\section*{6. For APPLICANTS IN $\left\{ \begin{array} { l } \text { COMPUTER SCIENCE } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \\ \text { COMPUTER SCIENCE \& PHILOSOPHY } \end{array} \right\}$ ONLY.}
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Four people $A , B , C , D$ are performing a dance, holding hands in the arrangement shown above. Each dancer is assigned a 1 or a 0 to determine their steps, and there must always be at least a 1 and a 0 in the group of dancers (dancers cannot all dance the same kind of steps). A dancer is off-beat if their assigned number plus the numbers assigned to the people holding hands with them is odd. The entire dance is an off-beat dance if every dancer is off-beat.\\
(i) In how many ways can the four dancers perform an off-beat dance? Explain your answer.

A new dance starts and two more people, $E$ and $F$, join the dance such that each dancer holds hands with their neighbours to form a ring.\\
(ii) In how many ways can the ring of six dancers perform an off-beat dance? Explain your answer.\\
(iii) In a ring of $n$ dancers explain why an off-beat dance can only occur if $n$ is a multiple of 3 .\\
(iv) For a new dance a ring of $n > 4$ dancers, each holds hands with dancers one person away from them round the ring (so $C$ holds hands with $A$ and $E$ and $D$ holds hands with $B$ and $F$ and so on). For which values of $n$ can the dance be off-beat?

On another planet the alien inhabitants have three (extendible) arms and still like to dance according to the rules above.\\
(v) If four aliens dance, each holding hands with each other, how many ways can they perform an off-beat dance?\\
(vi) Six aliens standing in a ring perform a new dance where each alien holds hands with their direct neighbours and the alien opposite them in the ring. In how many ways can they perform an off-beat dance?


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