  For a given number of colours, $s$, the guessing number of a graph is the
base $s$ logarithm of the size of the largest family of colourings of the
vertex set of the graph such that the colour of each vertex can be determined
from the colours of the vertices in its neighbourhood. An upper bound for the
guessing number of the $n$-vertex cycle graph $C_n$ is $n/2$. It is known that
the guessing number equals $n/2$ whenever $n$ is even or $s$ is a perfect
square \cite{Christofides2011guessing}. We show that, for any given integer
$s\geq 2$, if $a$ is the largest factor of $s$ less than or equal to
$\sqrt{s}$, for sufficiently large odd $n$, the guessing number of $C_n$ with
$s$ colours is $(n-1)/2 + \log_s(a)$. This answers a question posed by
Christofides and Markstr\"{o}m in 2011 \cite{Christofides2011guessing}. We also
present an explicit protocol which achieves this bound for every $n$. Linking
this to index coding with side information, we deduce that the information
defect of $C_n$ with $s$ colours is $(n+1)/2 - \log_s(a)$ for sufficiently
large odd $n$. Our results are a generalisation of the $s=2$ case which was
proven in \cite{bar2011index}.
