 Hello everyone, this is Alice Gao. In this video, I will go through the process of solving the question in lecture 3 on slide 29. In this question, we relax the 8 puzzle by removing one constraint. Originally, when we move a tile from square A to square B, A and B must be adjacent and B must be empty. In this relaxed problem, the target square B does not need to be empty. The optimal solution to this relaxed problem gives us an admissible heuristic function for the 8 puzzle. The correct answer to this question is A, the Manhattan distance heuristic. Let's solve this relaxed problem. Given any state of the relaxed problem, we need to determine the smallest number of moves to transform the state to a gold state. This number of moves is the heuristic value for the given state. Let's try an example first. Consider this state of the 8 puzzle. What is the minimum number of moves we need to take to change it to a gold state? We need to move each tile to its gold position. This is much easier in this relaxed problem since the target square doesn't need to be empty. For example, I am allowed to move tile 1 up one step to overlap with tile 6. For tile 1, its gold position is the top left square. We need 4 steps to move 1 to its gold position. For example, up, up, left and left. It's fine for 1 to overlap with 6, 3 and 5 along the way since the target square doesn't have to be empty. Let me ask you a question. For each tile that's not in its gold position, how many moves do we need to move it into its gold position? Since we are in a grid, the number of moves is equal to the Manhattan distance between its current position and its gold position. For example, for tile 1 we need 4 moves and that is the Manhattan distance between the bottom right square which is the current position of tile 1 and the top left square which is the gold position of tile 1. For another example, consider tile 2. The current position of tile 2 is bottom left and the gold position of tile 2 is top middle. So we need 3 moves which is the Manhattan distance between the bottom left position and the top middle position. We are effectively calculating the total Manhattan distance between its current position and its gold position for each tile that's not in its gold position. Therefore, the cost of the optimal solution of the relaxed problem must be equal to the Manhattan distance heuristic. The correct answer is A, the Manhattan distance heuristic. Thank you very much for watching. I will see you in the next video. Bye for now.