我仍然很擅长编码，我正在尝试稍微更难的主题，例如为数独修改递归回溯程序的解决方案。原始解决方案适用于大小为3x3的数独游戏，我希望我能使用正常大小的数独游戏（9x9）。找到3x3解决方案here。
我觉得我对算法的理解非常好：对于网格中的每个列表（保存该单元格的可能值），在每一步尝试每个数字，确保板仍然有效，移到下一个列表，分配一个可能的数字，直到其有效等等，回溯到最后一块仍然有效的板，如果当前的板不正确。
编辑：添加了一些函数来缩小每个空单元格的可能值，以及填充只有一个可能值的单元格。我已经验证这些功能是否正常工作。
仅在初始化数独时调用fill_zeros。 make_nodes构建了董事会的最新节点。 node_to_board以节点格式接收板并以行格式返回板。 refine_by函数不言自明。
def make_nodes(board):
nodes = [[board[0][0], board[0][1], board[0][2],
board[1][0], board[1][1], board[1][2],
board[2][0], board[2][1], board[2][2]],
[board[0][3], board[0][4], board[0][5],
board[1][3], board[1][4], board[1][5],
board[2][3], board[2][4], board[2][5]],
[board[0][6], board[0][7], board[0][8],
board[1][6], board[1][7], board[1][8],
board[2][6], board[2][7], board[2][8]],
[board[3][0], board[3][1], board[3][2],
board[4][0], board[4][1], board[4][2],
board[5][0], board[5][1], board[5][2]],
[board[3][3], board[3][4], board[3][5],
board[4][3], board[4][4], board[4][5],
board[5][3], board[5][4], board[5][5]],
[board[3][6], board[3][7], board[3][8],
board[4][6], board[4][7], board[4][8],
board[5][6], board[5][7], board[5][8]],
[board[6][0], board[6][1], board[6][2],
board[7][0], board[7][1], board[7][2],
board[8][0], board[8][1], board[8][2]],
[board[6][3], board[6][4], board[6][5],
board[7][3], board[7][4], board[7][5],
board[8][3], board[8][4], board[8][5]],
[board[6][6], board[6][7], board[6][8],
board[7][6], board[7][7], board[7][8],
board[8][6], board[8][7], board[8][8]]
]
return nodes
def fill_zeros(board):
nodes = make_nodes(board)
allnums = [1, 2, 3, 4, 5, 6, 7, 8, 9]
for i in range(9):
for j in range(9):
possible = [h for h in allnums if h not in nodes[i]]
if nodes[i][j] == 0:
nodes[i][j] = possible
return nodes
def node_to_board(nodes):
board = []
for y in range(3):
for i in range(0,9, 3):
for node in nodes[(y*3):((y+1)*3)]:
for x in range(3):
board.append(node[i+x])
board = [board[pos:pos + 9] for pos in range(0, 9 * 9, 9)]
return board
def refine_empty_by_col(board):
for col in range(9):
col_done = []
for row in range(9):
if type(board[row][col]) == int:
col_done.append(board[row][col])
for row in range(9):
if type(board[row][col]) == list:
board[row][col] = [x for x in board[row][col] if x not in col_done]
for row in range(9):
if type(board[row][col]) == list:
if len(board[row][col]) == 1:
k = board[row][col].pop()
board[row][col] = k
return board
def refine_empty_by_row(board):
for row in range(9):
row_done = []
for col in range(9):
if type(board[row][col]) == int:
row_done.append(board[row][col])
for col in range(9):
if type(board[row][col]) == list:
board[row][col] = [x for x in board[row][col] if x not in row_done]
for col in range(9):
if type(board[row][col]) == list:
if len(board[row][col]) == 1:
k = board[row][col].pop()
board[row][col] = k
return board
def refine_empty_by_node(board):
