By Rotman J.J.

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Selecting 1, 2, 3 flags leaves the respective states W (7), W (6), W (5) all of which are winning states. So state 8 is a losing state L(8). On the other hand, states 9, 10, 11 are winning states since there is some choice that leaves the opponent in the losing state L(8). 7, which shows winning and losing states. In particular, n = 21 is a winning state: taking 1 flag will leave the opponent in the losing state L(20). In general, the optimal strategy is to take enough flags to leave the opponent in a losing state: an integer multiple of 4.

Are there several solutions using the minimum number of pourings? (c) Answer the questions in part (b) for measuring out exactly 6 pints. 24. Suppose we have three jugs with capacities of 9, 8, and 2 pints, respectively. We begin with the smaller two jugs completely full, giving a total of 10 pints. We want to divide these 10 pints so that there are two jugs having 4 pints each. That is, we wish to reach the final configuration of (4, 4, 2). Exercises 23 (a) Create a BFS tree for this problem, extended far enough to solve this problem.

Also, each timer can be turned over (flipped) to measure the time already passed. For example, if there are 3 minutes remaining in the 5-minute timer, it can be turned over to measure 2 minutes. So, there are other (solid) edges to indicate flipping each of the timers individually or both at the same time; no elapsed time is associated with such edges. These vertices and edges define a graph that can be searched in a breadth-first manner. 15 shows the BFS tree obtained. As done earlier, we do not show edges that lead to states previously found in the BFS.

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