棋盘覆盖问题可视化

棋盘覆盖问题是指在一个大小为2^n * 2^n的棋盘上去掉一个方格后，用L型骨牌（形状类似“日”字）将其覆盖。可以使用可视化工具来演示这个问题的解决过程。以下是一个简单的Python示例代码：

import numpy as np
import matplotlib.pyplot as plt

def plot_board(board):
    # 绘制棋盘
    plt.imshow(board, cmap='binary')
    plt.xticks([])
    plt.yticks([])

def plot_domino(x, y, c):
    # 绘制L型骨牌
    xs = [x, x+1, x+1, x]
    ys = [y, y, y+1, y+1]
    plt.fill(xs, ys, c)

def cover(board, lab, top, left, side):
    # 递归解决棋盘覆盖问题
    if side == 1:
        return
    s = side // 2
    t = lab
    lab += 1
    mid = top + s - 1
    if x < mid and y < mid:
        cover(board, lab, top, left, s)
        plot_domino(left+s-1, top+s-1, 'r')
        board[mid][mid-1] = board[mid-1][mid] = lab
        plot_domino(left+s-1, top+s, 'b')
        cover(board, lab, top, left+s, s)
        plot_domino(left+s, top+s-1, 'b')
        cover(board, lab, top+s, left, s)
        plot_domino(left+s, top+s, 'r')
        cover(board, lab, top+s, left+s, s)
    elif x < mid and y >= mid:
        cover(board, lab, top, left+s, s)
        plot_domino(left+s, top+s-1, 'b')
        board[mid-1][mid-1] = board[mid][mid] = lab
        plot_domino(left+s-1, top+s-1, 'r')
        cover(board, lab, top, left, s)
        plot_domino(left+s-1, top+s, 'r')
        cover(board, lab, top+s, left, s)
        plot_domino(left+s, top+s, 'b')
        cover(board, lab, top+s, left+s, s)
    elif x >= mid and y < mid:
        cover(board, lab, top+s, left, s)
        plot_domino(left+s-1, top+s, 'r')
        board[mid-1][mid-1] = board[mid][mid] = lab
        plot_domino(left+s, top+s-1, 'b')
        cover(board, lab, top, left, s)
        plot_domino(left+s-1, top+s-1, 'b')
        cover(board, lab, top, left+s, s)
        plot_domino(left+s, top+s, 'r')
        cover(board, lab, top+s, left+s, s)
    else:
        cover(board, lab, top+s, left+s, s)
        plot_domino(left+s, top+s, 'r')
        board[mid-1][mid] = board[mid][mid-1] = lab
        plot_domino(left+s-1, top+s-1, 'b')
        cover(board, lab, top, left, s)
        plot_domino(left+s-1, top+s, 'b')
        cover(board, lab, top, left+s, s)
        plot_domino(left+s, top+s-1, 'r')
        cover(board, lab, top+s, left, s)
        lab = t

n = 4
board = np.zeros((2**n, 2**n), dtype=int)
board[3][3] = -1  # 去掉一个方格
plot_board(board)
cover(board, 1, 0, 0, 2**n)
plt.show()

这段代码使用了matplotlib库来绘制棋盘和骨牌。首先使用`plot_board`函数绘制棋盘，然后使用`plot_domino`函数绘制L型骨牌。`cover`函数用递归的方式解决棋盘覆盖问题。在每次递归调用时，根据覆盖区域的位置和大小，决定放置哪个方向的L型骨牌，并用颜色标记不同的骨牌。最后调用`plt.show()`函数显示绘制结果。