nodes = make_nodes(board)
allnums = [1, 2, 3, 4, 5, 6, 7, 8, 9]
for node in range(9):
taken = [x for x in allnums if x in nodes[node]]
for item in range(9):
if type(nodes[node][item]) == list:
nodes[node][item] = [x for x in nodes[node][item] if x not in taken]
if len(nodes[node][item]) == 1:
k = nodes[node][item].pop()
taken.append(k)
nodes[node][item] = k
node_list_nums = []
for item in range(9):
if type(nodes[node][item]) == list:
node_list_nums += nodes[node][item]
node_list_nums = [x for x in node_list_nums if node_list_nums.count(x) == 1]
for item in range(9):
if type(nodes[node][item]) == list:
if any(nodes[node][item]) in node_list_nums:
for x in node_list_nums:
if x in nodes[node][item]:
nodes[node][item] = x
node_list_nums.remove(x)
board = node_to_board(nodes)
return board
def refine_easy_style(board):
board_after = board
board_orig = []
while board_orig != board_after:
board_orig = board_after
board_after = refine_empty_by_col(board_orig)
board_after = refine_empty_by_row(board_after)
board_after = refine_empty_by_node(board_after)
return board_after
以下是（编辑过的）辅助功能：
def is_distinct(passed):
used = []
for item in passed:
if type(item) == list:
continue
if item in used:
return False
used.append(item)
return True
def is_valid(board):
for i in range(9):
row = [board[i][row] for row in range(9)]
if not is_distinct(row):
return False
col = [board[row][i] for row in range(9)]
if not is_distinct(col):
return False
for node in make_nodes(board):
if not is_distinct(node):
return False
return True
is_distinct传递一个列表，它会循环通过以确保每个项目都是唯一的（return True），否则返回False。
is_valid包含电路板的细分部分：行，列和节点（每个3x3平方）。然后它将每块板子传递给is_distinct。
（编辑过的）递归函数是
def solve_puzzle(board):
empties = 0
for row in board:
for item in row:
if type(item) == list:
empties += 1
if empties == 0:
if is_valid(board):
return board
else:
print('0 empties')
return False
for i in range(9):
for j in range(9):
if type(board[i][j]) == list:
board2 = board
for k in board2[i][j]:
print('trying = {}, cell-list ={}, coords = {},{}'.format(k, board2[i][j], i, j))
board2[i][j] = k
if is_valid(board2) and solve_puzzle(board2):
return True
else:
print('{} at slot {}, {} didnt work.'.format(k, i , j))
print('Backtracking here...')
board2[i][j] = board[i][j]
return False
设置是：
#below this line works
inp = [int(x) for x in input().split()]
empties = inp.count(0)
board = [inp[pos:pos+9] for pos in range(0, 9*9, 9)]
board = node_to_board(fill_zeros(board))
print(board)
board = refine_easy_style(board)
print(board)
#above this line works
# below this line is testing
board = solve_puzzle(board)
print(board)
编辑：根据Prune的建议，我投入了几个打印调用来输出一些有用的信息。在跟踪输出后，我认为问题在于程序在运行每个可能的输出后如何回溯。例如：在输出中，电路板有效直到达到（3,8），然后它会通过输出而不是回溯到较早的单元，而是移动到下一个空（4,1）。此外，看起来回溯可能会走得太远;在输出结束时，它表示即使第一个猜测（1）是正确值，它也会在（0,1）处尝试新值。
我试过看别人＆＃39;类似风格的数独解决方案，但似乎无法弄清楚我哪里出错了。
