What Color is My Hat?

May 25, 2001 10:22 AM

What Color is My Hat? I [heart] these mathematical conundrums -- simple, easy-to-state, seemingly obvious logic problems that have solutions that completely defy common sense. Here's another you can spring on a friend: "You want to fry up three pieces of french toast. You have a frying pan that is just large enough to accomodate two pieces of bread at a time. If it takes you 30 seconds to fry one side of bread, and each piece of must be fried on both sides, how long will it take you to cook up three pieces (assuming that the act of flipping a piece or adding/ removing it to or from the pan takes no time). Think about it. Answer inside.
posted by Shadowkeeper (24 comments total)

French toast answer: 90 seconds. You have three pieces of bread: 1, 2 and 3. Each piece of bread has two sides: 'a' and 'b'. To cook them all in 90 seconds you:

* Put A and B in the pan. Cook 1a & 2a. 30 seconds;
* Flip piece 1, remove piece 2 and add piece 3. Cook 1b & 3a. 30 seconds;
* Remove piece 1, flip piece 3, add piece 2. Cook 2b & 3b. 30 seconds.

Elapsed time: 90 seconds.

Of course no discussion of "seemingly simple, common-sense defying conundrums" would be complete without mention of the now infamous "Monty Hall Problem". The question: "Suppose that Monty Hall (on TV's Let's Make A Deal) asks you to choose between three doors: #1, #2, and #3. Behind a random door is a new Rolls Royce. Behind each of the other two doors is a goat. Let's assume that you would prefer a Rolls Royce to a goat. You choose a door. Now, Monty, who knows which door hides the Rolls Royce, shows you a goat behind one of the two doors that you did not choose. He then gives you the opportunity to change your choice. Assume that Monty always does this, regardless of your guess. Should you change your choice?"

I assume most of you already know the answer, but if you don't please read this. And check out this article from the New York Times about what happened to Marilyn vos Savan after she answered the puzzle in her column.

Anyone know any other conumdrums like this they'd like to share?
posted by Shadowkeeper at 10:26 AM on May 25, 2001

Actually - it took me 120 seconds. The problem didn't specify that the goal was to accomplish this in the shortest possible time. It asked me how long it would take me to do it, and I did it the lazy, non-brain-pretzel way.

Logic is nice, and some people like puzzlers, but indistinct language causes all sorts of problems. Especially when it's deliberate...
posted by Irontom at 10:36 AM on May 25, 2001

I actually did get the 90 sec. solution but, based on my dismal seccess with these things in the past, I assumed that there was probably a 60 sec. (or any other time less than 90) solution that I wasn't aware of.
posted by Octaviuz at 10:49 AM on May 25, 2001

Ah crap, you're right.

Um, "In the shortest possible", people. I mean. And now that the horse is gone, would'ja'all mind closing that barn door while you're at it?

Oh well. And I wholeheartedly agree that deliberate indistinctness sucks, Irontom, although that wasn't the case here.
posted by Shadowkeeper at 10:50 AM on May 25, 2001

I actually did get the 90 sec. solution but, based on my dismal success with these things in the past, I assumed that there was probably a 60 sec. (or any other time less than 90) solution that I wasn't aware of.
posted by Octaviuz at 10:54 AM on May 25, 2001

Please do not be alarmed by the sounds of a struggle, that's just me in the corner assaulting myself as punishment for stupidity.
posted by Octaviuz at 10:57 AM on May 25, 2001

The Monty Hall thing bothered me enough when I heard the solution that I wrote a simulator for it, even though I believed the reasoning.

It turns out you really should switch doors....
posted by crunchburger at 11:12 AM on May 25, 2001

Here's a similar puzzle a friend told me recently:

You have two fuses and a way to ignite them. Each fuse takes exactly one hour to burn entirely when lit at one end, but they don't burn at a consistent rate (i.e. cutting one fuse in half doesn't guarantee that each half will burn in exactly one half hour).

You need to measure out a span of exactly 15 minutes with only the fuses and the ignition source available to you. How?
posted by jfirman at 11:47 AM on May 25, 2001

I got 60 seconds. If you cut the bread into little triangles you can fit more on the pan. Yum!
posted by greensweater at 11:50 AM on May 25, 2001

How about you light one of the fuses at one end and light the other fuse at both ends. The fuse that's lit at both ends will expire in 30 minutes. You now know you have 30 minutes left on the fuse lit on one end. Then, light the other end of this fuse and it will expire in your desired 15 minute window.
posted by irishcreme at 12:28 PM on May 25, 2001

A great source for logic puzzles like this one is techInterview, run by Michael Pryor of Fog Creek Software (the company started by Joel of Joel on Software fame).
posted by mw at 12:57 PM on May 25, 2001

You tie a mass to one of the strings and start it swinging right as you light the other string at both ends. Count the number of oscillations in the 15 minute time period and that will tell you the period of one oscillation. Now you can measure any length of time to a fairly reasonable precision by using the callibrated pendulum.

