Self Referential Quiz

Answer to the set is $(C,A,B,B,A,B,E,B,E,D)$

Define the notations for the numbers $1,2,3,4,5,6,7,8,9,10$ to denote their respective answers, and not their numerical values. For example $5 = 4 = A$ implies Question 4 and Question 5 to both have an answer $A$ and $8 \ne B$ implies that answer for Question 8 is not $B$ .

Let's start with consonants and vowels (Question 8 and 9), because $(a,e,i,o,u)$ are only vowels in the alphabets, $N(\text{consonant}) + N(\text{vowels}) = 10$ , so the solution for $(8,9) = (A,D) \text{ or } (B,E)$ only.

After this, we eliminate the obvious contradictions first:

$6 \ne D$ , $7 \ne C$ , and $1 \ne D$

Because $6 \ne D$ , this implies that there exist at least one of the answers to be $D$ so $6 \ne E \Rightarrow 7 \ne B$

Consider $1 = A \Rightarrow 4 = A \Rightarrow 3 = A \Rightarrow 1 \ne A$ which is absurd. so $1 \ne A$

Consider $1 = B \Rightarrow 2 \ne A \text{ and } 3 = A \Rightarrow 4 = A \Rightarrow 2 = A$ which is aburd. so $1 \ne B$

So, $1 = C \text { or } E$ only

Because $8 = A \text{ or } B$ only, $7 \ne D$

With that $7 = A \text { or } E$ only, so $10 \ne E$

Suppose $7 = A \Rightarrow 5 = E \Rightarrow 9 = C$ which is absurd. So $7 = E$ only. Which gives $9 = E \Rightarrow 8 = B$

This gives $2 \ne E \text{ and } 2 \ne D$ and $5 \ne D \text { and } 5 \ne E$ and $3 \ne D \text { and } 3 \ne E$

Now suppose $3 = A \Rightarrow 4 = A \Rightarrow 2 = A \text{ and } B$ which is absurd, so $3 \ne A$ . Which means $3 = B \text { or } C$ only.

Again, suppose: $3 = C \Rightarrow 6 = A \text { and } 4 \ne A, 5 \ne A$ . Because $6 = A$ with $1 \ne D, 2 \ne D, 3 \ne D, 5 \ne D$ , then $4 = D \Rightarrow 2 \ne B \text{ and } 2 \ne A$ , so $2 = C \Rightarrow 5 = 6 = A \Rightarrow 1 = C \Rightarrow 2 = A \ne A$ , which is absurd. So $3 \ne C \Rightarrow 3 = B$ only.

With $3 = B \Rightarrow 5 = A \Rightarrow 1 = C \Rightarrow 2 = A \Rightarrow 4 = 3 = B$ which restrict the answer of Question 6.

$\Rightarrow 6 \ne A \text { and } 6 \ne C \Rightarrow 6 = B \Rightarrow 10 = D$ , hence the answer as above.

This question can be solved without solving the entire quiz. Consider questions 8 and 9. The answers to both have to add to ten because each question has a solution that is either a vowel or a consonant. That means 8 must be either A or B and 9 must be either D or E . Take a look at 7. A cannot be true because that would mean 9 is C . B cannot be true because then 6 would contradict itself. C cannot be true because that would also be contradictory. D cannot be true because that would mean 8 is E . This leaves E as the final answer. Thus, 9 is E .

Best logical question,I had ever seen. Answer is 1(c),2(a),3(b),4(b),5(a),6(b),7(e),8(c),9(e),10(d)

1C,2A,3B,4B,5A,6B,7E,8C,9E,10D