编辑：示例：3 0 6 5 0 8 4 0 0 5 2 0 0 0 0 0 0 0 0 8 7 0 0 0 0 3 1 0 0 3 0 1 0 0 8 0 9 0 0 8 6 3 0 0 5 0 5 0 0 9 0 6 0 0 1 3 0 0 0 0 2 5 0 0 0 0 0 0 0 0 7 4 0 0 5 2 0 6 3 0 0
示例：
[[3, [1, 4, 9], 6, 5, [1, 2, 3, 4, 6, 7, 9], 8, 4, [2, 5, 6, 7, 8, 9], [2, 5, 6, 7, 8, 9]], [5, 2, [1, 4, 9], [1, 2, 3, 4, 6, 7, 9], [1, 2, 3, 4, 6, 7, 9], [1, 2, 3, 4, 6, 7, 9], [2, 5, 6, 7, 8, 9], [2, 5, 6, 7, 8, 9], [2, 5, 6, 7, 8, 9]], [[1, 4, 9], 8, 7, [1, 2, 3, 4, 6, 7, 9], [1, 2, 3, 4, 6, 7, 9], [1, 2, 3, 4, 6, 7, 9], [2, 5, 6, 7, 8, 9], 3, 1], [[1, 2, 4, 6, 7, 8], [1, 2, 4, 6, 7, 8], 3, [2, 4, 5, 7], 1, [2, 4, 5, 7], [1, 2, 3, 4, 7, 9], 8, [1, 2, 3, 4, 7, 9]], [9, [1, 2, 4, 6, 7, 8], [1, 2, 4, 6, 7, 8], 8, 6, 3, [1, 2, 3, 4, 7, 9], [1, 2, 3, 4, 7, 9], 5], [[1, 2, 4, 6, 7, 8], 5, [1, 2, 4, 6, 7, 8], [2, 4, 5, 7], 9, [2, 4, 5, 7], 6, [1, 2, 3, 4, 7, 9], [1, 2, 3, 4, 7, 9]], [1, 3, [2, 4, 6, 7, 8, 9], [1, 3, 4, 5, 7, 8, 9], [1, 3, 4, 5, 7, 8, 9], [1, 3, 4, 5, 7, 8, 9], 2, 5, [1, 6, 8, 9]], [[2, 4, 6, 7, 8, 9], [2, 4, 6, 7, 8, 9], [2, 4, 6, 7, 8, 9], [1, 3, 4, 5, 7, 8, 9], [1, 3, 4, 5, 7, 8, 9], [1, 3, 4, 5, 7, 8, 9], [1, 6, 8, 9], 7, 4], [[2, 4, 6, 7, 8, 9], [2, 4, 6, 7, 8, 9], 5, 2, [1, 3, 4, 5, 7, 8, 9], 6, 3, [1, 6, 8, 9], [1, 6, 8, 9]]]
[[3, [1, 9], 6, 5, 7, 8, 4, [2, 9], [2, 9]], [5, 2, [1, 9], [1, 3, 4], [3, 4], [1, 4], [7, 8, 9], [6, 9], [6, 7, 8, 9]], [4, 8, 7, 6, 2, 9, 5, 3, 1], [[2, 6, 7], [4, 6, 7], 3, [4, 7], 1, [2, 4, 5, 7], [7, 9], 8, [2, 7, 9]], [9, [1, 4, 7], [1, 2, 4], 8, 6, 3, [1, 7], [1, 2, 4], 5], [[2, 7, 8], 5, [1, 2, 4, 8], [4, 7], 9, [2, 4, 7], 6, [1, 2, 4], [2, 3, 7]], [1, 3, [4, 8, 9], [4, 7, 9], [4, 8], [4, 7], 2, 5, [6, 8, 9]], [[2, 6, 8], [6, 9], [2, 8, 9], [1, 3, 9], [3, 5, 8], [1, 5], [1, 8, 9], 7, 4], [[7, 8], [4, 7, 9], 5, 2, [4, 8], 6, 3, [1, 9], [8, 9]]]
trying = 1, board-list =[1, 9], coords = 0,1
trying = 2, board-list =[2, 9], coords = 0,7
trying = 2, board-list =[2, 9], coords = 0,8
2 at slot 0, 8 didnt work.
Backtracking here...
trying = 9, board-list =2, coords = 0,8
trying = 1, board-list =[1, 9], coords = 1,2
1 at slot 1, 2 didnt work.
Backtracking here...
trying = 9, board-list =1, coords = 1,2
trying = 1, board-list =[1, 3, 4], coords = 1,3
trying = 3, board-list =[3, 4], coords = 1,4
trying = 1, board-list =[1, 4], coords = 1,5
1 at slot 1, 5 didnt work.
Backtracking here...
trying = 4, board-list =1, coords = 1,5
trying = 7, board-list =[7, 8, 9], coords = 1,6
trying = 6, board-list =[6, 9], coords = 1,7
trying = 6, board-list =[6, 7, 8, 9], coords = 1,8
6 at slot 1, 8 didnt work.
Backtracking here...
trying = 7, board-list =6, coords = 1,8
7 at slot 1, 8 didnt work.
Backtracking here...
trying = 8, board-list =7, coords = 1,8
trying = 2, board-list =[2, 6, 7], coords = 3,0
trying = 4, board-list =[4, 6, 7], coords = 3,1
trying = 4, board-list =[4, 7], coords = 3,3
4 at slot 3, 3 didnt work.
