the start of the game are as follows:

1 BB----
2 -BB---
3 --BB--
4 ---BB-
5 ----BB
6 B----B

As player 1 didn't die we know it was not scenarios 1 and 6, eliminating those rows, and the first column as that chamber is no longer in the game - we're left with this:

2 BB---
3 -BB--
4 --BB-
5 ---BB

Giving us a 1 in 4 chance of dying. By re spinning the gun it would have been 1 in 3. So in the case of 2 consecutive bullets it would not be in your interests to re-spin the chamber.

Assumptions

A fundamental assumption is that your objective is to survive the game. After that we're down to the usual, that the gun is randomised and not influenced by the weight of the bullets. The gun never gets jammed, you never miss. All very reasonable perhaps so implicit that you wouldn't think to mention them.



Neither AI could answer this question. Google was a bit sanctimonious about it too.

If you're curious what Bard made of this puzzle...

Show/Hide Google's Bard AI's Answer

If you're curious what ChatGPT made of this puzzle...

Show/Hide ChatGPT's Answer