Let's do another one.

[mcfbridge]

Since the last one got so much interest, here's a slightly different one.


You are playing an old-fashioned shell game. You are also playing the only honest one in the world.

There are three shells with a ball under a single one.


You have absolutely no idea which shell the ball is under.

To win the game, you must guess which shell the ball is under.


You pick a shell.

The man running the game lifts another shell and shows you that the ball was not under that one.

He now gives you a chance to switch from the shell you chose, to the other remaining shell.


Are you better off:

A: Keeping the shell you have?
B: Switching to the other shell?
C: It makes no difference?

Please explain how you came up with your answer.


If there is no consensus in a few hours, I'll post the answer.

 

[Colleen Thomas]

This sounds suspiciously like an odds & percentages game.

When you chose the first time, you had a 1 in 3 shot of being correct, or 33.3% chance. If he removes a shell, you now have a 1 in 2 of 50% chance of being right if you reselect. However, if you hold your originall selection you are actually seeing 2 of three or 66.6% of the original wager's odds. So staying with your original selection would seem wisest.

However, previous odds do not play into the current selection, anymore than the previous three rolls on a craps table have any bearing on the current roll. Ten 7's in a row does not increase or decrease the chance that the next roll will be a seven.

So I would say it makes no difference. You have a 50% chance of making the correct selection if you stay pat or change.

-Colly

 

[Tanuki]

I say stay with your original pick, because if it were wrong, he would probably have lifted your shell to show you. He showed you the other shell to try to make you change your guess.

 

[Guest]

Hats puzzle

I thought I'd add one of my own. Some of you might have seen this, if you have, please don't state the answer, yet.

Four men have been buried all the way to the neck, only their heads stick out. They cannot turn their heads, so they can see only in front of them. A wall has been placed between A and B, so that A cannot see the other 3 (B, C, D), and viceversa. All of them know in which position the others have been buried. So, for example, B knows that C and D can see him, even though he can't see them.

A hat has been placed on top of each man's head. All of them know that there are two black hats and two white hats, but no one is told the colour of the hat he's wearing.

http://www.psyche-erotica.co.uk/images/hats.jpg
(Excuse the poor drawing, I was in a rush.)

They will all be saved if at least one of them can safely say what colour is the hat he's wearing. Otherwise they'll all be decapitated.

Which one of them saved the day? And, most importantly, how?

-----------------------------------------------------------------------------------

I solved this one after about ten minutes, but no one else in my family could (none of us are very bright. )

Lou

P.S. I'm with Colly on my answer to Mike's shell riddle. I pick, C. It makes no difference.

 

[smutpen]

I'd say your odds are twice as good if you switch.

Call the shells A, B, and C.

You've chosen A.

But he gave you the information that B is empty.

B and C together have a two thirds chance of containing the ball, whether he showed you the empty or not. He did you the favor of giving you that two thirds chance by revealing which is the right one to pick out of those two.

 

[Colleen Thomas]

Re: Hats puzzle

I thought I'd add one of my own. Some of you might have seen this, if you have, please don't state the answer, yet.

Four men have been buried all the way to the neck, only their heads stick out. They cannot turn their heads, so they can see only in front of them. A wall has been placed between A and B, so that A cannot see the other 3 (B, C, D), and viceversa. All of them know in which position the others have been buried. So, for example, B knows that C and D can see him, even though he can't see them.

A hat has been placed on top of each man's head. All of them know that there are two black hats and two white hats, but no one is told the colour of the hat he's wearing.

http://www.psyche-erotica.co.uk/images/hats.jpg
(Excuse the poor drawing, I was in a rush.)

They will all be saved if at least one of them can safely say what colour is the hat he's wearing. Otherwise they'll all be decapitated.

Which one of them saved the day? And, most importantly, how?

-----------------------------------------------------------------------------------

I solved this one after about ten minutes, but no one else in my family could (none of us are very bright. )

Lou

P.S. I'm with Colly on my answer to Mike's shell riddle. I pick, C. It makes no difference.

Ick. Y'all are gonna kill me here, working brain teasers on pain meds. K.

C has to answer eventually and he has to answer he is wearing a black hat.

How does he know? Well. A can't see anyone so he can't answer. B can't see anyone, so he can't answer either. D can see B & C, but he can't answer because they are wearing different color hats. C can see B has a white hat on. But if D dosen't answer after a while, then it should become obvious to C that D can't answer. the only reason D would be unable to anser is if he was seeing 2 different colored hats. Since B has on a white hat, C's hat has to be black.

Unless this is like my first answe to Mike's question and I am reading too much into it?

-Colly

 

[Guest]

Re: Re: Hats puzzle

Ick. Y'all are gonna kill me here, working brain teasers on pain meds. K.

C has to answer eventually and he has to answer he is wearing a black hat.