Backtracking here...
trying = 7, board-list =4, coords = 3,3
trying = 2, board-list =[2, 4, 5, 7], coords = 3,5
2 at slot 3, 5 didnt work.
Backtracking here...
trying = 4, board-list =2, coords = 3,5
4 at slot 3, 5 didnt work.
Backtracking here...
trying = 5, board-list =4, coords = 3,5
trying = 7, board-list =[7, 9], coords = 3,6
7 at slot 3, 6 didnt work.
Backtracking here...
trying = 9, board-list =7, coords = 3,6
trying = 2, board-list =[2, 7, 9], coords = 3,8
2 at slot 3, 8 didnt work.
Backtracking here...
trying = 7, board-list =2, coords = 3,8
7 at slot 3, 8 didnt work.
Backtracking here...
trying = 9, board-list =7, coords = 3,8
9 at slot 3, 8 didnt work.
Backtracking here...
trying = 1, board-list =[1, 4, 7], coords = 4,1
1 at slot 4, 1 didnt work.
Backtracking here...
trying = 4, board-list =1, coords = 4,1
4 at slot 4, 1 didnt work.
Backtracking here...
trying = 7, board-list =4, coords = 4,1
7 at slot 4, 1 didnt work.
Backtracking here...
trying = 1, board-list =[1, 2, 4], coords = 4,2
1 at slot 4, 2 didnt work.
Backtracking here...
trying = 2, board-list =1, coords = 4,2
2 at slot 4, 2 didnt work.
Backtracking here...
trying = 4, board-list =2, coords = 4,2
4 at slot 4, 2 didnt work.
Backtracking here...
trying = 1, board-list =[1, 7], coords = 4,6
1 at slot 4, 6 didnt work.
Backtracking here...
trying = 7, board-list =1, coords = 4,6
7 at slot 4, 6 didnt work.
Backtracking here...
trying = 1, board-list =[1, 2, 4], coords = 4,7
1 at slot 4, 7 didnt work.
Backtracking here...
trying = 2, board-list =1, coords = 4,7
2 at slot 4, 7 didnt work.
Backtracking here...
trying = 4, board-list =2, coords = 4,7
4 at slot 4, 7 didnt work.
Backtracking here...
trying = 2, board-list =[2, 7, 8], coords = 5,0
2 at slot 5, 0 didnt work.
Backtracking here...
trying = 7, board-list =2, coords = 5,0
7 at slot 5, 0 didnt work.
Backtracking here...
trying = 8, board-list =7, coords = 5,0
8 at slot 5, 0 didnt work.
Backtracking here...
trying = 1, board-list =[1, 2, 4, 8], coords = 5,2
1 at slot 5, 2 didnt work.
Backtracking here...
trying = 2, board-list =1, coords = 5,2
2 at slot 5, 2 didnt work.
Backtracking here...
trying = 4, board-list =2, coords = 5,2
4 at slot 5, 2 didnt work.
Backtracking here...
trying = 8, board-list =4, coords = 5,2
8 at slot 5, 2 didnt work.
Backtracking here...
trying = 4, board-list =[4, 7], coords = 5,3
4 at slot 5, 3 didnt work.
Backtracking here...
trying = 7, board-list =4, coords = 5,3
7 at slot 5, 3 didnt work.
Backtracking here...
trying = 2, board-list =[2, 4, 7], coords = 5,5
2 at slot 5, 5 didnt work.
Backtracking here...
trying = 4, board-list =2, coords = 5,5
4 at slot 5, 5 didnt work.
Backtracking here...
trying = 7, board-list =4, coords = 5,5
7 at slot 5, 5 didnt work.
Backtracking here...
trying = 1, board-list =[1, 2, 4], coords = 5,7
1 at slot 5, 7 didnt work.
Backtracking here...
trying = 2, board-list =1, coords = 5,7
2 at slot 5, 7 didnt work.
Backtracking here...
trying = 4, board-list =2, coords = 5,7
4 at slot 5, 7 didnt work.
Backtracking here...
trying = 2, board-list =[2, 3, 7], coords = 5,8
2 at slot 5, 8 didnt work.
Backtracking here...
trying = 3, board-list =2, coords = 5,8
3 at slot 5, 8 didnt work.
Backtracking here...
trying = 7, board-list =3, coords = 5,8
7 at slot 5, 8 didnt work.
