$[7$ marks$]$ I want to pave a $2n$ rectangle with

$12$ blocks which come in two colours, white and grey.

Let $w_n$ be the number of different ways this can be

done. For example, it can be shown that $w_8=8704$

and the diagram at the right displays one of these

pavings.

Find a recursive equation for $w_n$ and provide enough initial conditions so that the values of the $w_n$

will be determined by the recursion. Use the values of $w_1$ and $w_2$ and the recursive equation to

find the value of $w_3$ and then check by direct enumeration of the pavings of a $23$ rectangle.

$[$Note: $w_3$ is large enough that you do not want to present all of the pavings. But you can classify

them into different types and enumerate the pavings of each type easily enough. That is$,$ it's worth

spending some time thinking about presentation.$]$