A computer science professor decides to play a game with his brightest (nearly perfectly intelligent) students. He sits them in a circle with their backs to the center such that they cannot see each other but they can hear each other and he can watch them. He hands them each a folded up sheet of paper and says the following:

"I have chosen 2 numbers such that $2\le x \le y \le 100$. I have written the value of $D$ which equals $Y-X$ and given it to Donna. I have written the value of $S$ which equals $Y+X$ and given it to Sarah. I have written the value of $R$ which equals $Y/X$ and given it to Richard. All of these five of these numbers are whole numbers. You are welcome to talk but the first to correctly determine the values of $x$ and $y$ wins. Anyone who says something they could have deduced to be false from what they already know loses. You may begin."

After a few minutes the following exchange occurs:

• Sarah: Well, Richard doesn’t know the numbers.

• Donna: No, he doesn’t.

• Richard: Can I take it from the silence you both don’t know the numbers? … Yes?... Well in that case, I’ll admit I don’t know the numbers. But that admission alone doesn’t provide either of you with any useful information.

• Donna: Wow... Really?... That means Sarah knows the answer. I still don't though.

• Sarah: You must have made a mistake Donna; I didn't know the numbers.

• Richard: Neither of you know the answer; but I do and I never even unfolded my paper.

Both Sarah and Donna assumed Richard knew the value of $R$, Richard knew that they would assume that, and no one made any other mistakes. What are the numbers?

Hint 1:

If Abby says “Bob ate the pie” it can reasonably be translated to “According to Abby's calculations, assumptions, and aquired data: Bob ate the pie and Abby knows Bob ate the pie.”

Hint 2:

When Richard says that the other two couldn’t figure out the numbers during the long pause or they would have already said them, he is correct and the others recognize it. At no other time should you use the assumption “she would say she knew if she knew”.

Hint 3:

Richard’s bolded line would have been interpreted by Sarah and Donna to mean “Richard knows we both already knew Richard didn’t know.” Richard and the professor know it means “I don’t have any new information to give as I don’t know R or any information you haven’t already been told.”

A computer science professor decides to play a game with his three brightest (nearly perfectly intelligent) students. He sits them in a circle with their backs to the center such that they cannot see each other but they can hear each other and he can watch them. He hands them each a folded up sheet of paper and says the following:

"I have chosen two numbers $X$ and $Y$, such that $2\le X \le Y \le 100$. I have written the value of $D$ which equals $Y-X$ and given it to Donna. I have written the value of $S$ which equals $Y+X$ and given it to Sarah. I have written the value of $R$ which equals $Y/X$ and given it to Richard. All of these five of these numbers are whole numbers. You are welcome to talk but the first to correctly determine the values of $X$ and $Y$ wins. Anyone who says something they could have deduced to be false from what they already know loses. You may begin."

After a few minutes the following exchange occurs:

• Sarah: Well, Richard doesn’t know the numbers.

• Donna: No, he doesn’t.

• Richard: Can I take it from the silence you both don’t know the numbers? … Yes?... Well in that case, I’ll admit I don’t know the numbers. But that admission alone doesn’t provide either of you with any useful information.

• Donna: Wow... Really?... That means Sarah knows the answer. I still don't though.

• Sarah: You must have made a mistake Donna; I didn't know the numbers.

• Richard: Neither of you knows the answer; but I do and I never even unfolded my paper.

Both Sarah and Donna assumed Richard knew the value of $R$, Richard knew that they would assume that, and no one made any other mistakes. What are the numbers?

Hint 1:

If Abby says “Bob ate the pie” it can reasonably be translated to “According to Abby's calculations, assumptions, and aquired data: Bob ate the pie and Abby knows Bob ate the pie.”

Hint 2:

When Richard says that the other two couldn’t figure out the numbers during the long pause or they would have already said them, he is correct and the others recognize it. At no other time should you use the assumption “she would say she knew if she knew”.

Hint 3:

Richard’s bolded line would have been interpreted by Sarah and Donna to mean “Richard knows we both already knew Richard didn’t know.” Richard and the professor know it means “I don’t have any new information to give as I don’t know R or any information you haven’t already been told.”