Backtracking here...
trying = 4, board-list =[4, 8, 9], coords = 6,2
4 at slot 6, 2 didnt work.
Backtracking here...
trying = 8, board-list =4, coords = 6,2
8 at slot 6, 2 didnt work.
Backtracking here...
trying = 9, board-list =8, coords = 6,2
9 at slot 6, 2 didnt work.
Backtracking here...
trying = 4, board-list =[4, 7, 9], coords = 6,3
4 at slot 6, 3 didnt work.
Backtracking here...
trying = 7, board-list =4, coords = 6,3
7 at slot 6, 3 didnt work.
Backtracking here...
trying = 9, board-list =7, coords = 6,3
9 at slot 6, 3 didnt work.
Backtracking here...
trying = 4, board-list =[4, 8], coords = 6,4
4 at slot 6, 4 didnt work.
Backtracking here...
trying = 8, board-list =4, coords = 6,4
8 at slot 6, 4 didnt work.
Backtracking here...
trying = 4, board-list =[4, 7], coords = 6,5
4 at slot 6, 5 didnt work.
Backtracking here...
trying = 7, board-list =4, coords = 6,5
7 at slot 6, 5 didnt work.
Backtracking here...
trying = 6, board-list =[6, 8, 9], coords = 6,8
6 at slot 6, 8 didnt work.
Backtracking here...
trying = 8, board-list =6, coords = 6,8
8 at slot 6, 8 didnt work.
Backtracking here...
trying = 9, board-list =8, coords = 6,8
9 at slot 6, 8 didnt work.
Backtracking here...
trying = 2, board-list =[2, 6, 8], coords = 7,0
2 at slot 7, 0 didnt work.
Backtracking here...
trying = 6, board-list =2, coords = 7,0
6 at slot 7, 0 didnt work.
Backtracking here...
trying = 8, board-list =6, coords = 7,0
8 at slot 7, 0 didnt work.
Backtracking here...
trying = 6, board-list =[6, 9], coords = 7,1
6 at slot 7, 1 didnt work.
Backtracking here...
trying = 9, board-list =6, coords = 7,1
9 at slot 7, 1 didnt work.
Backtracking here...
trying = 2, board-list =[2, 8, 9], coords = 7,2
2 at slot 7, 2 didnt work.
Backtracking here...
trying = 8, board-list =2, coords = 7,2
8 at slot 7, 2 didnt work.
Backtracking here...
trying = 9, board-list =8, coords = 7,2
9 at slot 7, 2 didnt work.
Backtracking here...
trying = 1, board-list =[1, 3, 9], coords = 7,3
1 at slot 7, 3 didnt work.
Backtracking here...
trying = 3, board-list =1, coords = 7,3
3 at slot 7, 3 didnt work.
Backtracking here...
trying = 9, board-list =3, coords = 7,3
9 at slot 7, 3 didnt work.
Backtracking here...
trying = 3, board-list =[3, 5, 8], coords = 7,4
3 at slot 7, 4 didnt work.
Backtracking here...
trying = 5, board-list =3, coords = 7,4
5 at slot 7, 4 didnt work.
Backtracking here...
trying = 8, board-list =5, coords = 7,4
8 at slot 7, 4 didnt work.
Backtracking here...
trying = 1, board-list =[1, 5], coords = 7,5
1 at slot 7, 5 didnt work.
Backtracking here...
trying = 5, board-list =1, coords = 7,5
5 at slot 7, 5 didnt work.
Backtracking here...
trying = 1, board-list =[1, 8, 9], coords = 7,6
1 at slot 7, 6 didnt work.
Backtracking here...
trying = 8, board-list =1, coords = 7,6
8 at slot 7, 6 didnt work.
Backtracking here...
trying = 9, board-list =8, coords = 7,6
9 at slot 7, 6 didnt work.
Backtracking here...
trying = 7, board-list =[7, 8], coords = 8,0
7 at slot 8, 0 didnt work.
Backtracking here...
trying = 8, board-list =7, coords = 8,0
8 at slot 8, 0 didnt work.
Backtracking here...
trying = 4, board-list =[4, 7, 9], coords = 8,1
4 at slot 8, 1 didnt work.
Backtracking here...
trying = 7, board-list =4, coords = 8,1
7 at slot 8, 1 didnt work.
