# Question which is sort of puzzle from mathematics Exam

I am self studying and trying questions of quantitative aptitude in my mathematics exam and I am unable to solve this problem. Of three persons $$A$$, $$B$$ and $$C$$, one always lies while the others always speak the truth. $$C$$ asked $$A$$, "Do you always speak the truth, yes or no?" He said something that $$C$$ could not hear. So, $$C$$ asked $$B$$. "what did $$A$$ say?" $$B$$ replied, "$$A$$ said No".

So, who is the liar?

$$B$$.

Unfortunately, I am completely clueless on how to approach this question. Any help would be really appreciated!

• It would help if you gave the question to which “B” is the answer. I assume the question is “Who always lies?” – Damila Jun 21 at 5:04
• @Damila I am really sorry. I cropped by mistake the last line" So, who is the liar" ? Kindly accept apologies!! – user795826 Jun 21 at 5:10
• It’s ok! If you don’t mind, I hid the answer in your question for future readers. – Damila Jun 21 at 5:16
• @Damila since I cannot see your edit suggestion, I added the spoiler tag myself. (at)user795826 can you please provide the source of this problem? The title of the book from which this puzzle came from should be enough – melfnt Jun 21 at 8:58
• @melfnt its from previous year papers of an exam. Not from a book. – user795826 Jun 22 at 7:45

If the question posed is "Who always lies?" then the answer is indeed.

$$B$$.

We can approach this problem by noticing that

$$A$$ must have said "yes," yet $$B$$ claimed that $$A$$ said "no."

This can be deduced like this:

$$A$$ is asked "Do you speak the truth?"

- If $$A$$ is truthful, they will truthfully say "yes."
- If $$A$$ always lies, they will lie and say "yes."

Therefore, $$A$$'s answer to the first question is always "yes." Hence, by claiming that $$A$$ said "no," $$B$$ is the liar.