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thenonsequitur
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06-22-2006, 10:19 PM
Post subject: A logic puzzle
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#1 (permalink)
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Full House
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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?"
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Warpe
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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?
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thenonsequitur
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Full House
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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?
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Warpe
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the dots were put on on in a dark room...
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thenonsequitur
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Full House
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Quote:
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Originally Posted by Warpe
the dots were put on on in a dark room...
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Yes, but by "dark room" I just meant a room without light, not a photo-development room. It's just so that they wouldn't be able to see their dot as it was being put on. It could just as well have been they were blindfolded, or otherwise prevented from seeing while the dots were being placed.
And remember, the solution to these puzzles does have some relevence to poker (don't focus on this though, this clue might lead one down the wrong path).
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yorib
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Flush
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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.
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BlueBull
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Join Date: Jun 2006
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The logic master had to observe the reactions of the other two to deduce the correct answer... relevance to Poker... Position! But... but... but...
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thenonsequitur
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Full House
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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.
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Warpe
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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.
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thenonsequitur
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Full House
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Quote:
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Originally Posted by Warpe
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.
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Heh, we arrived at the same conclusion at the same time...You stated it better than me though (and besides I was in the middle of revising my post as you posted, so technically you got it right first).
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Warpe
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Quote:
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Originally Posted by thenonsequitur
Quote:
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Originally Posted by Warpe
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.
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Heh, we arrived at the same conclusion at the same time...You stated it better than me though (and besides I was in the middle of revising my post as you posted, so technically you got it right first).
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I'd be interested to hear your thoughts on this problem's relevance to poker. I have to chew this over a bit.
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Warpe
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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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givememyleg
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WHO YA GONNA CALL?!??
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Quote:
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Originally Posted by Warpe
In the meantime, let's start a thought experiment:
You're at a poker table where the players only have the option of four 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 hold cards are in the process.
Flop is dealt...KKK.
Villain pushes.
Call or fold?
Give reasons for your decision.
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?
Did you mean postflop errr what ?
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ark4748
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Quote:
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Originally Posted by Warpe
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.
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So I'm trying to wrap my head around this... Initially, the puzzle is unsolveable because the reactions will be the same for either two or three red dots. But if we assume that the grandmaster has devised a fair test for the masters, then there has to be three red dots, otherwise one student would have an easy solution to the problem..... Can we assume that the grandmaster is fair? I guess we need to otherwise the puzzle isnt solveable.
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Warpe
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Quote:
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Originally Posted by givememyleg
Quote:
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Originally Posted by Warpe
In the meantime, let's start a thought experiment:
You're at a poker table where the players only have the option of four 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 hold cards are in the process.
Flop is dealt...KKK.
Villain pushes.
Call or fold?
Give reasons for your decision.
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?
Did you mean postflop errr what ?
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My bad. Fixed.
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thenonsequitur
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Full House
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Quote:
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Originally Posted by ark4748
So I'm trying to wrap my head around this... Initially, the puzzle is unsolveable because the reactions will be the same for either two or three red dots. But if we assume that the grandmaster has devised a fair test for the masters, then there has to be three red dots, otherwise one student would have an easy solution to the problem..... Can we assume that the grandmaster is fair? I guess we need to otherwise the puzzle isnt solveable.
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That's sort of the point, and is why I see this as related to poker.
Playing just your cards (i.e. taking into consideration only the color of the dots you see and the hands you see up), this puzzle is not solvable.
Just like in poker, you have to make some assumptions about how others think to arrive at the best solution (i.e. playing the player, why I posted in this forum). So in this puzzle you have to make assumptions about either the grandmaster or the logic masters or both.
Above because we know that the other two people are logic masters, it's safe to assume they've reasoned it out, and by not saying anything, it's either because a) the answer is not immediately obvious to them and they are in the same predicament as the third, or b) the answer is immediately obvious to both of them and they are colluding against the third. Add in the assumption that the logic grandmaster devised a fair game, and that eliminates the collusive possibility, leaving you with just one right answer.
