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Logic Puzzles


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I've been in a logic puzzle mood lately. Anyone out there have any good ones, and/or want to take a crack at these?

 

1. You're blindfolded and incapable of telling the sides of a coin apart by touch for some reason. In front of you is a square, with a quarter in each corner. Each turn, the square is rotated an unknown amount. You can then flip, but not move, whichever quarter(s) you want, if any. Once you say you're done flipping, you win if all four quarters are the same (heads or tails). Otherwise, a new turn starts. How many turns do you need to be allowed in order to guarantee victory?

 

2. You're blindfolded and still incapable of telling the sides of a coin apart by touch. In front of you are placed 99 quarters head-up, and an unknown amount of quarters tails-up. How do you separate all of the quarters into two piles with the same number of quarters heads-up? (This time you can move and flip the quarters freely. The two piles need not have an equal number of coins, just an equal number of coins showing heads.)

 

3. You are the most eligible bachelor in the kingdom, and as such the King has invited you to his castle so that you may choose one of his three sons to marry. The eldest prince is honest and always tells the truth. The youngest prince is dishonest and always lies. The middle prince is mischievous and tells the truth sometimes and lies the rest of the time. If the method matters to your thinking, the moment he realizes you are asking a yes/no question, he tunes out anything you say, and picks an answer at random.

 

As you will be forever married to one of the princes, you want to marry the eldest (truth-teller) or the youngest (liar) because at least you know where you stand with them.

 

The problem is that you cannot tell which brother is which just by their appearance, and the King will only grant you ONE yes or no question which you may only address to ONE of the brothers. What yes or no question can you ask which will ensure you do not marry the middle brother?

 

Clarification: Your question MUST have a clear yes/no answer no matter which brother you are addressing. Assume the rules of this comic are in place - if you try to create a paradox or a question that one or two of the brothers can't answer, you will be stabbed by a guard for causing discomfort to a prince.

 

4. Once again 100 prisoners are locked up with a quirky chance of being released. This time the guard tells them he will bring them one at a time into a room with 100 boxes. Each box contains one of their names (they all have unique names and each of their names is in some box). Each prisoner may look inside up to 50 of the boxes to try to find his name. If every single prisoner ends up finding his name, they will all be released. The prisoners are allowed to discuss before any of them are sent into the room, and may take a sheet of notes in with them if the strategy involves any memorization, but after the guessing starts, there is no more collusion. The room is reset to its original state after each prisoner's turn. What should their strategy be to maximize their chance of success?

 

Hint:

Although each prisoner will only have a 50/50 chance of getting their name, your goal is to link the success or failure of as many prisoners together as possible. Consider: Assume there were 2 prisoners each opening 1 of 2 boxes. If they each pick a box at random, there's no correlation between whether the first one succeeds and whether or not the second one succeeds, and the odds of winning are 25%. If they each pick box 1, each one's success is perfectly negatively correlated with the other, and their odds of winning are zero. If they agree that one will take box 1 and the other box 2, each one's success is perfectly positively correlated with the other, and their odds of winning are 50%. How can you make 100 prisoners' chances as strongly correlated as possible?

 

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On the first one, I don't see any way you can GUARANTEE victory. The problem is you don't find out if you've won until you've decided to not take any more turns. Each turn, there's no way to know what quarter is where (even if you knew the original setting) because the random amount the square is turned. Essentially, you don't know the state of any coins until you've stopped, and that's going to mean you've got no way to use any strategy.

 

I must be missing something....

Edited by Graeme
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On the first one, I don't see any way you can GUARANTEE victory. The problem is you don't find out if you've won until you've decided to not take any more turns. Each turn, there's no way to know what quarter is where (even if you knew the original setting) because the random amount the square is turned. Essentially, you don't know the state of any coins until you've stopped, and that's going to mean you've got no way to use any strategy.

 

I must be missing something....

 

At the end of each turn, if all four quarters are the same, the game ends and you win. If not, only then does a new turn begin.

 

Edit: For example, there's no reason not to declare your first turn over immediately. If the board starts all heads or all tails, you win and the game is over. Otherwise, the square is rotated, and turn two begins.

Edited by BlueSoxSWJ
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I used to go for a walk at lunchtime. There was a 1km straight stretch with a bridge at the end. Over the weeks I calculated the middle point by pacing, then the quarter points, then the eighth points and so on. Then it occurred to me that on my way to the bridge I would walk half the distance to the bridge, then when I'd reached the half way point I would be walking half the remaining distance from the half way point, then half the remaining distance from that point, then half - well you get the idea. So if during my walk I was always walking half the remaining distance (which I was) - how did I ever reach the bridge?