A computer science professor decides to play a game with his brightest (nearly perfectly intelligent) students. He sits them in a circle with their backs to the center such that they cannot see each other but they can hear each other and he can watch them. He hands them each a folded up sheet of paper and says the following:

"I have chosen 2 numbers such that $2\le x< y \le 100$. I have written the value of $D$ which equals $Y-X$ and given it to Donna. I have written the value of $S$ which equals $Y+X$ and given it to Sarah. I have written the value of $R$ which equals $Y/X$ and given it to Richard. All of these five of these numbers are natural numbers. You are welcome to talk but the first to correctly determine the values of $x$ and $y$ wins. Anyone who says something they could have deduced to be false from what they already know loses. You may begin."

After a few minutes the following exchange occurs:

• Sarah: Well, Richard doesn’t know the numbers.

• Donna: No, he doesn’t.

• Richard: Can I take it from the silence you both don’t know the numbers? … Yes?... Well in that case, I’ll admit I don’t know the numbers. But that admission alone doesn’t provide either of you with any useful information.

• Donna: Wow... Really?... That means Sarah knows the answer. I still don't though.

• Sarah: You must have made a mistake Donna; I didn't know the numbers.

• Richard: Neither of you know the answer; but I do and I never even unfolded my paper.

Both Sarah and Donna assumed Richard knew the value of $R$, Richard knew that they would assume that, and no one made any other mistakes. What are the numbers?

Hint 1:

If Abby says “Bob ate the pie” it can reasonably be translated to “According to Abby's calculations, assumptions, and aquired data: Bob ate the pie and Abby knows Bob ate the pie.”

Hint 2:

When Richard says that the other two couldn’t figure out the numbers during the long pause or they would have already said them, he is correct and the others recognize it. At no other time should you use the assumption “she would say she knew if she knew”.

Hint 3:

Richard’s bolded line would have been interpreted by Sarah and Donna to mean “Richard knows we both already knew Richard didn’t know.” Richard and the professor know it means “I don’t have any new information to give as I don’t know R or any information you haven’t already been told.”

A computer science professor decides to play a game with his brightest (nearly perfectly intelligent) students. He sits them in a circle with their backs to the center such that they cannot see each other but they can hear each other and he can watch them. He hands them each a folded up sheet of paper and says the following:

"I have chosen 2 numbers such that $2\le x \le y \le 100$. I have written the value of $D$ which equals $Y-X$ and given it to Donna. I have written the value of $S$ which equals $Y+X$ and given it to Sarah. I have written the value of $R$ which equals $Y/X$ and given it to Richard. All of these five of these numbers are whole numbers. You are welcome to talk but the first to correctly determine the values of $x$ and $y$ wins. Anyone who says something they could have deduced to be false from what they already know loses. You may begin."

After a few minutes the following exchange occurs:

• Sarah: Well, Richard doesn’t know the numbers.

• Donna: No, he doesn’t.

• Richard: Can I take it from the silence you both don’t know the numbers? … Yes?... Well in that case, I’ll admit I don’t know the numbers. But that admission alone doesn’t provide either of you with any useful information.

• Donna: Wow... Really?... That means Sarah knows the answer. I still don't though.

• Sarah: You must have made a mistake Donna; I didn't know the numbers.

• Richard: Neither of you know the answer; but I do and I never even unfolded my paper.

Both Sarah and Donna assumed Richard knew the value of $R$, Richard knew that they would assume that, and no one made any other mistakes. What are the numbers?

Hint 1:

If Abby says “Bob ate the pie” it can reasonably be translated to “According to Abby's calculations, assumptions, and aquired data: Bob ate the pie and Abby knows Bob ate the pie.”

Hint 2:

When Richard says that the other two couldn’t figure out the numbers during the long pause or they would have already said them, he is correct and the others recognize it. At no other time should you use the assumption “she would say she knew if she knew”.

Hint 3:

Richard’s bolded line would have been interpreted by Sarah and Donna to mean “Richard knows we both already knew Richard didn’t know.” Richard and the professor know it means “I don’t have any new information to give as I don’t know R or any information you haven’t already been told.”