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Topic: Choose Your Cards (Read 693 times) |
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ThudnBlunder
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Choose Your Cards
« on: Apr 7th, 2003, 3:15pm » |
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Every member of a set of cards has a digit on one side and a letter of the alphabet on the other side. Four cards are placed on a table so that we can see the sequence D, F, 2, 3. You need to find out if every card in the set with a 'D' on one side also has a '3' on the other side. You are allowed to turn over two cards. Which two cards would you turn over, and why?
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| « Last Edit: Apr 7th, 2003, 11:07pm by ThudnBlunder » |
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towr
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Re: Choose Your Cards
« Reply #1 on: Apr 7th, 2003, 11:46pm » |
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First of all, I would forget about the 'you can turn over two cards' bit, it's not necessary, and you'll probably get more interesting mistakes from people.
..It's an exercise in abstraction, most people get it wrong. But a problem which is equivalent in every way, but stated in different terms, almost every one gets right:
Suppose the legal drinking age is 21, you have to check that no one at a bar under 21 is drinking an alcoholic beverage.
One of them is 18, one is 22, one is drinking a beer, one is drinking a soda. Which do you have to check?
Every one under 21 needs to be drinking something other than alcohol, and everyone drinking alcohol needs to be over 21.
Of course abstractly it's easy enough as well, D -> 3 => ~3 -> ~D. Every D needs a 3 on the other side, and every not-3 needs a not-D on the other side.
..
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ThudnBlunder
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Re: Choose Your Cards
« Reply #2 on: Apr 8th, 2003, 3:13am » |
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Hey, I was supposed to point out the easy analogy. 
Where did you learn of it?
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| « Last Edit: Apr 8th, 2003, 3:24am by ThudnBlunder » |
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wowbagger
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Re: Choose Your Cards
« Reply #3 on: Apr 8th, 2003, 3:47am » |
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Well, I know it's subjective but I feel the easy section would be more appropriate for this one. Granted, a lot of people may get it wrong, but come on! It is quite obvious, isn't it? (Hope you believe me that the only reason why I didn't post before towr is that I don't get up as early as he does.)
On a more fundamental point, one should give a completely different reply to answer the question most precisely.
Consider the following hidden quote from the problem statement:
You need to find out if every card in the set with a 'D' on one side also has a '3' on the other side. (emphasis mine)
Do we really know enough (after turning over some of the four cards) to answer "Yes" or "No?
I suppose that the deck has more than four cards, so we can only falsify the claim D ~ 3 based on a sample, but we can't verify it without looking at at least one side of all cards.
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| « Last Edit: Apr 8th, 2003, 3:49am by wowbagger » |
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towr
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Re: Choose Your Cards
« Reply #4 on: Apr 8th, 2003, 8:43am » |
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on Apr 8th, 2003, 3:13am, THUDandBLUNDER wrote:Hey, I was supposed to point out the easy analogy.
Where did you learn of it? |
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I think most logic courses here cover it. But more in general I'm a student in (applied) Cognitive Science, which deals with, among other things, how people think/solve problems. So I actually came across it at least two or three times before.
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ThudnBlunder
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Re: Choose Your Cards
« Reply #5 on: Apr 8th, 2003, 12:56pm » |
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Wowbagger, if it were a new puzzle I think it might belong in the Medium section, but - as you say - it is subjective. But I would certainly not describe it as 'obvious'. Hindsight is always 20/20, as they say. It seems that these days one needn't bother posting 'oldies', no matter how nice they may be, because they are so well-known. (However, I intend to get my revenge with a couple of unusual chess puzzles which I have sent to William.) 
As for the problem statement:
1) I should have stated that the set contains more than four cards.
2) I wrote "You need to find out if every card in the set..." when "You need to find out whether or not every card in the set..." is better. It is not stated that one can find out, merely that one needs to. It is surely up to the person doing it to work out that it's impossible to prove it's true, so they may as well try to prove it is untrue.
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| « Last Edit: Apr 8th, 2003, 1:15pm by ThudnBlunder » |
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ThudnBlunder
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Re: Choose Your Cards
« Reply #6 on: Apr 8th, 2003, 12:58pm » |
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Quote:| I think most logic courses here cover it. But more in general I'm a student in (applied) Cognitive Science, which deals with, among other things, how people think/solve problems. So I actually came across it at least two or three times before. |
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towr, I believe the analogy is due to Halmos.
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| « Last Edit: Apr 9th, 2003, 4:52am by ThudnBlunder » |
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towr
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Re: Choose Your Cards
« Reply #7 on: Apr 8th, 2003, 1:50pm » |
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I have know idea who Halmos is, but I'll take your word for it 
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aero_guy
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Re: Choose Your Cards
« Reply #8 on: Apr 8th, 2003, 6:35pm » |
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Just my two cents, I've never heard this one before but I think it fits a little better with what Wu describes as an easy problem.
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wowbagger
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Re: Choose Your Cards
« Reply #9 on: Apr 9th, 2003, 4:29am » |
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on Apr 8th, 2003, 12:56pm, THUDandBLUNDER wrote:| Wowbagger, if it were a new puzzle I think it might belong in the Medium section, but - as you say - it is subjective. But I would certainly not describe it as 'obvious'. |
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Maybe my phrasing was a little bit exaggerated. I still consider it easy, though.
Quote:| It seems that these days one needn't bother posting 'oldies', no matter how nice they may be, because they are so well-known. |
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Well, I can see the point of posting well-known puzzles. After all, the newbies and youngsters could very well be unaware of them. Keep it up!
Quote:| (However, I intend to get my revenge with a couple of unusual chess puzzles which I have sent to William.) |
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I'm already looking forward to them! 
Quote:As for the problem statement:
1) I should have stated that the set contains more than four cards.
2) I wrote "You need to find out if every card in the set..." when "You need to find out whether or not every card in the set..." is better. It is not stated that one can find out, merely that one needs to. It is surely up to the person doing it to work out that it's impossible to prove it's true, so they may as well try to prove it is untrue. |
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The first point isn't so important, I'd say. Only in case you want to ask the follow-up question:
"How many sides of N cards do you have to examine in order to be sure that D ~ 3, given that there are ND cards with a 'D' and N3 with a '3' in the set?"
I agree on the second point. It's a good way to check for hasty decisions.
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| « Last Edit: Apr 9th, 2003, 4:30am by wowbagger » |
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ThudnBlunder
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Re: Choose Your Cards
« Reply #10 on: Apr 9th, 2003, 5:32am » |
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Quote:| I have know idea who Halmos is, but I'll take your word for it |
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Halmos is quite a well-known mathematician.
But I checked and the analogy actually comes from "The Narrative of Numbers" by Paulos. (At least, that's where I first saw it.) He cites research in evolutionary psychology that suggests that our brains have evolved to spot cheats, and hence a mathematical problem that exploits this talent is easier for us to deal with than an abstract version of the same problem.
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| « Last Edit: Apr 9th, 2003, 12:47pm by ThudnBlunder » |
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ThudnBlunder
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Re: Choose Your Cards
« Reply #11 on: Apr 9th, 2003, 6:08am » |
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Quote:[I'm already looking forward to them!
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One, called Chess Game Reconstruction, is already available. The other, easier but not Easy, is a retrograde analysis puzzle where one must work out the last move that was played.
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| « Last Edit: Jul 7th, 2008, 1:08am by ThudnBlunder » |
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