 Hello everyone, this is Alice Gal. In this video, I will go through the process of solving the question in lecture 3 on slide 33. This question asks us to compare two heuristic functions for the eight puzzle, using the concept of the dominating heuristic. Does one function dominate the other one, or does neither dominate the other one? The correct answer is A. The Manhattan distance heuristic dominates the misplaced tile heuristic. To verify this relationship, we need to verify the two conditions in the definition of dominating heuristic. Condition 1. We need to show that, for every state, the Manhattan distance heuristic value is greater than or equal to the misplaced tile heuristic value. Let's consider the state on the right as an example. Consider a tile that's not in its goal position. Since the tile is not in its goal position, the misplaced tile heuristic will add 1 to the heuristic value for this tile. What about the Manhattan distance heuristic? The Manhattan distance heuristic will add at least 1 to the heuristic value for this tile. If the current position of the tile and its goal position are adjacent, then the Manhattan distance heuristic will add exactly 1 to the heuristic value. Otherwise, the Manhattan distance heuristic will add more than 1 to the heuristic value. For example, for tile 3, the Manhattan distance heuristic adds 1 to the heuristic value. For tile 1, the Manhattan distance heuristic adds 4 to the heuristic value. Therefore, for every state, the Manhattan distance heuristic value has to be greater than or equal to the misplaced tile heuristic value. Remember that it's possible for the two heuristic values to be the same. Consider the state on the left. Only one tile, tile 8, is out of place, and its current position and its goal position are adjacent. For this state, the values of both heuristic functions are 1. Next, we need to verify the second condition. We need to find a state such that the Manhattan distance heuristic produces a strictly higher value than the misplaced tile heuristic. We already have an example, the state on the right. The Manhattan distance heuristic value is 16, whereas the misplaced tile heuristic value is 7. Therefore, there exists one state such that the Manhattan distance heuristic produces a strictly higher value than the misplaced tile heuristic. Again, the correct answer is A. The Manhattan distance heuristic dominates the misplaced tile heuristic. Thank you very much for watching. I will see you in the next video. Bye for now.