How does he know? Well. A can't see anyone so he can't answer. B can't see anyone, so he can't answer either. D can see B & C, but he can't answer because they are wearing different color hats. C can see B has a white hat on. But if D dosen't answer after a while, then it should become obvious to C that D can't answer. the only reason D would be unable to anser is if he was seeing 2 different colored hats. Since B has on a white hat, C's hat has to be black.

Unless this is like my first answe to Mike's question and I am reading too much into it?

-Colly

I knew you'd get it.

Yep, that answer is spot on. Well done, Colly!

Lou

 

[Colleen Thomas]

this one is a little tricky, but not too difficult.

You have just walked into a dark two-storey house. There are three light switches on the first floor 1, 2 and 3. Only one of the switches works, and it only turns on the light on the second floor. You cannot see the second floor light from the first floor and there are no windows in the building. You can only make one trip up the stairs, and you can only flip one switch at a time. How do you determine which of the three switches turns on the light on the second floor ?


-Colly

 

[mcfbridge]

Colly, still working on yours.

Here's a classic.

You are at a fork in the road. Down one fork is a city where all residents always tell the truth. Down the other fork is a city where all the residents always lie.

There is a man standing at the fork. You don't know which city he came from. What can you ask him that will insure that his answer will tell you which city lies down which road?

 

[Guest]

this one is a little tricky, but not too difficult.

You have just walked into a dark two-storey house. There are three light switches on the first floor 1, 2 and 3. Only one of the switches works, and it only turns on the light on the second floor. You cannot see the second floor light from the first floor and there are no windows in the building. You can only make one trip up the stairs, and you can only flip one switch at a time. How do you determine which of the three switches turns on the light on the second floor ?


-Colly

My guess is, it has got something to do with feeling the light bulb.

Ok, here goes. You flip switch 1 and wait for 30 seconds. Then turn switch 1 off, then turn on switch 2 and leave that switched on.

You then go upstairs. If the light is on, you know it was switch 2. If the light is off, you know it isn't switch 2. This is where feeling the lightbulb comes in. If it is cold, you know that switch 1 didn't turn it on, and, therefore, as pure deduction, you know that switch 3 will work. If the lightbulb is still warm, you know it was switch 1 that turned it on.

Is that right?

Lou

 

[Colleen Thomas]

My guess is, it has got something to do with feeling the light bulb.

Ok, here goes. You flip switch 1 and wait for 30 seconds. Then turn switch 1 off, then turn on switch 2 and leave that switched on.

You then go upstairs. If the light is on, you know it was switch 2. If the light is off, you know it isn't switch 2. This is where feeling the lightbulb comes in. If it is cold, you know that switch 1 didn't turn it on, and, therefore, as pure deduction, you know that switch 3 will work. If the lightbulb is still warm, you know it was switch 1 that turned it on.

Is that right?

Lou

Bravo Lou lou

Spot on

*HUGS*

-Colly

 

[Carl East]

Colly, still working on yours.

Here's a classic.

You are at a fork in the road. Down one fork is a city where all residents always tell the truth. Down the other fork is a city where all the residents always lie.

There is a man standing at the fork. You don't know which city he came from. What can you ask him that will insure that his answer will tell you which city lies down which road?

Are you from the city that tells the truth? If he says no, ask him which fork to take and then take the opposite one.

Carl

 

[Guest]

Bravo Lou lou

Spot on

*HUGS*

-Colly

Woohoo! I got one right!

Lou

 

[mcfbridge]

Are you from the city that tells the truth? If he says no, ask him which fork to take and then take the opposite one.

Carl

Sorry, guess I didn't spell it out. Only one question may be asked.

Besides, he'll never say no to that question. If he's from the city that tells the truth, he will tell the truth and say YES.

If he's from the city that lies, he will lie and say YES.

 

[Carl East]

Sorry, guess I didn't spell it out. Only one question may be asked.

Besides, he'll never say no to that question. If he's from the city that tells the truth, he will tell the truth and say YES.

If he's from the city that lies, he will lie and say YES.

And I thought my answer was so clever. Actually, I'm crap at riddles. This one was in the film Labyrinth, but I can't remember the solution. lol

Carl

 

[Guest]

Since the last one got so much interest, here's a slightly different one.


You are playing an old-fashioned shell game. You are also playing the only honest one in the world.

There are three shells with a ball under a single one.


You have absolutely no idea which shell the ball is under.

To win the game, you must guess which shell the ball is under.


You pick a shell.

The man running the game lifts another shell and shows you that the ball was not under that one.

He now gives you a chance to switch from the shell you chose, to the other remaining shell.


Are you better off:

A: Keeping the shell you have?
B: Switching to the other shell?
C: It makes no difference?

Please explain how you came up with your answer.


If there is no consensus in a few hours, I'll post the answer.

It's a 50/50 situation at the best of times.

But the guy doesn't know I know his trick. So, obviously, I expose the trick.

I win ten bucks. Which is all right.

 

[mcfbridge]

I'll post the answer to the shell game problem. Colleen, you thought yourself right away from it.