Now if you replace the grandmaster with a 50/50 random number generator (or random color generator if you will), and replace the logic masters with idiots, then maybe they're not saying anything because they haven't even figured out how to read their cards correctly, much less that they might be practicing second-level thinking. And the grandmaster being a random generator means you can't make any assumptions about the fairness of the game. So once again, it is unsolvable.
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thenonsequitur
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Full House
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Quote:
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Originally Posted by Warpe
let's start a thought experiment
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For this, are we to assume that the stacks are both very deep?
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Warpe
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Quote:
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Originally Posted by thenonsequitur
Quote:
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Originally Posted by Warpe
let's start a thought experiment
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For this, are we to assume that the stacks are both very deep?
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That sounds like fun. 200BB should do it.
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givememyleg
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WHO YA GONNA CALL?!??
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Quote:
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Originally Posted by Warpe
Villain calls, while also correctly surmising what your hole cards are in the process.
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Does he say "so you got AA huh?" and then calls? So he tells you that he has a read, and it happens to be correct? You could be re-raising a much wider range on the button.
This is pretty tricky. You certainly are wither wa/wb (dar!)..
How is villian? To be honest, if he was a better player I'd be more inclined to call here moreso than if he was a worse player.
Also, this experiment is kind of silly. Just the fact that the only options are "check, call, push, fold" sort of skews this. Is this a built in rule at this game? If that was the case, I would be more inclined to think we were ahead. Villian only has 2 optins here, check or push.
If this was a normal game, with normal betting and villian open pushed here how would you think your reaction should change?
I don't see how Hero can fold here.... would yoooooou?
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Warpe
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Quote:
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Originally Posted by thenonsequitur
Quote:
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Originally Posted by Warpe
let's start a thought experiment
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For this, are we to assume that the stacks are both very deep?
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Quote:
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Originally Posted by Warpe
That sounds like fun. 200BB should do it. 
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Rethink: Let's scratch stack sizes. Assume they're even but otherwise unimportant. I just want us to focus on the decsion making process.
Quote:
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Originally Posted by givememyleg
Also, this experiment is kind of silly. Just the fact that the only options are "check, call, push, fold" sort of skews this. Is this a built in rule at this game? If that was the case, I would be more inclined to think we were ahead. Villian only has 2 optins here, check or push.
If this was a normal game, with normal betting and villian open pushed here how would you think your reaction should change?
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You're right. We don't need the weird rules. They just kind of came to me while I was coming up with the problem.
It's the decision that's important here. What is going on in Villain's mind? If he has AK, how does he get the chips in? If he doesn't, how does he get you to fold? He's decided to push. Why?
Think about the red dots.
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Warpe
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Quote:
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Originally Posted by givememyleg
I don't see how Hero can fold here.... would yoooooou?
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On the face of it, quads are normally/often slowplayed, right? So it follows that if Villain has a K, pushing is the last thing we should expect him to do. But if our thinking villain knows that that is our expectation...
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krimson
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Quote:
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Originally Posted by yorib
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).
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This counter-argument is completely irrelevant. The statement of the problem makes it clear that you are rewarded for giving a correct answer, and makes no reference to any rewards if the others give an incorrect answer. Anyone who sees a blue dot has an incentive to claim the win immediately, and zero incentive to "slow-play".
In particular the solution to the puzzle has nothing to do with assuming that the grand-master has set a fair puzzle, and would be equally valid if the grand-master was replaced with a random colour generator that gives a cookie to whoever figures his dot out first.
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thenonsequitur
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Full House
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Well all that discussion was assuming a corollary puzzle where there is a reward for giving a correct answer, but an even bigger reward for getting another person to give an incorrect answer. If there is only a reward for being the first to get the right answer, then we agree that this counter-argument is not relevent. But I find the correlary puzzle much more interesting.
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thenonsequitur
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Full House
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Quote:
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Originally Posted by Warpe
Quote:
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Originally Posted by givememyleg
I don't see how Hero can fold here.... would yoooooou?