Backtracking here...
trying = 9, board-list =7, coords = 8,1
9 at slot 8, 1 didnt work.
Backtracking here...
trying = 4, board-list =[4, 8], coords = 8,4
4 at slot 8, 4 didnt work.
Backtracking here...
trying = 8, board-list =4, coords = 8,4
8 at slot 8, 4 didnt work.
Backtracking here...
trying = 1, board-list =[1, 9], coords = 8,7
1 at slot 8, 7 didnt work.
Backtracking here...
trying = 9, board-list =1, coords = 8,7
9 at slot 8, 7 didnt work.
Backtracking here...
trying = 8, board-list =[8, 9], coords = 8,8
8 at slot 8, 8 didnt work.
Backtracking here...
trying = 9, board-list =8, coords = 8,8
9 at slot 8, 8 didnt work.
Backtracking here...
9 at slot 3, 6 didnt work.
Backtracking here...
5 at slot 3, 5 didnt work.
Backtracking here...
trying = 7, board-list =5, coords = 3,5
7 at slot 3, 5 didnt work.
Backtracking here...
7 at slot 3, 3 didnt work.
Backtracking here...
4 at slot 3, 1 didnt work.
Backtracking here...
trying = 6, board-list =4, coords = 3,1
6 at slot 3, 1 didnt work.
Backtracking here...
trying = 7, board-list =6, coords = 3,1
7 at slot 3, 1 didnt work.
Backtracking here...
2 at slot 3, 0 didnt work.
Backtracking here...
trying = 6, board-list =2, coords = 3,0
6 at slot 3, 0 didnt work.
Backtracking here...
trying = 7, board-list =6, coords = 3,0
7 at slot 3, 0 didnt work.
Backtracking here...
8 at slot 1, 8 didnt work.
Backtracking here...
trying = 9, board-list =8, coords = 1,8
9 at slot 1, 8 didnt work.
Backtracking here...
6 at slot 1, 7 didnt work.
Backtracking here...
trying = 9, board-list =6, coords = 1,7
9 at slot 1, 7 didnt work.
Backtracking here...
7 at slot 1, 6 didnt work.
Backtracking here...
trying = 8, board-list =7, coords = 1,6
8 at slot 1, 6 didnt work.
Backtracking here...
trying = 9, board-list =8, coords = 1,6
9 at slot 1, 6 didnt work.
Backtracking here...
4 at slot 1, 5 didnt work.
Backtracking here...
3 at slot 1, 4 didnt work.
Backtracking here...
trying = 4, board-list =3, coords = 1,4
4 at slot 1, 4 didnt work.
Backtracking here...
1 at slot 1, 3 didnt work.
Backtracking here...
trying = 3, board-list =1, coords = 1,3
3 at slot 1, 3 didnt work.
Backtracking here...
trying = 4, board-list =3, coords = 1,3
4 at slot 1, 3 didnt work.
Backtracking here...
9 at slot 1, 2 didnt work.
Backtracking here...
9 at slot 0, 8 didnt work.
Backtracking here...
2 at slot 0, 7 didnt work.
Backtracking here...
trying = 9, board-list =2, coords = 0,7
9 at slot 0, 7 didnt work.
Backtracking here...
1 at slot 0, 1 didnt work.
Backtracking here...
trying = 9, board-list =1, coords = 0,1
9 at slot 0, 1 didnt work.
Backtracking here...
