"Each card has a number on one side and a color on the other side."
So 5 could not have an even number on the back given the rules. But your reasoning would make sense if you missed that statement.
If you can take it as absolute truth that each card has a number on one side and color on the other, then you only have to check which color. If on the other hand you must verify both that each card has one color/one number and that even cards have blue on the back, then you must turn over every card to ensure that each has the expected quality of opposite side (i.e. that if you see a number, the opposite side has a color and vice versa).
The confusion comes from making an assumption that if a card is odd then it cannot also be blue.
if even then blue //what is being asked
if even then blue && if odd then !blue //what people assumeThe situation follows:
1) There are four cards with a number on one side and a color on the other.
2) If a card has an even number on one side, then it's blue on the other.
You are only allowing for confusion about the commutative nature of the conditional in (2); it is not commutative, but a common false assumption is that it is.
What the parent of this thread is referring to is that there can be confusion regarding whether (1) must also be verified. If (1)must be verified then every card must be flipped.
"If Jenny has 5 apples and Bob has 5 apples and Jenny gives Bob 2 apples - how many apples does Jenny now have?"
We don't question if Jenny went and bought n more apples to give to Bob and might have 3 Apples or 5 Apples or 5+n where n is the number of apples purchased minus 2 because it isn't part of the problem and isn't part of what is being asked. Maybe she won an apple lottery and that is why she gave Bob 2 apples. Maybe she gave Bob 2 apples and later lost her other 3 apples in a fire.
If you begin by questioning the validity of the question - Jenny can have 0...n apples. Which doesn't do us any good for solving the problem.
"There are four cards. Each card has a number on one side and a color on the other side." These details are taken as facts as a basis for solving the problem.
If you are questioning if there is a color and number on the sides. Are there really 4 cards? What if there is a 5th card I'm not told about? What if we took 4...n cards and I found a card that invalidates the problem? It's ignoring the problem and coming up with a new one to solve.
We are told two things about the situation. You are assuming the first thing we are told can be trusted and the second can't.
I understand that the logic puzzle is only to verify the second thing we are told. I am not confused. The way that the puzzle is presented, though, allows for confusion, because we are presented with three things: 1) A physical setup which can be trusted (4 cards exist, we see 1 side of each card) 2) A statement that each card has one color and one number 3) Even -> blue on other side.
Given that one of the things we are expected to trust is a statement made to us and the thing we are expected to verify is a statement made to us, there is potential for confusion as to which statement(s) we are to verify. This potential is all that OP raised and all that I have been trying to explain.
Edit: To summarize in brief, as the problem is presented, it is not 100% clear that some statements told to you by the experimenter can/should be trusted.
The actual situation as given to us is not that there are 4 visible cars on a table. It is that there are 4 visible cars on a table that contain a number on one side and a color on the other side.
We are being told to verify the statement:
"Of the 4 cards placed on the table, which have a number on one side and a color on the other side, if the number is even then the other side must be blue."
If the color/number statement was left out of the original problem/video (it isn't) then your argument would be true, as it would be an assumption that is not listed and we would need to flip the 5 as well.
E:
And I suppose the blue card since it could have green on the other side.
I don't retract anything I said, but apparently his confusion was different.
Actually, only the 5,8 and green. The blue is good no matter what.
I saw this pedanticism in a Martin Gardner book originally, IIRC.
To verify (1) you must prove that each card has a color on one side and a number on the other. The only way to do this is to see both sides of each card.
If you doubt (1), how is it that you need to verify that the 5 has a color, but not that the blue has a number on the other side?
To verify (2) you only need to prove that every even card has a blue back. An odd may be blue or green, but an even must be blue. Hence the 5 is fine, it can have any color. The blue is fine, because it can have any number. But that 8 must have a blue back, and that green must have an odd number.
If you can't be sure that the cards have a color and a numbers, there's nothing outlawing a 5-6 card, which breaks (2).
You trust that if a card has a color on one side, it has a number on the other side.
You do not trust that if a card has a number on one side, it has a color on the other side.
Is this correct?
However, regardless of what's on the back of blue, (2) will still be correct; the consequent is true, therefore the conditional is true.
(1) may or may not be true. Understanding that we cannot trust (1), but that we don't care to verify it, then the 5 is necessary to prove (2).
That is much more subtle than I originally interpreted from you original post. Thanks for your patient explanation.
You are not supposed to test for truth the problem premises. Just the if "even then blue" statement.
If you also had to test the premises then there would be no question and no asnwer -- because those require premises.
Short version:
You are presented with 4 cards on a table. You can see one side of each card.
You are told two things:
1) Each card has a number on one side and a color on the other side.
2) If a card has an even number, then its color is blue.
It is not unreasonable assume that you are expected to verify all things you have been told (i.e. all things you cannot verify by what you observe).
Since you know you can't trust the second thing you are told about the cards, why should you assume that the first thing you are told about the cards is true?
OP is saying that you cannot trust anything you've been told, and is attempting to verify (2) without trusting (1). I misinterpreted and assumed that OP was saying the goal to verify all statements that you've been told.
1) Each card has a number on one side and a color on the other side.
2) Pick the minimum numbers of cards needed to verify the statement "If a card has an even number, then its color is blue".
>Since you know you can't trust the second thing you are told about the cards, why should you assume that the first thing you are told about the cards is true?
This reading doesn't make any sense. If you were indeed given (1) and (2) in that form, and indeed couldn't trust them both, then there would be no quiz at all. Just 2 random sentenses that might or might not be true.
As OP of this thread pointed out, the card with 5 must be checked, because it's possible that it has an even number on its reverse side.
I originally misinterpreted the OP as saying that both (1) and (2) must be verified. You can see the two sibling branches that explore both of these in much more depth and back and forth.
>So 5 could not have an even number on the back given the rules. //
Having "a number" on one side doesn't mean it only has one. Nor does having a colour on the other side mean it doesn't have more than one colour, nor indeed that it doesn't have colours on both sides.
It's like me saying "I have a coin in my pocket" it only tells you the number is >= 1 assuming I'm being truthful.
If you expect a precise answer you need to ask the question in a precise manner.
"Will you hand me a pencil" doesn't mean I want you to hand me at least 1 pencil but possibly 5.
I see where you are trying to come from with that angle though. You can feel free to apply different semantics to "a ___ "but I feel the general consensus is that it is equivalent to "one ____".
Note that "a colored shirt" is not the same as "a shirt with a color". As "A colored shirt" could be a shirt with multiple colors due to the ambiguity of "colored".
E:
From the article:
>In one version of the task, one subject (always one—he spurned testing subjects in groups) is presented with four cards lying flat on a table, each with a single-digit number on one face and one of two colors on the other.
It specifies "one" for the colors.
There's a difference between your examples and the question:
"Give me a shirt." is a request for one shirt.
"You've got a shirt in the wardrobe." is a statement that doesn't tell you how many shirts are in the wardrobe, just that there is one.
Similarly:
"Will you hand me a pencil." is asking for one pencil.
"I have a pencil." is telling you that I have at least one pencil.
Based on en-gb native use.
From the article. All the needed information is present.
But if you take them at their word that each card has only one number, then yeah, it's 8 and green.
I guess I'm saying the article didn't make it clear that you could trust anything you didn't see in front of you.