 

Btw turns out I wasn't the first to think of this problem :)

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I used to go for a walk at lunchtime. There was a 1km straight stretch with a bridge at the end. Over the weeks I calculated the middle point by pacing, then the quarter points, then the eighth points and so on. Then it occurred to me that on my way to the bridge I would walk half the distance to the bridge, then when I'd reached the half way point I would be walking half the remaining distance from the half way point, then half the remaining distance from that point, then half - well you get the idea. So if during my walk I was always walking half the remaining distance (which I was) - how did I ever reach the bridge?

 

Btw turns out I wasn't the first to think of this problem Posted Image

 

Indeed, this is ...

 

Xeno's Paradox. The trick is that each time you cover half the remaining distance, it only takes half as long. So covering each half is an infinite geometric series, which has a finite sum (as long as r < 1).

 

 

Seems like there isn't as much interest as I thought there would be ... but I'll toss out some hints if people are interested but just want a tip to begin each puzzle.

 

1:

Because we don't care about heads vs. tails, and because we can't tell two possibilities apart if they are rotationally symmetric, there are only 4 possible board states:

1. H H 2. H H 3. H T 4. H H

_. H T _. T T _. T H _. H H

 

Obviously, only the opening turn can start in state 4, or the last turn would've won. Similarly, there are only 3 possible moves, except for the first turn (doing nothing): Flip 1 quarter, Flip 2 quarters that are next to each other, Flip 2 quarters that are diagonally apart from each other.

 

So the game has a surprisingly small set of possibilities ...

 

 

2.

If you split the coins into two piles and one has h # of heads, how many heads are there in the other pile? How can you take advantage of this?

 

 

3.

It isn't possible to be sure whether or not you're addressing the middle brother, no matter what you ask. Thus, you should ask your question to the ugliest brother, and your solution should never involve picking the brother to whom you address your question.

 

 

4.

Although the prisoners aren't allowed to communicate once they establish their strategy, each prisoner does gain new information during his turn - specifically, each time he opens a box, he either wins, or sees another prisoner's name. You need to take advantage of that fact in order to maximize your odds of winning.

 

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Indeed, this is ...

 

That's the mathematical answer but this approach to infinity seems not to be sufficient for the philosophers Posted Image

 

 

 

http://plato.stanfor...radox-zeno/#Dic

Applying the Mathematical Continuum to Physical Space and Time: Following a lead given by Russell (1929, 182–198), a number of philosophers—most notably Grünbaum (1967)—took up the task of showing how modern mathematics could solve all of Zeno's paradoxes; their work has thoroughly influenced our discussion of the arguments. What they realized was that a purely mathematical solution was not sufficient: the paradoxes not only question abstract mathematics, but also the nature of physical reality. So what they sought was an argument not only that Zeno posed no threat to the mathematics of infinity but also that that mathematics correctly describes objects, time and space. The idea that a mathematical law—say Newton's law of universal gravity—may or may not correctly describe things is familiar, but some aspects of the mathematics of infinity—the nature of the continuum, definition of infinite sums and so on—seem so basic that it may be hard to see at first that they too apply contingently. But surely they do: nothing guarantees a priori that space has the structure of the continuum, or even that parts of space add up according to Cauchy's definition. (Salmon offers a nice example to help make the point: since alcohol dissolves in water, if you mix the two you end up with less than the sum of their volumes, showing that even ordinary addition is not applicable to every kind of system.) Our belief that the mathematical theory of infinity describes space and time is justified to the extent that the laws of physics assume that it does, and to the extent that those laws are themselves confirmed by experience. While it is true that almost all physical theories assume that space and time do indeed have the structure of the continuum, it is also the case that quantum theories of gravity likely imply that they do not. While no one really knows where this research will ultimately lead, it is quite possible that space and time will turn out, at the most fundamental level, to be quite unlike the mathematical continuum that we have assumed here.

 

http://www.mathpages.../s3-07/3-07.htm

... these arguments merely confirm Zeno's position that the physical world is not scale-invariant or infinitely divisible (noting that Planck’s constant h represents an absolute scale). Thus, we haven't debunked Zeno, we've merely conceded his point. Of course, this point is not, in itself, paradoxical. It simply indicates that at some level the physical world must be regarded as consisting of finite indivisible entities. We arrive at Zeno's paradox only when these arguments against infinite divisibility are combined with the complementary set of arguments (The Arrow and The Stadium) which show that a world consisting of finite indivisible entities is also logically impossible, thereby presenting us with the conclusion that physical reality can be neither continuous nor discontinuous.

 

 

Edited by Zombie
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