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On the face of it, quads are normally/often slowplayed, right? So it follows that if Villain has a K, pushing is the last thing we should expect him to do. But if our thinking villain knows that that is our expectation...
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But you can take this game of rock-paper-scissors as far as you want.
Villian knows that quads are often slowplayed, and that if he pushes, we might think he doesn't have quads for that reason. But if he knows we know this, he might push without quads trying to get us to fold, hoping that we will think he is pushing with quads (which we think because quads are normally slowplayed, and he is a thinking opponent that might push to disguise this).
But then he knows we know the above, so he might be pushing with quads to get us to call based on the above.
So what it comes down is whether his push means quads or not depends on how many levels deep we believe he is thinking.
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givememyleg
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WHO YA GONNA CALL?!??
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lol I was just thinking the same thing... we can think he will do that because we think he won't, but then we can think he is thinking we are thinking that he will shove because it is what we think he would do to make us think... if ya get me!
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pgil
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Full House
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the guy at the front says, i have a red hat. here goes:
there must be at least 1 red hat on one of the two front people, otherwise the last guy can simply say he has a red hat as there are only 2 blues. the middle guy knows this, and so if he sees that the guy in front of him has a blue hat, then his must be red (unless the guy in back has already spoken). therefore, the guy in front must be wearing a red hat.
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"If you can't say f*ck, you can't say f*ck the government" - Lenny Bruce
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IowaSkinsFan
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Quote:
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Originally Posted by pgil
the guy at the front says, i have a red hat. here goes:
there must be at least 1 red hat on one of the two front people, otherwise the last guy can simply say he has a red hat as there are only 2 blues. the middle guy knows this, and so if he sees that the guy in front of him has a blue hat, then his must be red (unless the guy in back has already spoken). therefore, the guy in front must be wearing a red hat.
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wow, that was the final piece for me, I realized that if the front guy had a blue hat that either the far back guy or the middle would know what hat he was wearing, but I didn't realize the fact that if they both didn't figure it out that the front guy would know his hat was red, good puzzle.
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IowaSkinsFan
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Here's another puzzle:
A man has 12 metal balls in front of him. 11 are the same weight however one is heavier or lighter (you are not sure which)than the other balls. All you have with you is a balance to weigh the balls against each other, the balance does not give the weight of the balls. In 3 tries, how can you figure out which ball does not weigh the same as the others. Note there are multiple answers and you have to answer many possible happening when coming to the answer.
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IowaSkinsFan
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Here's another puzzle:
A man has 12 metal balls in front of him. 11 are the same weight however one is heavier or lighter (you are not sure which)than the other balls. All you have with you is a balance to weigh the balls against each other, the balance does not give the weight of the balls. In 3 tries, how can you figure out which ball does not weigh the same as the others. Note there are multiple answers and you have to answer many possible happening when coming to the answer.
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Ragnar4
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Full House
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I read the logic puzzle.. haven't looked at the answer.
3 blue dots, no hands go up
2 blue dots 1 red dot, 2 hands go up
1 blue dot, 2 red dots, all 3 hands go up
finally 3 red dots all 3 hands go up.
Since all 3 hands went up, we are left with the final two choices. BUT since all 3 paused, and had to think about the answer for a considerable time, one of the dots could not have been blue. Because the moment they saw a blue dot, and all 3 hands up, they had to immediately assume that their dot must be red. It's the PAUSE, and the considerable THINKING that brings about the logical conclusion.
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The older I get, the more I start wondering; Just what in the hell is going on here?
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jyms
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If his dot was blue. The other two would know the colour of their own dots by seeing the hands go up on the other monk, obviously if one of the other monks hands went up and his dot was blue the others would know the colur of thier dot. They had to be all red.
edit: Sorry to repeat the answers, I wanted to answer it first before reading the thread. Now onto the next one. The hat one I figured out a while ago when somebody gave me that one at work.
