棋盘覆盖问题可视化
时间: 2023-08-12 15:07:43 浏览: 207
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:
```python
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。
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