In Questions 6-10 of Sheet 3 the strategy is to use the necessary conditions of PMP to synthesize the optimal solution. What I mean by "synthesize" is that you write down all the information that PMP says is necessary (in terms of $u(t)$ maximizing, $H(x,u,\lambda)$ to $-\lambda_0(t)$, and $\dot \lambda_i=-\partial H/\partial x_i$, and any boundary and transversality conditions) until you convince yourself that the optimal policy can only be one thing. Question 7 is a bit like proving the shortest distance between two points is a straight line. The monster's optimal strategy is fairly obvious, but we can prove it is optimal by applying the necessary conditions of PMP to rule out the possibility of any other solution.