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biondino
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The aces puzzle: If we assume the guy has AK 1/2 the time and rags the other 1/2 (let's ignore the split pot):
The villain knows without a shadow of a doubt that if he checks, you will bet. So, if he has rags, he is folding, if he has the king, he is calling. Therefore if he check the flop, he will win 50% of the time.
If he pushes the flop, however, he still wins 50% of the time you call; he also wins every time you fold. Therefore, you have to call because then it is a flip; if you fold, you are 50% of the time folding to a worse hand.
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oteezy03
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Join Date: Apr 2008
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Quote:
Warpe
PostPosted: Fri, 23 Jun 2006, 10:28am
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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I like the last answer, but it also seems obvious that he wouldn't call your preflop raise w/ AK if he knows for sure that you have Aces. Then again, what WOULD he call it with...
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tom
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Straight
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Quote:
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Originally Posted by IowaSkinsFan
Here's another puzzle:
A man has 12 metal balls in front of him. 11 are the same weight however one is heavier or lighter (you are not sure which)than the other balls. All you have with you is a balance to weigh the balls against each other, the balance does not give the weight of the balls. In 3 tries, how can you figure out which ball does not weigh the same as the others. Note there are multiple answers and you have to answer many possible happening when coming to the answer.
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1) split the balls into 2 groups of 6 and weigh them against each other
2) whichever group of 6 is heaver, split into 2 groups of 3 and weigh them against each other
3) take the group of 3 that is the heaviest and weigh just 2 of the balls. If they are the same weight, the ball you didn't weigh is the heaviest. If they are not the same weight, whichever ball is heavier of the 2 is your answer.
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IowaSkinsFan
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Straight Flush
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Quote:
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Originally Posted by tom
Quote:
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Originally Posted by IowaSkinsFan
Here's another puzzle:
A man has 12 metal balls in front of him. 11 are the same weight however one is heavier or lighter (you are not sure which)than the other balls. All you have with you is a balance to weigh the balls against each other, the balance does not give the weight of the balls. In 3 tries, how can you figure out which ball does not weigh the same as the others. Note there are multiple answers and you have to answer many possible happening when coming to the answer.
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1) split the balls into 2 groups of 6 and weigh them against each other
2) whichever group of 6 is heaver, split into 2 groups of 3 and weigh them against each other
3) take the group of 3 that is the heaviest and weigh just 2 of the balls. If they are the same weight, the ball you didn't weigh is the heaviest. If they are not the same weight, whichever ball is heavier of the 2 is your answer.
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tom
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Straight
Join Date: May 2005
Location: ATX
Posts: 158
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it only took me the better part of 2 years and a lot of intense concentration to figure it out
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BankItDrew
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4-of-a-Kind
Join Date: Oct 2005
Location: Losing Prop Bets
Posts: 2,789
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Quote:
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Originally Posted by IowaSkinsFan
Quote:
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Originally Posted by tom
Quote:
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Originally Posted by IowaSkinsFan
Here's another puzzle:
A man has 12 metal balls in front of him. 11 are the same weight however one is heavier or lighter (you are not sure which)than the other balls. All you have with you is a balance to weigh the balls against each other, the balance does not give the weight of the balls. In 3 tries, how can you figure out which ball does not weigh the same as the others. Note there are multiple answers and you have to answer many possible happening when coming to the answer.
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1) split the balls into 2 groups of 6 and weigh them against each other
2) whichever group of 6 is heaver, split into 2 groups of 3 and weigh them against each other
3) take the group of 3 that is the heaviest and weigh just 2 of the balls. If they are the same weight, the ball you didn't weigh is the heaviest. If they are not the same weight, whichever ball is heavier of the 2 is your answer.
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This doesn't work if the 12th ball that is unlike the others, is lighter. Because the lighter group of six were thrown out of the equation at the beginning.
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pantherhound
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Flush
Join Date: Mar 2005
Location: Love me for a season
Posts: 492
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yah thats what i thought
new answer needed
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JKDS
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Full House
Join Date: Feb 2008
Location: Chandler, AZ
Posts: 1,024
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rest them all on the balance and let them roll off at the same time onto the same surface. Do this three times. The one that bounces higher or lower than the rest weighs different since it would have a greater or weaker gravitational force and thus a greater or weaker normal force once hitting the ground.
ya i cheated but ship it.
