2. This question should be answered on the sheet provided.

A pool player is to play in four tournaments. Some of the tournaments take place simultaneously and the player has to choose one of each of the following:

$$\begin{array} { l l } 
1 ^ { \text {st } } \text { tournament: } & A , B \text { or } C , \\
2 ^ { \text {nd } } \text { tournament: } & D , E \text { or } F , \\
3 ^ { \text {rd } } \text { tournament: } & G , H \text { or } I , \\
4 ^ { \text {th } } \text { tournament: } & J , K \text { or } L
\end{array}$$

Each tournament has six rounds and the player estimates how well he will do in each tournament based on which tournament he plays before it. The table below shows his expectations with each number indicating the round he expects to reach and a "7" indicating he expects to win the tournament.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{12}{|c|}{Expected performance in tournament} \\
\hline
 &  & A & B & C & $D$ & E & $F$ & $G$ & $H$ & I & J & $K$ & $L$ \\
\hline
\multirow{10}{*}{Previous tournament} & None & 5 & 3 & 3 &  &  &  &  &  &  &  &  &  \\
\hline
 & A &  &  &  & 6 & 3 & 7 &  &  &  &  &  &  \\
\hline
 & B &  &  &  & 5 & 5 & 4 &  &  &  &  &  &  \\
\hline
 & C &  &  &  & 7 & 5 & 5 &  &  &  &  &  &  \\
\hline
 & D &  &  &  &  &  &  & 5 & 3 & 3 &  &  &  \\
\hline
 & E &  &  &  &  &  &  & 3 & 5 & 6 &  &  &  \\
\hline
 & $F$ &  &  &  &  &  &  & 3 & 6 & 5 &  &  &  \\
\hline
 & G &  &  &  &  &  &  &  &  &  & 2 & 4 & 1 \\
\hline
 & H &  &  &  &  &  &  &  &  &  & 3 & 2 & 2 \\
\hline
 & $I$ &  &  &  &  &  &  &  &  &  & 2 & 5 & 3 \\
\hline
\end{tabular}
\end{center}

He wishes to choose the tournaments such that his worst performance is as good as possible. Use dynamic programming to find which tournaments he should play.\\
(10 marks)

\hfill \mbox{\textit{Edexcel D2  Q2 [10]}}