The world is divided into two species, vampires and werewolves. Vampires always tell the truth when talking about a vampire, but always lie when talking about a werewolf. Werewolves always tell the truth when talking about a werewolf, but always lie when talking about a vampire. (Note that this does not imply that creatures necessarily lie when speaking to creatures of the other species. Note also that "Zaccaria is a vampire" is a statement about Zaccaria, rather than necessarily about a vampire.) These facts are well known to both sides, and creatures can tell instinctively which species an individual belongs to. In your answers to the questions below, you may abbreviate "vampire" and "werewolf" to "V" and "W", respectively. (i) Azrael says, "Beela is a werewolf." Explain why Azrael must be a werewolf, but that we cannot tell anything about Beela. (ii) Cesare says, "Dita says 'Elith is a vampire.'" What can we infer about any of the three from this statement? Explain your answer. (iii) Suppose $N$ creatures (where $N \geq 2$ ) are sitting around a circular table. Each tells their right-hand neighbour, "You lie about your right-hand neighbour." What can we infer about $N$ ? What can we infer about the arrangement of creatures around the table? Explain your answer. (iv) Consider a similar situation to that in part (iii) (possibly for a different value of $N$ ), except that now each tells their right-hand neighbour, "Your right-hand neighbour lies about their right-hand neighbour." Again, what can we infer about $N$ and the arrangement of creatures around the table? Explain your answer. If you require additional space please use the pages at the end of the booklet
(i) [2 marks] If Beela is a werewolf, then the statement is true, so Azrael must be a werewolf too. If Beela is a vampire, then the statement is false, so Azrael again must be a werewolf. Beela could be either a werewolf or a vampire. [0pt]
\section*{6. For APPLICANTS IN $\left\{ \begin{array} { l } \text { COMPUTER SCIENCE } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \\ \text { COMPUTER SCIENCE \& PHILOSOPHY } \end{array} \right\}$ ONLY.}
The world is divided into two species, vampires and werewolves. Vampires always tell the truth when talking about a vampire, but always lie when talking about a werewolf. Werewolves always tell the truth when talking about a werewolf, but always lie when talking about a vampire. (Note that this does not imply that creatures necessarily lie when speaking to creatures of the other species. Note also that "Zaccaria is a vampire" is a statement about Zaccaria, rather than necessarily about a vampire.)
These facts are well known to both sides, and creatures can tell instinctively which species an individual belongs to.
In your answers to the questions below, you may abbreviate "vampire" and "werewolf" to "V" and "W", respectively.\\
(i) Azrael says, "Beela is a werewolf." Explain why Azrael must be a werewolf, but that we cannot tell anything about Beela.\\
(ii) Cesare says, "Dita says 'Elith is a vampire.'" What can we infer about any of the three from this statement? Explain your answer.\\
(iii) Suppose $N$ creatures (where $N \geq 2$ ) are sitting around a circular table. Each tells their right-hand neighbour, "You lie about your right-hand neighbour." What can we infer about $N$ ? What can we infer about the arrangement of creatures around the table? Explain your answer.\\
(iv) Consider a similar situation to that in part (iii) (possibly for a different value of $N$ ), except that now each tells their right-hand neighbour, "Your right-hand neighbour lies about their right-hand neighbour." Again, what can we infer about $N$ and the arrangement of creatures around the table? Explain your answer.
If you require additional space please use the pages at the end of the booklet