C Blank Cards

• Consider the blank card: it does not matter whether or not there is a number on the other side because the statement is still true in either case.

• Consider the C card: You need to check the C card because the statement will be false if there is a number on the back.

• Therefore, the C card is the only card you need to flip.

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Edit for clarity :

Examine the table below of all possible scenarios:

1. There is a number on the back of a non-blank card. Therefore, the statement is false. Verification is only needed on the C Card.

2. There is a number on the back of a non-blank card. Therefore, the statement is false. Verification is only needed on the C Card.

3. All cards that are not blank do not have numbers on the back. Therefore, the statement is true. Verification is only needed for the C Card.

4. All cards that are not blank do not have numbers on the back. Therefore, the statement is true. Verification is only needed for the C Card.

In all the scenarios, you only need to check the C Card. If you have any questions about this specific edit, tag me in your comment.

There are two kinds of people in a party - those who are drinking and those who are not.

As the security officer, you must ensure the following:

If a person is under 18 years, then they should not be drinking.

Which people should you ask their ages?

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Compare the above scenario with the problem.

C corresponds to drinking, Blank Corresponds to not drinking. Having a number corresponds to being below 18, not having a number corresponds to being at least 18.

A coomon misconception is that people should flip the blank card. The blank card can either have a number or not, and the statement is still true either way.

The card you should flip is the C card. If it has a number then it's false.

EDIT: Flipping the C card is to check that the statement is not false. That means getting a blank or a letter does not mean the statement is false nor is it true.

https://www.youtube.com/watch?v=AOAtz8xWM0w

If the second card has a number on the other side, the statement can still be true.

If the second card does not have a number on the other side, the statement can still be true.

If the first card has a number on the other side, the statement cannot be true.

If the first card does not have a number on the other side, the statement can still be true.

The first card is the only card which might contradict the statement, so we need to flip that card only.

I'm going to try and address a range of reasons for why some people have been getting this problem wrong, and then I will solve the problem.

1. Confusion/misconceptions about checking that the statement IS true.
2. Understanding conditional statement logic.
3. Explicit vs Implicit
4. Incorrectly using converse logic.
5. Solving the problem

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1. Confusion/misconceptions about checking that the statement IS true :

• Checking that something IS true, is the same as checking that it IS NOT false.
• In a different context, a substance is considered to be pure by the absence of impurities.

Only when you have eliminated the possibility of falsities existing can you determine that something is true.

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2. Understanding conditional statement logic

A conditional (if, then) statement is asserted in this problem. Conditional statements have the following details:

• Adheres to Material Condiational logic.
• The 'if' part is called the Antecedent ( $p$ ).
• The 'then' part is called the Consequent ( $q$ ).
• $p \rightarrow q$
  • $\rightarrow$ is the annotation for implies .
• The material conditional only states that $q$ is true when (but not necessarily only when) $p$ is true, and makes no claim that $p$ causes $q$ .
• A conditional statement is only false when the condition of $p$ is true, and $q$ is false.
  • When $p$ is false, the statement is vacuously true .

Here's an example:

Conditional statement: " If it is raining, I will stay at home. "

1. If it is raining, and I stay at home; I have not broken my rule - The statement remains $\color{#20A900}true$ .
2. If it is raining, and I go out; I will have broken my rule - The statement is now $\color{#D61F06}false$ .
3. If it is not raining, and I stay at home; I have not broken my rule - The statement remains $\color{#20A900}true$ .
4. If it is not raining, and I go out; I have not broken my rule - The statement remains $\color{#20A900}true$ .

How does this relate to the problem?

• $p$
  • In my example, $p$ is true when it is raining , and false when it is not raining.
  • In this problem, $p$ is true when a card has a number on one side , and false when there is not a number on one side.
• $q$
  • In my example, $q$ is true when I stay at home , and false when I do not stay at home.
  • In this problem, $q$ is true when the other side is blank , and false when the other side is not blank.

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3. Explicit vs Implicit

In all cases, anything other than what is explicitly stated will render the condition as false.
It's implicit that we don't need to state rules for every possible thing before we can apply conditional logic to them.

For example:

• The letter C is not a number ( $p$ = false), and it is not blank ( $q$ = false).
• A tree is not a number ( $p$ = false), and it is not blank ( $q$ = false).
• Blank is not a number ( $p$ = false).

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4. Incorrectly using converse logic.

Converse logic can be used when $p$ and $q$ are equivalent.
For $p \rightarrow q$ , when $p$ and $q$ are equivalent, it allows us to assert $q \rightarrow p$ .

Here's an example:

• "If I am a triangle, then I am a three-sided polygon"
• "If I am a three-sided polygon, then I am a triangle"

The definition of "triangle" is "three-sided polygon", thus the use of converse logic for the above statements is correct - Both are always true.

