6 The Devil's Dice are four cubes with faces coloured green, yellow, blue or red.\\
Cube 1 has three green faces and one each of yellow, blue and red.

\begin{itemize}
  \item Two of the green faces are opposite one another.
  \item The other green face is opposite the yellow face.
  \item The blue face is opposite the red face.
\end{itemize}

This information is represented using the graph in Fig. 1.

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Cube 1}
  \includegraphics[alt={},max width=\textwidth]{372c062a-793f-4fb8-a769-957479f5fce7-08_359_330_685_957}
\end{center}
\end{figure}

Fig. 1\\
(i) Cube 2 has a green face opposite a blue face, another green face opposite a red face and a second red face opposite a yellow face. Draw a graph to represent this information.

The graph in Fig. 2 represents opposite faces in cube 3.

Cube 3

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{372c062a-793f-4fb8-a769-957479f5fce7-08_350_326_1398_986}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}

(ii) How many yellow faces does cube 3 have?

Cube 4 has one green face, two yellow faces, one blue face and two red faces. The graph in Fig. 3 is an incomplete representation of opposite faces in cube 4 .

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Cube 4}
  \includegraphics[alt={},max width=\textwidth]{372c062a-793f-4fb8-a769-957479f5fce7-08_257_273_2115_1018}
\end{center}
\end{figure}

Fig. 3\\
(iii) Complete the graph in your answer book.

The Devil's Dice puzzle requires the cubes to be stacked to form a tower so that each long face of the tower uses all four colours.

The puzzle can be solved using graph theory. First the graphs representing the opposite faces of the four cubes are combined into a single graph. The edges of the graph are labelled $1,2,3$ or 4 to show which cube they belong to.

The labelled graph in Fig. 4 shows cube 1 and cube 3 together.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{372c062a-793f-4fb8-a769-957479f5fce7-09_630_689_625_689}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{center}
\end{figure}

(iv) Complete the copy of the labelled graph in your answer book to show all four cubes.

A subgraph is a graph contained within a given graph.\\
From the graph representing all four cubes a subgraph needs to be found that will represent the front and back faces of the tower. Each face of the tower uses each colour once. This means that the graph representing the front and back faces must be a subgraph of the answer to part (iv) with four edges labelled $1,2,3$ and 4 and four nodes each having order two.\\
(v) Explain why if the loop labelled 1 joining G to G is used, it is not possible to form a subgraph with four edges labelled 1, 2, 3 and 4 and nodes each having order two.

Suppose that the edge labelled 1 that joins B and R is used.\\
(vi) Draw a subgraph that has the required properties and uses the edge labelled 1 that joins B and R .\\
(vii) Using your answer to part (vi), show the two possible colourings of the back of the tower.

\hfill \mbox{\textit{OCR D1 2015 Q6 [10]}}