A logic problem by Sunny Dhondkar

This problem may firstly puzzle, but its answer is quite simple. The factor that makes it harder is the human tendency to make simple things look more complicated. I designed this problem myself, psychologically to make this question look like a paradox.

                     **Question 1**: Is Question 2 unanswerable?
                     **Ans**: No, it is answerable.

                     Question 2: Does question 1 have an answer?
                     Ans: Yes, question 1 has an answer

Hence, question 1 has an answer that question 2 is answerable.

So, the answer to question 2 is Yes .

Question 1: Are both "yes" and "no" invalid answers to question 2?

Answer: $\color{#D61F06}\text{No}$ , it has a valid answer.

Question 2: Is either "yes" or "no" a valid answer to question 1?

Answer: $\color{#20A900}\boxed{\text{Yes.}}$

Using the logic of a multiple choice question, we can ignore question 1. If the answer to question 2 is "no," then that means there is no valid answer to this problem at all. Therefore, that means that the answers "No" and "This is an inconsistent scenario" are the same answer. Since there can only be one answer, that leaves us with the only possible solution being "Yes."

This strategy can be very helpful on standardized tests on any subject.

As long as there is hope for consistency , we can't claim inconsistency .

Saying 'Yes' to Question- $1$ leads to inconsistency. Because, saying 'Yes' to Q- $1$ is granting that both 'Yes' and 'No' are invalid answers to Q- $2$ . But both 'Yes' and 'No' cannot be invalid at the same time. So, for the sake of hope of consistency , $\text{we have to answer Question-}1 \text{ with} \boxed{\text{No}}$ .

Similarly, saying 'No' to Question- $2$ leads to inconsistency. Because, saying 'No' to Q- $2$ is granting that none of 'Yes' and 'No' is valid answer to Q- $1$ . But at least and exactly one of 'Yes' and 'No' has to be valid at any given time. So, for the sake of hope of consistency , $\text{we have to answer Question-}2 \text{ with} \boxed{\text{Yes}}$ .

So, so far, the only hope for consistency is: saying 'No' to Question- $1$ and saying 'Yes' to Question- $2$ .

The last thing we have to be sure about is: saying 'No' to Question- $1$ doesn't contradict with saying 'Yes' to Question- $2$ . The fact that the Question- $2$ is satisfied with whatever is used to answer Question- $1$ as long as that 'whatever' is either 'Yes' or 'No' ensures us that there is no chance of inconsistency. So, our hope is rewarded.

Therefore, the answer is 'Yes' to Question- $2$ .

This question, when looked upon carefully, is a very simple one. What confuses is their framing of the questions. The first question asks if yes or no is a valid solution for question 2, while question 2 asks if either yes or no are possible answers for question 1. Since question 1 does not ask for a definite answer, yes or no, both are possible answers. Hence the answer for the second question is YES

Brute force/comprehensive solution:

Case 1:

A: Assume the answer to the second question to be YES

A>B: That means either"YES" or "NO" is a valid answer to question one

Case 1-1: Following statement B, assume the answer to the first question to be YES

> A>B>C: This means both "YES" and "NO" are invalid answers to the second question

>> STATEMENTS A AND C CONTRADICTS, proceed to case 1-2

Case 1-2: Following statement B, assume the answer to the first question to be NO

> A>B>D: This means either "YES" or "NO" is a valid answer to the second question

>> Coherent solution, no contradictions among statements A B and D, therefore "Yes" is a sufficient answer to the second question, proceed to case 2

Case 2:

E: Assume the answer to the second question to be NO

E>F: That means neither "yes" nor "no" are valid answers to question one, meaning the only answer to question one is that it has no valid answer

E>G: However, assuming the answer to the second question to be NO means the answer to the first question should be YES

> STATEMENTS F AND G CONTRADICTS, AS BOTH STATEMENTS STEM FROM STATEMENT E, STATEMENT E IS FALSE

>> "No" is not an acceptable answer to the second question,, proceed to case 3

Case 3:

H: Assume the second question has no valid answer

H>I: This means the first question's answer would be YES

H>I>J: However, this means the answer to the second question should be YES

> STATEMENTS H AND J CONTRADICTS, AS STATEMENT J STEMS FROM STATEMENT H, STATEMENT H is false

Conclusion: The answer to the second question simply has to be "Yes".

The answer to the 1st question must be "No," otherwise there is no way to answer question 2. Then, because the answer to question 1 is "No," the answer to question 2 is "Yes."

Alternatively, I converted this logic problem to a Boolean Algebra one i.e

1. Let A = "Yes" = True
2. Let B = "No" = False

Then:

Q1 = True <=> Invalid ( Yes and No ) = True <=> ! ( A && B) = True

From Boolean Algebra:

!True = False <=> !A = B <=> B = !A

So, to our solution:

Q2 = Yes or No <=> Q2 = A || B <=> Q2 = !B || !A <=> Q2 = !(B && A) <=> Q2 = Q1 <=> Q2 = True

For brevity, let’s call a yes-no question unanswerable if neither answer is consistent, and answerable if at least one answer is consistent.

For example:

Is it false that the answer to this question is ‘yes’?

is unanswerable, while:

Is the answer to this question ‘yes’?

is answerable (with either answer).

Applying to the question at hand:

Q1: Is Q2 unanswerable?

Q2: Is Q1 answerable?

If we assume that both questions are answerable, then by definition the answer to Q1 must be ‘no’, and the answer to Q2 must be ‘yes’.

Conversely, if the answers are ‘no’ and ‘yes’ respectively, then we conclude from the meaning of the questions that Q2 is answerable and Q1 is answerable.

Hence we have shown that the questions are answerable if and only if the answers are ‘no’ and ‘yes’, respectively.

Draw an and logic table Q1: T and F = F.
T and T = T F and F = F F and T = F Do the same for Q2: Then realise that both Qs wants T and F When you do this it will seem like the answer is wrong however, the questions are reversed make q1 q2 and q2 q1 Hence it is true

I’ve used discrete mathematics for this problem. Let question 1 be p and question 2 be q. p is false because q allows both parameters to be an answer. Since p implies q and p is false, it must be true.

Answer one question: for question 1 say yes. Then that means question 2 works. So it is consistent. I don't know how right I am but yeah

“Valid” <> (“true” OR “false”) “Valid” = (“Yes” OR “No”)