Need help with a hard combinatorics problem I can't solve the following combinatorics problem from the european kangaroo competition.
Four cars enter a roundabout at the same time, each one from a different
direction, as shown in the diagram. Each of the cars drives less than a full
round of the roundabout, and no two cars leave the roundabout in the
same direction. How many different ways are there for the cars to leave
the roundabout?
Can anyone suggest an approach to solving problems of this kind?

 A: You need a derangement of the cars.  If you think about lining up the cars in the order they enter, then lining them up in the order they leave, there need not to be any matches among them.
A: Let's answer an equivalent question:
In how many ways can you arrange $\{1,2,3,4\}$ so that no number is at its "natural" position?

The answer is $9$:


*

*$\color{red}1\color{red}2\color{red}3\color{red}4$

*$\color{red}1\color{red}2           4           3$

*$\color{red}1           3           2\color{red}4$

*$\color{red}1           3           4           2$

*$\color{red}1           4           2           3$

*$\color{red}1           4\color{red}3           2$

*$           2           1\color{red}3\color{red}4$

*$           2           1           4           3$ a valid combination

*$           2           3           1\color{red}4$

*$           2           3           4           1$ a valid combination

*$           2           4           1           3$ a valid combination

*$           2           4\color{red}3           1$

*$           3           1           2\color{red}4$

*$           3           1           4           2$ a valid combination

*$           3\color{red}2           1\color{red}4$

*$           3\color{red}2           4           1$

*$           3           4           1           2$ a valid combination

*$           3           4           2           1$ a valid combination

*$           4           1           2           3$ a valid combination

*$           4           1\color{red}3           2$

*$           4\color{red}2           1           3$

*$           4\color{red}2\color{red}3           1$

*$           4           3           1           2$ a valid combination

*$           4           3           2           1$ a valid combination
