\section*{6. For APPLICANTS IN $\left\{ \begin{array} { l } \text { COMPUTER SCIENCE } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \\ \text { COMPUTER SCIENCE \& PHILOSOPHY } \end{array} \right\}$ ONLY.}
Alice plays a game 5 times with her friends Sam and Pam. In each game Alice chooses two integers $x$ and $y$ with $1 \leqslant x \leqslant y$. She whispers the sum $x + y$ to Sam, and the product $x \times y$ to Pam, so that neither knows what the other was told. Sam and Pam then have to try to work out what the numbers $x$ and $y$ are. Sam and Pam are well known expert logicians.\\
(i) In the first game, Pam says "I know $x$ and $y$."

What can we deduce about the values of $x$ and $y$ ? Explain your answer.\\
(ii) In the second game, Pam says "I don't know what $x$ and $y$ are."

Sam then says "I know $x$ and $y$."\\
Suppose the sum is 4 . What are $x$ and $y$ ? Explain your answer.\\
(iii) In the third game, Pam says "I don't know what $x$ and $y$ are."

Sam then says "I don't know what $x$ and $y$ are."\\
Pam then says "I now know $x$ and $y$."\\
Suppose the product is 4 . What are $x$ and $y$ ? Explain your answer.\\
(iv) In the fourth game, Pam says "I don't know what $x$ and $y$ are."

Sam then says "I already knew that."\\
Pam then says "I now know $x$ and $y$."\\
Suppose the product is 8 . What are $x$ and $y$ ? Explain your answer.\\
(v) Finally, in the fifth game, Pam says "I don't know what $x$ and $y$ are."

Sam then says "I don't know what $x$ and $y$ are."\\
Pam then says "I don't know what $x$ and $y$ are."\\
Sam then says "I now know $x$ and $y$."\\
Suppose the sum is 5 . What are $x$ and $y$ ? Explain your answer.