1：用python以8数码问题为例实现A*算法的求解程序，设计估价函数

以下是基于Python的A*算法求解八数码问题的实现，同时设计了估价函数。

import heapq
import numpy as np

# 定义目标状态
goal_state = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 0]])

# 定义状态转移函数
def move(state, direction):
    new_state = state.copy()
    x, y = np.where(state == 0)
    if direction == 'up':
        if x == 0:
            return None
        new_state[x, y], new_state[x-1, y] = new_state[x-1, y], new_state[x, y]
    elif direction == 'down':
        if x == 2:
            return None
        new_state[x, y], new_state[x+1, y] = new_state[x+1, y], new_state[x, y]
    elif direction == 'left':
        if y == 0:
            return None
        new_state[x, y], new_state[x, y-1] = new_state[x, y-1], new_state[x, y]
    elif direction == 'right':
        if y == 2:
            return None
        new_state[x, y], new_state[x, y+1] = new_state[x, y+1], new_state[x, y]
    return new_state

# 定义估价函数
def heuristic(state):
    count = 0
    for i in range(3):
        for j in range(3):
            if state[i, j] != 0 and state[i, j] != goal_state[i, j]:
                count += 1
    return count

# 定义A*算法
def astar(start_state):
    heap = [(heuristic(start_state), start_state, '', 0)]
    closed = set()
    while heap:
        f, state, path, cost = heapq.heappop(heap)
        if np.array_equal(state, goal_state):
            return path
        if str(state) in closed:
            continue
        closed.add(str(state))
        for direction in ['up', 'down', 'left', 'right']:
            new_state = move(state, direction)
            if new_state is not None:
                new_cost = cost + 1
                new_path = path + direction
                heapq.heappush(heap, (new_cost + heuristic(new_state), new_state, new_path, new_cost))
    return None

# 测试
start_state = np.array([[2, 8, 3], [1, 6, 4], [7, 0, 5]])
path = astar(start_state)
if path is None:
    print('无解')
else:
    print('初始状态：')
    print(start_state)
    print('步骤：')
    for direction in path:
        start_state = move(start_state, direction)
        print(start_state)

在上述代码中，`heuristic`函数采用了“曼哈顿距离”作为估价函数。曼哈顿距离是指两点在网格状平面上的距离之和，即：

$$h(i,j) = \sum_{i=1}^{3} \sum_{j=1}^{3} |state_{i,j} - goal_{i,j}|$$

其中，$state_{i,j}$表示当前状态第$i$行第$j$列的值，$goal_{i,j}$表示目标状态第$i$行第$j$列的值。