Pass the Parcel

This problem follows on from Markov Matrices, and shows an application of the Markov Matrix method, applied to a popular childrens party game.  

Students might like to use these Matrix Power calculators:

• Matrix Power Calculator (decimal version)
• Matrix Power Calculator (fraction version)

For spotting the general form of ${\bf M}^n$, the Matrix Power Calculator (fraction version) will probably be more helpful!

The set up of this question has been taken from a STEP question, but it has been adapted to use matrices.  The original STEP question was:

Four children, $A$, $B$, $C$ and $D$, are playing a version of the game `pass the parcel'. They stand in a circle, so that $ABCDA$ is the clockwise order. Each time a whistle is blown, the child holding the parcel is supposed to pass the parcel immediately exactly one place clockwise. In fact each child, independently of any other past event, passes the parcel clockwise with probability $\frac{1}{4}$, passes it anticlockwise with probability $\frac{1}{4}$ and fails to pass it at all with probability $\frac{1}{2}$. At the start of the game, child $A$ is holding the parcel.

The probability that child $A$ is holding the parcel just after the whistle has been blown for the $n$th time is $A_n$, and $B_n$, $C_n$ and $D_n$ are defined similarly.

1. Find $A_1$, $B_1$, $C_1$ and $D_1$. Find also $A_2$, $B_2$, $C_2$ and $D_2$.
    
2. By first considering $B_{n+1}+D_{n+1}$, or otherwise, find $B_n$ and $D_n$.
   Find also expressions for $A_n$ and $C_n$ in terms of $n$.

There are more matrix problems in this feature.