# 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?” Jun 21, 2020 at 5:04
• @Damila I am really sorry. I cropped by mistake the last line" So, who is the liar" ? Kindly accept apologies!! Jun 21, 2020 at 5:10
• It’s ok! If you don’t mind, I hid the answer in your question for future readers. Jun 21, 2020 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 Jun 21, 2020 at 8:58
• @melfnt its from previous year papers of an exam. Not from a book. Jun 22, 2020 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.