The two perfect logicians $A$ and $C$ meet their colleague $B$ at an exam, and they are not aware of the fact that $B$ is a moron and a horrible mathematician. At the exam, the examiner informs them that he has chosen two different integers $x$ and $y$ with $2\le x<y\le100$, such that $y$ is a multiple of $x$.
The examiner then tells the difference $d=y-x$ to the first mathematician $A$, the ratio $r=y/x$ to the second mathematician $B$, and the sum $s=x+y$ to the third mathematician $C$ in such a way that none knows which numbers have been whispered to the others.
The three mathematicians then start this implying conversation:
- $A$: I don't know the numbers, and I know that you both know this.
- $B$: I already knew the numbers when the examiner told me their ratio.
- $A$ and $C$ simultaneously: Aha! We have just deduced the two numbers.
- $B$: Oops, damn, I am sorry! I think I made a mistake in my calculation. As a matter of fact, the ratio $r$ does not allow me to deduce the two numbers.
Question1: What are those two numbers $x$ and $y$??
Question2: What are those two numbers $x$ and $y$, if the following conversation took place instead from the third line on??
- $C$: Aha! I have just deduced the two numbers.
- $A$: I strongly recommend for $B$ to recalculate his ratio because he was surely wrong according to my data.
For hints and notes, refer to the following links:
Deducing Two Numbers based on their Difference and Ratio
What are the numbers?