This is a very early stage exercise from J.B. Fraleigh, so it should be quite elementary, but I can't find a way to complete the Cayley table below in order for $*$ to be associative. I have googled the subject and found Light's method, but it's quite expensive in terms of computation, although not as much as writing down all possibilities. I wonder if there's some way to deduce the missing entries for this particular $*$ by writing down some conditions.
For example, I have the this feeling that this last line could be filled with $d$'s or $c$'s, but I cannot prove it.
$$ \begin{array}{l | cccc} * & a & b & c & d \\ \hline a & a & b & c & d \\ b & b & a & c & d \\ c & c & d & c & d \\ d & & & & \\ \end{array} $$