A logic puzzle

Here's an interesting logic puzzle that I recently came accross. The solution has an interesting relevence to poker.

"Three competing masters of logic wanted to decide who was the best. So they asked the grandmaster to propose a challenge that would determine this. So in a dark room he placed a dot (either red or blue) on each of their foreheads. Then he told them to walk out into the light where they could see the colors on each of the other two people. He said that if they saw at least one red dot, to raise their hand. Then the first to correctly identify the color of the dot on their head wins.

So this is what happend. One of the logic masters saw a red dot on each of the other two people (and thus raised his hand, seeing at least on red dot), and he also saw that both other people were also raising their hands. After a sizable amount of time spent thinking, he correctly deduced (didn't guess) that the dot on his forehead was also red. How did he figure this out with such certainty?"

3 blue dots - no hands go up
2 blue dots, 1 red dot - 2 hands go up
2 red dots, 1 blue dot - 3 hands go up
3 red dots - 3 hands go up

Since the result will be the same whether there are 2 red dots or 3 red dots, logically, there is no point in using a blue dot. The simplest solution is to just use 3 red dots. Amirite?

That was my first guess when this puzzle was presented to me. And while tihs is a good guess, it turns out the logic master has even more conclusive reason to believe he has a red dot than this.

Hint: Give the logic grandmaster more credit for devising a puzzle for his pupils that is more complex than it seems at first.

Also, here's a follow-up puzzle along the same lines. If you find the solution to one, the solution to the other is pretty easy. It's a little easier to follow this one to the correct solution though:

The same three participants above were at the wrong place at the wrong time and ended up being taken prisoner by an Indian tribe. They begged to be let go, and the tribe leader decided to let their freedom hinge on their ability to solve a problem.

The tribe leader had five hats...three red ones and two blue ones. In the dark, he put one hat on each of the three people and left two unused. Then he lined up the prisoners in the light outside such that each person could only see the people ahead. The guy in front could see neither of the other two. The second guy only saw the color of the hat of the guy in front. The third guy saw the colors of the hats of both others.

The tribe leader told them that if any of them gave a correct answer to the color of the hat they were wearing, he'd let all of them free. If any of them gave an incorrect answer they'd all remain prioners. They were not allowed to move or to speak unless they were giving their final answer, but they had as much time to ponder as they wanted.

So that's what they did. They pondered. After a few minutes, one of them correctly reasoned out the color of his hat. Which one figured it out, and how did he figure it out?

the dots were put on on in a dark room...

What's really interesting is how the wrong answer can be applicable to poker as well, or, at least the rationalle for it.

Given that there's a logic grandmaster, there must be a non-trivial answer. That means everyone must have their hand up. Since anyone who sees someone with their hand down knows with 100% certainty that their dot is blue. Therefore everyone must raise their hands, and there can be at most one blue dot.

Now, given that their is, at most, one blue dot, anyone who sees a blue dot knows that their dot must be red. The only way there can be any hesitation is if all the dots are red. (Which also puts everyone on an exactly equal footing for figuring out the puzzle.)

The counter-argument is this: If there were a severe punishment for getting the answer wrong, then if you saw a red dot and a blue dot you would know the answer, but if you were to "slow play" you could convince the person with a blue dot, that his dot were, in reality red. The problem is, that two people have to agree to slowplay (kind of like checking down a small stack on the bubble).

The grandmaster would (in theory) have all the dots red, but would expect the students to realize this, therefore he makes one dot blue hoping that the students will think one step beyond and trick their cohort.

The logic master had to observe the reactions of the other two to deduce the correct answer... relevance to Poker... Position! But... but... but...

It also just occurred to me with regard to the first puzzle is that the third player that the other two would be colluding against would also realize that this could happen, and understanding the predicament this would put him in. So if all three dots were red, all three people would be in the same boat (did I correcly reason this out, or the other two colluding against me?). So it really ends as unsolvable if you add in a punishment for getting it wrong. Or rather than unsolvable, the solution is really that one of them will realize the predicament that putting a red dot on all of them would put them in, and will say red expecting the grandmaster to do something like that to them.

Three red dots get the same reaction as two red dots and one blue (three hands in the air), therefore one cannot know by eveyone's reaction what colour one's dot is. It could be blue or it could be red. But, if there is a correct solution, and each logic master has an equal opportunity to solve it, then each logic master needs to see two red dots to face the same conundrum. Anyone seeing one red dot and one blue dot and three hands in the air would know their dot is red, so that can't be correct. They all have to be red.

Edit: Damn, somone beat me to it. I didn't read before posting.

*Last edited by Warpe on Fri, 23 Jun 2006, 1:10pm; edited 2 times in total*

In the meantime, let's start a thought experiment:

You're at a poker table where the players only have the option of four postflop actions: check, fold, call or push. We also assume that there will be no action on any streets beyond the flop.

You are dealt AA on the Button. Villain, a thinking player, is UTG and raises 6xBB preflop, you reraise to 18xBB, Villain calls, while also correctly surmising what your hole cards are in the process.

Flop is dealt...KKK.

Villain thinks for a bit, then pushes.

Call or fold?

Give reasons for your decision.

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A logic puzzle