You had 1/3 chance you were right when you chose. In this case that does stay with you. The fact that the man shows you one that was empty, does not change the fact one 2 times out of 3 your original choice was wrong. Therefore, if you change, you have a 2/3 chance of being right.

To demonstrate.

Shells are 1, 2, and 3.

You choose shell 3.

If the ball is under number 1, dealer shows you number 2, you switch and win.

If the ball is under number 2, dealer shows you number 1, you switch and win.

If the ball is under number 3, dealer show you either shell, you switch and lose.

So, 2 out 3 times you switch you will win.

 

[smutpen]

I'll post the answer to the shell game problem. Colleen, you thought yourself right away from it.

You had 1/3 chance you were right when you chose. In this case that does stay with you. The fact that the man shows you one that was empty, does not change the fact one 2 times out of 3 your original choice was wrong. Therefore, if you change, you have a 2/3 chance of being right.

To demonstrate.

Shells are 1, 2, and 3.

You choose shell 3.

If the ball is under number 1, dealer shows you number 2, you switch and win.

If the ball is under number 2, dealer shows you number 1, you switch and win.

If the ball is under number 3, dealer show you either shell, you switch and lose.

So, 2 out 3 times you switch you will win.

This answer looks remarkably similar to mine.

The thing that fools people is probably that we're used to trying to boil down probability questions to a coin toss.

Since this is a three-part problem, the revelation of the empty shell does not complete the equivalent of one coin toss, because you still haven't seen the result.

But we're used to the binary way of thinking about it, so our instinct is that any answer resets the odds.

 

[Lauren Hynde]

Colly, still working on yours.

Here's a classic.

You are at a fork in the road. Down one fork is a city where all residents always tell the truth. Down the other fork is a city where all the residents always lie.

There is a man standing at the fork. You don't know which city he came from. What can you ask him that will insure that his answer will tell you which city lies down which road?

Ask him which way to his city. Whatever he answers will be the city where all residents tell the truth.

 

[mcfbridge]

Congrats Lauren, absolutely correct.

 

[MagicFingers]

This Week's Puzzler: Walking and Riding to Town-Is It Faster?

This puzzler was sent in by Doug Burnside. Of course, I did have to obfuscate and de-clarify.

Doug writes, "John and Fred have a dilemma. They need to get back to town as quickly as possible, but they only have one bicycle between them. And, of course, you can travel faster on the bike than you can by walking.

John says, "We'll leave at the same time. You start walking, and I'll ride the bike for a mile. Then I'll leave it by the side of the road, while you keep walking. When you get to the bike, you get on it, and you ride for a mile, and then you leave it by the side of the road. When I get to it, I'll ride another mile. We'll keep leap frogging like this until we get to town.

"Since we'll be riding part of the way and walking part of the way, each of us will have a higher average speed than if we walked, so we'll get there faster.

Fred says, "That isn't going to do any good, you dope! Think about it. Between the two of us, one or the other is going to walk every inch of the way from here to town. So, we can't possibly get there any faster than if we just walked all the way and didn't use the bike at all."

Who's right? Or, like us, are they both wrong?

What do YOU think? Drop us a note via

http://www.cartalk.com/email/email.html

Lots of interesting puzzlers here, one every week.

 

[Lauren Hynde]

John is closer to the truth, although not entirely right either.

They will get there a lot faster (i.e. in the time that would take to travel riding a bike half of the way and walking the other half) but only one of them will have to leave the bike by the side of the road (after the odd miles).

If it takes the biker one minute to ride a mile and the walker ten minutes to walk a mile, then after one minute the biker (A) will be at the 1 mile mark and the walker (B) at 1/10 of a mile. A will leave the bike at the side of the road and walk. Ten minutes after they started, B will get on the bike at mile 1; by then, A will have been walking for nine minutes and is 1/10 of a mile short of mile 2. One minute later, when B reaches the 2 mile mark, A will have just completed his full mile walking as well, so they can simply hand each other the bike.

0 min.: Both are at mile 0; A takes the bike, B walks.
1 min.: A is at mile 1, leaves the bike; B is at 1/10.
10 min.: B is at mile 1, gets on the bike; A is at 1 9/10
11 min.: Both are at mile 2. On foot, it would have taken them 20 minutes.

 

[mcfbridge]

Like the last puzzle, well done Lauren.

Here's another one purely for math nuts.


You have two variables X and Y, such that X = Y.

read X^2 as X squared.


Since X = Y XY = Y^2

So X^2 - XY = X^2 - Y^2

factorying X(X-Y) = (X +Y)(X - Y)

canceling X = X + Y

Substituting X = X + X

thus X = 2X

Divide both 2 = 1
sides by X


What, if anything, is the flaw in this proof that 2 = 1.

 

[Lauren Hynde]

factorying X(X-Y) = (X +Y)(X - Y)

canceling X = X + Y

You can't cancel. X-Y=0 and you can't divide a number by 0.

 

[ABSTRUSE]

Hi Lauren.