Three Dots

Scott Hardie | January 20, 2002
The "Three Dots" puzzle is solved! The "Three Dots" puzzle is solved!

If you have no idea what I'm talking about, let me explain. I was searching for puzzles back in early 2000 to use in the world game, a fantasy rpg that I used to run. I discovered an excellent puzzle with three characters in a fantasy setting, so it seemed great. The only problem was that I couldn't solve it as hard as I tried. I racked my brain for hours, using as much logic as I'd ever learned in class, and I just couldn't find the answer. So I put it into the game, expecting one of the players to solve it. None of them did, and boy were they mad at me when they found out that I didn't know either. The web page that gave me the problem in the first place didn't provide the answer, even if you asked the webmaster nicely, so I was out of luck. I've carried that puzzle with me for two years now, asking everybody I know who seems to me to be capable of solving it, from my mom to Jesse Newsom to Krishna Sham, and none of them have ever solved it. Let me print the puzzle here, in its entirety. It's taken from this web page.

You are travelling in a deep, dark forest when, suddenly, on a dark and stormy night, you are captured by the kind-but-not-so-bright forest people. They keep looking at you and saying to each other, "It be a smart person!"

You are marched to their village and seated in the square. Before you are blindfolded, you notice two other lucky souls seated in chairs facing you. Then you are told that the forest people king is about to die. He previously sent messengers throughout the land seeking the 3 smartest people. You are one, and two others have also been found. He now gives you all a task to see which one is the wisest.

He tells you, "I have seated you in an equilateral triangle so that each of you faces the other two. While you are blindfolded I will paint a dot on each of your foreheads. Each dot will be red or green so that there can be any combination of red and green dots, for example, 1 red and 2 greens, or all red, etc. When I remove the blindfolds each of you must raise your hand if you see _any_ green dots, i.e. 1 or 2 dots. As soon as you have figured out what color your own dot is, tell me how you knew."

So he paints a green dot on all three foreheads. When the blindfolds are removed, all three hands go up. After a long pause, the wisest person, (that's you!) say, "Your highness, I have a green dot. The reason I know this is that..."

How did you know?

So anyway, after the game session last night, I was talking with Kelly and Jon and Bill about riddles that have made their way into my games over the years, and of course I came to this one. I explained it to them, and Bill knew the answer! He'd read another puzzle that was very much like this one, and was able to remember how it worked.

The solution: Pretend that you are A, and the others are B and C. Imagine for a moment that you have a red dot, and B and C still have green dots. B sees one green and one red, and so does C, so all three of you raise your hands. B is then able to figure that he has green, since if he had red, C would not be raising his hand. The same goes for C being able to figure it out. They'd be able to figure this out rather quickly. The fact that a long pause has passed and no one has gotten it means that you cannot have red. Thus, you must have green.

I cannot express in mere words how relieved I am that I finally know this solution after two years of frustration. I need bodily noises. This is how you spell relief. Bill: Thank you, thank you, thank you.

So, would the world game players have been able to solve it? No, not as it was presented to them. In their version of the puzzle, the king took off the blindfolds, they all raised their hands, and the king waited for the answer. That was it. The long pause before "you" realize that you have green is an essential element of the solution. Brad, Kelly, Ryan: I'm sorry.

Scott Hardie | January 20, 2002
P.S. I know why I couldn't solve it with logic. I wasn't considering the fact that the other players were unable to answer. I was only computing various combinations of red and green, and didn't take the "long pause" into consideration.