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Quote:
Originally Posted by OngBonga
But no, jkds is lolvillager and anyone who wants to string him up is sighbad.
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BankItDrew
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4-of-a-Kind
Join Date: Oct 2005
Location: Losing Prop Bets
Posts: 2,789
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Quote:
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Originally Posted by JKDS
rest them all on the balance and let them roll off at the same time onto the same surface. Do this three times. The one that bounces higher or lower than the rest weighs different since it would have a greater or weaker gravitational force and thus a greater or weaker normal force once hitting the ground.
ya i cheated but ship it.
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Thinking outside the box are we? I like it.
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TLR
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Moderator
Join Date: Aug 2004
Posts: 5,007
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Quote:
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Originally Posted by tom
Quote:
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Originally Posted by IowaSkinsFan
Here's another puzzle:
A man has 12 metal balls in front of him. 11 are the same weight however one is heavier or lighter (you are not sure which)than the other balls. All you have with you is a balance to weigh the balls against each other, the balance does not give the weight of the balls. In 3 tries, how can you figure out which ball does not weigh the same as the others. Note there are multiple answers and you have to answer many possible happening when coming to the answer.
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1) split the balls into 2 groups of 6 and weigh them against each other
2) whichever group of 6 is heaver, split into 2 groups of 3 and weigh them against each other
3) take the group of 3 that is the heaviest and weigh just 2 of the balls. If they are the same weight, the ball you didn't weigh is the heaviest. If they are not the same weight, whichever ball is heavier of the 2 is your answer.
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You assumed the different ball is heavier, we do not know if is heavier or not.
My guess:
divide the balls into 3 groups of 4
1234 5678 9TJQ
weigh group1 vs group2:
If Group1=Group2 we know the odd ball is in group 3
weigh 9 vs T, if they are equal the odd ball is either J or Q
weigh 9 vs J, if equal odd ball is Q, otherwise it J
if 9 and T are not equal it is either 9 or T, weigh 9 vs J, if equal
its T otherwise its 9
If group1 <> group2 things become a little harder, lets assume group1 is heavier then group2 (if it is the other way the solution is the same with reverse groups)
weigh 125 vs 369
if they are equal it is either
4 heavy
7 light
8 light
weigh 7 vs 8 if equal it is 4 otherwise it is the one which is lighter
if 125 is lighter then 369 it means it is either 3 or 5 since those are the
balls that were shifted when the direction was changed,
weigh 3 vs Q if equal it is 5 otherwise it is 3
if 125 is heavier then 369 we are left with 3 options
1 is heavy
2 is heavy
6 is light
so we weigh 1 vs 2
if equal its 6
if 1 is heavier then it is 1
if 2 is heavier then it is 2
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Juked07
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3-of-a-Kind
Join Date: Apr 2008
Posts: 83
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In regards to the first puzzle:
Let's suppose the situation is (with person 1 being the one who deduces the correct answer)
1: B (hand up)
2: R (hand up)
3: R (hand up)
Then person 2 and 3 would see 1B, 1R.
From each of their perspectives, then, they would know that if their own hat were blue, the other would see 2 blue hats, and would not have his hand up. The situation would have to be:
1. B (hand up)
2. B (hand up)
3. R (hand down)
or
1. B (hand up)
2. R (hand down)
3. B (hand up)
Therefore, person 2 and person 3 would each know they were wearing a red hat. They would also know that the other red hat wearer also knows he is wearing a red hat, and would have no reason to slowplay.
Person 1, then, can know that if he is wearing a blue hat, person 2 or 3 would soon declare their own hat color as red.
Because this declaration is absent, he knows that his own hat must be red. The situation must be:
1. R (hand up)
2. R (hand up)
3. R (hand up)
Did I make mistakes in my thinking?
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