# Question 4:

## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Stage 1: state 0, action 0, working 4, minimax 4; state 1, action 0, working 3, minimax 3; state 2, action 0, working 2, minimax 2 | | Values only credited when seen in table |
| Stage 2, state 0: $\max(6,4)=6$, $\max(2,3)=3$, $\max(3,2)=3$, minimax 3 | | |
| Stage 2, state 1: $\max(2,4)=4$, $\max(4,3)=4$, $\max(5,2)=5$, minimax 4 | M1 | For calculating the maxima as 4, 4, 5 |
| | A1 | For calculating the minimax as 4 |
| Stage 2, state 2: $\max(2,4)=4$, $\max(3,3)=3$, $\max(4,2)=4$, minimax 3 | B1 | For completing 4, 3, 2 in the brackets |
| | M1 | For calculating the maxima as 4, 3, 4 (method) |
| | A1 | For calculating the minimax as 3 cao |
| Stage 3, state 0: $\max(5,3)=5$, $\max(5,4)=5$, $\max(2,3)=3$, minimax 3 | B1 | For using their minimax values from stage 2 |
| | M1 | For calculating the maxima for their values |
| | A1 | For calculating the maxima as 5, 5, 3 cao |
| | A1 | For calculating the minimax as 3 cao |

## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Value = 3 | M1 | For the value from their tabulation |
| | A1 | For 3 (irrespective of their tabulation) cao |
| Route: $(0;0) \to (1;1) \to (2;2) \to (3;0)$ (or in reverse) | M1 dep | For reading route from their tabulation |
| | A1 | For this route (irrespective of their tabulation) cao |

## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Graph structure correct with nodes $(2;0), (1;0), (3;0), (2;1), (1;1), (0;0), (2;2), (1;2)$ | B1 | For the graph structure correct |
| Substantially correct attempt at weights (no more than two definite errors or omissions) | M1 | For a substantially correct attempt at the weights |
| Weights unambiguously correct | A1 | For weights unambiguously correct cao |

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