# Riddle of Logic: Death by Firing Squad

1. Nov 1, 2009

### z3hr

This is a cool riddle of logic that I heard recently. It is supposed to be drawn out so I'll try to describe it as best as I can.

Four prisoners of war are sentenced to death by firing squad. They line them up from north to south, all of them facing north (so that they are in line facing the person in front of them, Directions are unimportant North South etc..) Each of them is wearing a hat, they are told that there are two black hats and two white hats. If ONE of them can say for sure what color hat they are wearing, they are all set free. The person in front, who cannot see the hat colors of any of his comrades, is wearing a white hat. The man behind him is wearing a black hat, and the man behind him is wearing a white hat. The last man is behind a wall wearing a black hat so he cannot see the hat colors of the men in front of him. All of them are aware of their physical positions; for example, the man in the front knows that there are two men behind him, and another behind a wall etc... The only thing they don't know is their own hat color and the hat colors of those behind them. It looks like this:

Black | White Black White

They can only see the hat colors of the people in --> direction from them, with the exception of the last man behind the wall.

They have exactly ten minutes before they are executed. Which one of them would be able to know their hat color and why?

2. Nov 2, 2009

### Caracrist

It is simple:
The second one will know it for sure, his hat is not the same color of the first, cause if it was the same color, the third one would know for sure his hat color. So he just have wait for a moment, and if the third one stays silent, he will know his hat color (different from the first one).

3. Nov 4, 2009

### Wallace

I like it! I'm kicking myself for not working it out though (I gave up and looked at Caracrist's spoiler).

4. Nov 8, 2009

### suganyanathan

The last black knows the colour . i think he looks his shadow in the wall .we all know that shadow will be black but he think his hat is black.

5. Nov 8, 2009

### suganyanathan

The last black knows the colour . i think he looks his shadow in the wall .we all know that shadow will be black but he think his hat is black.

6. Nov 8, 2009

### drizzle

Last edited by a moderator: Apr 24, 2017
7. Nov 9, 2009

### BobG

Actually, that's a key distinction. If the only way to be set free is to guess the color of your own hat and each prisoner gets an opportunity to guess their own hat color, then there's no benefit to delaying a guess unless someone else's prior guess would help you. If only one prisoner is going to be set free no matter what, then there's definitely no reason for anyone to delay.

The first two will reveal no new info to the 3rd that he doesn't already know and the 4th's guess reveals no info at all. The 3rd doesn't benefit from delaying, even if he doesn't know the color of his own hat.

Prisoners 1, 2, and 4 always benefit from letting the 3rd guess first, even if the 3rd is selfish and willing to guess wrong when he doesn't know. In fact, letting a selfish #3 guess first gives everyone a 67% chance of surviving.

If only one prisoner is going to be set free no matter what, then there's definitely no reason for anyone to delay. In that case, there's about a 54% chance that one of the four prisoners will survive.

I went ahead and put this in a spoiler tag, since it actually gives a really big hint as to the solution of the teaser in the op

Last edited by a moderator: Apr 24, 2017