# Logic Puzzle

Another little puzzle this week.

If you like puzzles, you might like to also read my posts on The Two Egg Problem and the Tribute to Martin Gardner.

### Bizarre gunman and the colored dots

 The Setup …   You, and two equally smart friends, have colored dots painted on your foreheads (don’t ask why, the reason is not part of the puzzle). You are told that there are just two colors in use:  Red  and  Blue . You can see the color of your two friends’ dots, but not your own.
 Suddenly, a masked man runs into the room, holding a gun. He orders each of you to raise your hand if you can see a  Red  dot on anyone else’s forehead. He tells you that if any of you speaks, or communicates any other information about the color of the dots, he will shoot you. He then tells you that you have one minute to determine the color of your own dot, or he will shoot you. If you correctly deduce the color of your own dot, he will set you free. What do you do in order to save your life? Can you work out the color of your own dot?

This is a logic puzzle plain and simple. So, please, no answers like “look in a mirror” or “use iPhone to text picture” …

### Solution

 Yes, you can work out the color of your own dot! Logic is our friend. Let’s walk through this puzzle step-by-step …

First, let’s enumerate all possible combinations of dots. As we can see, there are eight possible permutations.

YouFriend 1Friend 2
Red  Red  Red
Red  Red  Blue
Red  Blue  Red
Red  Blue  Blue
Blue  Red  Red
Blue  Red  Blue
Blue  Blue  Red
Blue  Blue  Blue
 Now let's mark this table with hands that are up. Remember, if anyone see's a  Red  dot they will hold their hand up. In the table below, I've marked everyone who has their hand up with a gold star ★

The column on the far right of the table shows the number of hands that are held up for each case.

YouFriend 1Friend 2# hands
Red   Red   Red  3 hands up
Red   Red   Blue  3 hands up
Red   Blue   Red  3 hands up
Red  Blue   Blue  2 hands up
Blue   Red   Red  3 hands up
Blue   Red  Blue  2 hands up
Blue   Blue   Red 2 hands up
Blue  Blue  Blue 0 hands up

Now we'll group them into collections with the same number of hands up. The first thing to note is that it is impossible for there to be one hand up. Thinking about it, this is obvious. If there is a  Red  dot out there, there have to be two people who will be able to see it!

### Zero hands up

 Blue Blue Blue 0 hands up

This is pretty easy. If there are no hands up, there are no  Red  dots. So you (and both of your friends) can deduce that you all have  Blue  dots. You're safe!

### Two hands up

 Red Blue  ★ Blue  ★ 2 hands up Blue  ★ Red Blue  ★ 2 hands up Blue  ★ Blue  ★ Red 2 hands up

If there are two hands up, it means there is just one  Red  dot out there. Take a look around; if you can see it, you know that you can't have it, so you must have a  Blue  dot. If you can't see a  Red  dot, you know must be the one who is wearing it. You're safe again!

### Three hands up

 Red  ★ Red  ★ Red  ★ 3 hands up Red  ★ Red  ★ Blue  ★ 3 hands up Red  ★ Blue  ★ Red  ★ 3 hands up Blue  ★ Red  ★ Red  ★ 3 hands up

OK, this is where it gets a little tricky. If there are three hands in the air, it can mean one of two things: Either there are two  Red  dots, or three  Red  dots.

Another way to describe the situation is "There are at least two red dots"

Take a look around you. Do you see both a  Red  and a  Blue  dot? (The middle two rows in the table above). Since you know that there needs to be "at least two red dots", then you know you must be wearing a red dot. Safe again!

The problem comes if you see two  Red  dots. What does this mean? It means that you could be wearing either a  Blue  or  Red  dot. What do you do?

Well, this is where you are glad that your two friends are equally as smart as you, and that you are given a minute to deduce your color. If nobody announces their color after a few seconds have passed then everyone must be experiencing the same dilema as you. This can only occur if there are three  Red  Dots. (Anyone who can see a  Blue  would have announced that they have  Red  already at this point).