Have I mentioned that I hate these problems?
posted by plinth at 1:26 PM on May 25, 2001

See, I can't do these things. Admittedly, I suck at math, but I like to think my brain freeze on stuff like this is a strictly home economics problem. If I'm frying bread, I want to finish the two pieces I started first, otherwise, taking one out will make it nasty and greasy, and even worse when you put it back in to finish it. (Then again, I guess the problem didn't include that all three pieces of fried bread should be -edible- when you finish.)

So, if one door always lies, and the other door always tells the truth, which one do you go through to avoid certain death?
posted by headspace at 3:33 PM on May 25, 2001

The problem with plinth's pendulum solution is that you have no way of measuring the 15 minutes span during which time you count the oscillations. If you did, the problem would be moot.

As for headspace's door question, ask the following: If I were to ask you if your door leads to safety, would you say "yes"?
posted by Monk at 4:46 PM on May 25, 2001

Sorry- misread. Make that 1/2 hour. I edited it to 15 minutes because I'm impatient. Did I mention that I hate these problems?
posted by plinth at 5:59 PM on May 25, 2001

i got 38 minutes. *phew*
posted by muppetboy at 6:48 PM on May 25, 2001

if doors could, in fact, talk:
1. go to either door;
2. ask it which door would the other door tell you is going to kill you;
3. go through the door of the answer.

or

1. go to either door;
2. ask it which door would the other door tell you is not going to kill you;
3. go through the door other than the door of the answer.
posted by elle at 6:52 PM on May 25, 2001

Here's a good one, from last month's Scientific American.
posted by jpoulos at 8:02 PM on May 25, 2001

But... but... I'm still missing a leap of logic here. (Because I am dense and taste good with cheese.) I ask door one which door to go through. It says, go through me, I do not lead to certain death. If I asked the truth telling door, I -should- do that. But how does that question determine whether I am talking to the lying door or the honest door?

(And good lord, what I have I been smoking to write a sentence like that??)
posted by headspace at 9:26 PM on May 25, 2001

I'm still missing a leap of logic here.

It's because you're phrasing the question wrong. You aren't asking one of the doors "Which way is safe?" or "Which way isn't safe" you're asking the doors "Which way would the other door say is safe?"

You have two doors, Door 1 (D1) and Door 2 (D2). One always lies, the other always tells the truth. You have two directions, East and West. One is safe, the other will kill you.

In elle's first answer you ask "Which way would the other door say is going to kill me?"

D1 would say "D2 would tell you East is the Door of Death."
D2 would say "D1 would tell you East is the Door of Death"

If D1 is telling the truth, than D2 is the liar. Since the liar would tell you East is going to kill you, you go East.

If D1 is lying, than D2 would not tell you East is going to kill you, which means east is safe.

In elle's second example you ask the doors "Which way would the other door say is safe?"

D1 would say "D2 would tell you West is safe."
D2 would would say "D1 would tell you West is safe."

If D1 is telling the truth, D2 is the liar, and you wouldn't want to go west.

If D2 is lying, then D2 (the truthsayer) would have really told you to go East.

No matter what, you go East.
posted by cCranium at 7:55 AM on May 26, 2001

Ohhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh... I think I get it now. I think. Thank you, cCranium! My head aches with a bursting of new knowledge!
posted by headspace at 12:11 PM on May 26, 2001

the monty hall problem becomes obvious
if you have, say, 100 doors instead of 3 doors.
BEFORE opening any doors, choose your door.
you have a 1/100 chance of hitting the wining door.
now, open 98 doors, leaving closed your door and
monty's door. your door still has only a 1/100 chance
of being correct. you can only your odds by switching.
posted by anocious at 2:19 PM on May 26, 2001

So you're on an island of knights and knaves. Knights always tell the truth, knaves always lie. You're at a crossroads where there is a knight or a knave waiting (you don't know which). Can you find out which road will take you into town the fastest by asking one question?

"Hey-did you hear that they're giving out free beer in town?" Then follow him in.

Have I mentioned that I really hate these problems?
posted by plinth at 7:07 AM on May 28, 2001

If goats are behind 2 doors and the car behind the third then the probability of choosing a goat is 2/3. Therefore, you should assume that you did pick a goat initially. Monty will then reveal the location of the second goat, so you can change your odds of winning the car from 1/3 to 2/3 by selecting the remaining door!! Am I right?

Now, where's my Rolls Royce?
posted by MarkC at 6:02 AM on May 29, 2001