False
更正：3 1 6 5 7 8 4 9 2 5 2 9 1 3 4 7 6 8 4 8 7 6 2 9 5 3 1 2 6 3 4 1 5 9 8 7 9 7 4 8 6 3 1 2 5 8 5 1 7 9 2 6 4 3 1 3 8 9 4 7 2 5 6 6 9 2 3 5 1 8 7 4 7 4 5 2 8 6 3 1 9
以数独格式正确回答：
3 1 6 5 7 8 4 9 2
5 2 9 1 3 4 7 6 8
4 8 7 6 2 9 5 3 1
2 6 3 4 1 5 9 8 7
9 7 4 8 6 3 1 2 5
8 5 1 7 9 2 6 4 3
1 3 8 9 4 7 2 5 6
6 9 2 3 5 1 8 7 4
7 4 5 2 8 6 3 1 9
答案 0 :(得分：0)
由OP解决。工作代码是：
def make_nodes(board):
'''
make_nodes converts a board compiled of 1x9 rows to a board compiled of 3x3 nodes
:param board:
:return nodes:
'''
nodes = [[board[0][0], board[0][1], board[0][2],
board[1][0], board[1][1], board[1][2],
board[2][0], board[2][1], board[2][2]],
[board[0][3], board[0][4], board[0][5],
board[1][3], board[1][4], board[1][5],
board[2][3], board[2][4], board[2][5]],
[board[0][6], board[0][7], board[0][8],
board[1][6], board[1][7], board[1][8],
board[2][6], board[2][7], board[2][8]],
[board[3][0], board[3][1], board[3][2],
board[4][0], board[4][1], board[4][2],
board[5][0], board[5][1], board[5][2]],
[board[3][3], board[3][4], board[3][5],
board[4][3], board[4][4], board[4][5],
board[5][3], board[5][4], board[5][5]],
[board[3][6], board[3][7], board[3][8],
board[4][6], board[4][7], board[4][8],
board[5][6], board[5][7], board[5][8]],
[board[6][0], board[6][1], board[6][2],
board[7][0], board[7][1], board[7][2],
board[8][0], board[8][1], board[8][2]],
[board[6][3], board[6][4], board[6][5],
board[7][3], board[7][4], board[7][5],
board[8][3], board[8][4], board[8][5]],
[board[6][6], board[6][7], board[6][8],
board[7][6], board[7][7], board[7][8],
board[8][6], board[8][7], board[8][8]]
]
return nodes
def fill_zeros(board):
'''
fill_zeros replaces every empty 0 with the possible values for that cell based off given values in its node
:param board:
:return board:
'''
nodes = make_nodes(board)
allnums = [1, 2, 3, 4, 5, 6, 7, 8, 9]
for i in range(9):
for j in range(9):
possible = [h for h in allnums if h not in nodes[i]]
if nodes[i][j] == 0:
nodes[i][j] = possible
return nodes
def node_to_board(nodes):
'''
node_to_board converts a board compiled of 3x3 nodes to a board of 1x9 rows
:param nodes:
:return board:
'''
board = []
for y in range(3):
for i in range(0,9, 3):
for node in nodes[(y*3):((y+1)*3)]:
for x in range(3):
board.append(node[i+x])
board = [board[pos:pos + 9] for pos in range(0, 9 * 9, 9)]
return board
def refine_empty_by_col(board):
'''
refine_empty_by_col takes a board with empty cells, and checks each number in the empty cells. If the number
is already a definitenumber in its own column, remove that number from the possible values for that cell
:param board:
:return board:
'''
for col in range(9):
col_done = []
for row in range(9):
if type(board[row][col]) == int:
col_done.append(board[row][col])
for row in range(9):
if type(board[row][col]) == list:
board[row][col] = [x for x in board[row][col] if x not in col_done]
for row in range(9):
if type(board[row][col]) == list:
if len(board[row][col]) == 1:
k = board[row][col].pop()
board[row][col] = k
return board
def refine_empty_by_row(board):
'''
refine_empty_by_row takes a board with empty cells, and checks each number in the empty cells. If the number is
already a definite number in its own row, remove that number from the possible values for that cell
:param board:
:return board:
'''
for row in range(9):
row_done = []
for col in range(9):
if type(board[row][col]) == int:
row_done.append(board[row][col])
for col in range(9):
if type(board[row][col]) == list:
board[row][col] = [x for x in board[row][col] if x not in row_done]
for col in range(9):
if type(board[row][col]) == list:
if len(board[row][col]) == 1:
k = board[row][col].pop()
board[row][col] = k
return board
def refine_empty_by_node(board):
'''
refine_empty_by_node takes a board with empty cells, and checks each number in the empty cells. If the number is
already a definite number in its own node, remove that number from the possible values for that cell
:param board:
:return board:
'''
nodes = make_nodes(board)
allnums = [1, 2, 3, 4, 5, 6, 7, 8, 9]
for node in range(9):
taken = [x for x in allnums if x in nodes[node]]
for item in range(9):
if type(nodes[node][item]) == list:
nodes[node][item] = [x for x in nodes[node][item] if x not in taken]
if len(nodes[node][item]) == 1:
k = nodes[node][item].pop()
taken.append(k)
nodes[node][item] = k
node_list_nums = []
for item in range(9):
if type(nodes[node][item]) == list:
node_list_nums += nodes[node][item]
node_list_nums = [x for x in node_list_nums if node_list_nums.count(x) == 1]
for item in range(9):
if type(nodes[node][item]) == list:
if any(nodes[node][item]) in node_list_nums:
for x in node_list_nums:
if x in nodes[node][item]:
nodes[node][item] = x
node_list_nums.remove(x)
board = node_to_board(nodes)
return board
def refine_easy_style(board):
'''
refine_easy_style takes a "fresh" board and repeatedly calls refine_empty_by_* until no more changes can be made
using this method.