However, here's an example of when using converse logic is wrong:

• "If I am in Paris, then I am in France."
• "If I am in France, than I am in Paris."

A person can be in France without being in Paris, thus the use of converse logic for the above statements is wrong - The 2nd statement can be true, but it isn't necessarily true.

For this problem, the use of converse logic is wrong, because blank is not equivalent to a number - A blank face does not necessarily imply that the other face is a number.

By implying that the a blank face results in a number on one side, you are then forced to prove that assertion - leading you to having to check the blank card.

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5. Solving the problem

With all of the above in mind, in order to check that the statement in this problem IS true, we must eliminate the possibility that falsities exist for the conditional statement within the given scenario.

Where in the scenario is there the possibility that the conditional statement could be false?

Neither of the card faces presented to us match the condition for the Antecedent ( $p$ ) being true (they are not numbers), so we can consider each of them as a Consequent ( $q$ ).

• A blank face means that $q$ is true.
  • This corresponds to cases 1 and 3 from the table above.
  • When $q$ is true, the conditional statement for that card is always true.

By this application of conditional logic, we know that the possibility for falsities for the conditional statement does not exist with the blank card. We gain no further sense of truth about the statement, so we don't need to check it.

• A non-blank face (C) means that $q$ is false.
  • This corresponds to cases 2 and 4 from the table above.
  • When $q$ is false, the conditional statement for that card could be either true or it could be false (depending on whether $p$ is true or false).

If either card is false (breaks the rule), it renders the statement covering both cards as false.

As the only card that has the possibility of falsities for the conditional statement, it's only necessary to check $\boxed{\text{Just the card with C}}$ to determine whether the statement is true (or not) for both cards.

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Bonus

Let's consider the statement in this problem in relation to a different scenario with just 1 card:

• "If a card has a number on one side, then it is blank on the other."

Is the statement true when there is a number on both (two) sides of the card?

This is a good example of a problem requiring rational thinking, an aspect of intelligence not tested on IQ tests. Most people would say that the blank card should be flipped over, thinking that revealing a number would prove the statement true and a letter would prove it false. However, this is not correct, because revealing a number would not prove the statement true in all cases, and revealing a letter would not prove it false, since the statement didn't say that all blank cards must have a number behind them, only that all numbers must have a blank behind them. So, the only way to prove this statement false is to find a letter with a number, since it would mean that not ALL numbers have a blank behind them. This is why the "C" card must be turned over instead.

The only way for this statement to be false is for a card to have a number on one side and a $C$ and a number on the other. By this logic, you should check the card with the C.

The blank card doesn't affect whether the statement is true or not. If it has a number on the other side, it is true. But if it doesn't have a number on the other side, it is still true.

Since neither card shows a number, one can instead use the logically equivalent contrapositive statement: "If it's not blank on one side, then there's not a number on the other side." So flipping over the card with the letter "C" will confirm whether or not the statement is true.

You need to check the statement and its contrapositive. Therefore, you need to flip any card that has a number and any card that is not blank.

Wiki: Truth Tables

In everyday use if ... then ... can have two meanings, and we usually understand which by the context. One is the conditional statement: if my house catches fire (then) I will go outside, the second is biconditional, if and only if abbreviated to iff : if a building catches fire (then) I will call the fire brigade.

In the context of a logic question we can assume a conditional statement, which is evaluated as true or false based on the truth values of the simple statements according to the following truth table.

That is, the statement is only false if P (a card has a number on one side) is true and Q (it is blank on the other) is false.

Whether the blank card has a number on the other side or not, the compound statement is true: TT and FT both evaluate to T.

If we check the C card and find it has a number on the other side we would have TF, which would prove the statement false.

The statement "If a card has a number on one side, then it is blank on the other" is different from "If a card is blank on one side, then it has a number on the other." Basically, we only have to check if there is a number on the card with a "C" to see if it has a number. After all, there is no rule saying that blank cards must have numbers, but there is one saying that a number card must be blank. This photo may help: It shows that ALL number cards must be blank, and that SOME blank cards are number cards.

• If the card is blank on one side then it is bound to have a number on the opposite.

• We only have to flip the card containing the "C" because we already know what is on the other side of the blank one (As said "A number").

• By flipping the "C" card we can see what is contains on the other side, and then compare them.

Lots of people here bring up good points about the possibility of not being able to fully verify the statements truth. If the lettered card is flipped and doesn't have a number, then you would need to flip the blank card to fully verify. I think a better query to this question would have been "Which action would render the following statement false"? OR Supplement the initial information to claim that a number exists at all.

Easy,just ignore that except that statement,then pick the card with C