:param board:
:return board:
'''
board_after = board
board_orig = []
while board_orig != board_after:
board_orig = board_after
board_after = refine_empty_by_col(board_orig)
board_after = refine_empty_by_row(board_after)
board_after = refine_empty_by_node(board_after)
return board_after
## above this line works
## below this line is testing
def is_distinct(passed):
'''
is_distinct checks to ensure the (node|row|column) is has only unique definite values (True) if not return False
:param passed:
:return True if all distinct definites, False otherwise:
'''
used = []
for item in passed:
if type(item) == list:
continue
if item in used:
return False
used.append(item)
return True
def is_valid(board):
'''
is_valid ensures the entire board is still valid by checking each row, column and node via is_distinct
:param board:
:return True if valid board, False otherwise:
'''
for i in range(9):
row = [board[i][row] for row in range(9)]
if not is_distinct(row):
return False
col = [board[row][i] for row in range(9)]
if not is_distinct(col):
return False
for node in make_nodes(board):
if not is_distinct(node):
return False
return True
def next_empty(board):
'''
next_empty finds the next empty cell on the board, based on how many possibilities that empty cell has in
ascending order
:param board:
:return x,y coords:
'''
min = [x for x in range(9)]
row = 0
col = 0
for i in range(9):
for j in range(9):
if type(board[i][j]) == list:
if len(board[i][j]) < len(min):
min = board[i][j]
row = i
col = j
return row, col
def solve_puzzle(board):
'''
solve_puzzle recursively calls itself until the board is solved or a solution is not found
:param board:
:return True if solved, False if no solution exists:
'''
# if there are no empty cells left, ensure the board is valid
empties = 0
for row in board:
for item in row:
if type(item) == list:
empties += 1
if empties == 0:
return is_valid(board)
# get the next empty cell coords
row, col = next_empty(board)
board2 = board
possible = board2[row][col]
# recursive step
# loop through every possible value until one is valid, backtracking until a valid board is found
for k in possible:
board2[row][col] = k
if is_valid(board2) and solve_puzzle(board2):
return True
board2[row][col] = possible
return False
# read input
inp = [int(x) for x in input().split()]
# make a board from the input (row-format)
board = [inp[pos:pos+9] for pos in range(0, 9*9, 9)]
# fill zeros with possible values, then put it back to row-format
board = node_to_board(fill_zeros(board))
# fill in what is possible without recursion
board = refine_easy_style(board)
# solve the remaining cells with recursion
solve_puzzle(board)
for row in board:
print(row)
示例： 3 0 6 5 0 8 4 0 0 5 2 0 0 0 0 0 0 0 0 8 7 0 0 0 0 3 1 0 0 3 0 1 0 0 8 0 9 0 0 8 6 3 0 0 5 0 5 0 0 9 0 6 0 0 1 3 0 0 0 0 2 5 0 0 0 0 0 0 0 0 7 4 0 0 5 2 0 6 3 0 0
示例：
[3, 1, 6, 5, 7, 8, 4, 9, 2]
[5, 2, 9, 1, 3, 4, 7, 6, 8]
[4, 8, 7, 6, 2, 9, 5, 3, 1]
[2, 6, 3, 4, 1, 5, 9, 8, 7]
[9, 7, 4, 8, 6, 3, 1, 2, 5]
[8, 5, 1, 7, 9, 2, 6, 4, 3]
[1, 3, 8, 9, 4, 7, 2, 5, 6]
[6, 9, 2, 3, 5, 1, 8, 7, 4]
[7, 4, 5, 2, 8, 6, 3, 1, 9]