notebook.community

Edit and run 





In [1]:



    


# N-queens Problem 
# backtracking algo



    











In [2]:



    


# problem params
num_queens = 4
# for now, board is num_queens*num_queens sized



    











In [3]:



    


def is_threatened(Q1, Q2):
    # given two positions as (row,col) tuples
    # returns true if the queens threaten each other
    
    # same row, column or diagonal
    return (Q1[0]==Q2[0]) or (Q1[1]==Q2[1]) or (sum(Q1)==sum(Q2)) or ((Q1[0]-Q1[1])==(Q2[0]-Q2[1]))



    











In [4]:



    


# test
Q1 = (0,0)
Q2 = (1,1)
print(is_threatened(Q1,Q2))



    


















    






True

















In [5]:



    


def n_queens_is_threatened(*args):
    # given any number of queens, returns true if any pair is threatening each other
    
    for i, Q1 in enumerate(args[:-1]):
#         print(Q1)
        for Q2 in args[i+1:]:
#             print(Q2)
            if is_threatened(Q1,Q2):
                return True
    
    # if none found
    return False



    











In [6]:



    


# test
import random
Q_list = [(random.randint(0,num_queens-1),random.randint(0,num_queens-1)) for _ in range(4)]
print(Q_list)
print(n_queens_is_threatened(*Q_list))



    


















    






[(0, 0), (1, 2), (0, 0), (3, 0)]
True

















In [72]:



    


# backtracking algo
import copy

def place_n_queens(num_queens):
    
    num_rows = num_queens
    num_cols = num_queens
    
    Q_result = []
    Q_list = []
    cur_row = 0
    cur_col = 0
    
    while cur_row >= 0:            
        # place new queen
#         print("adding new queen", (cur_row, cur_col))
        Q_list.append((cur_row,cur_col))
#         print(Q_list)

        # check
        if n_queens_is_threatened(*Q_list):
            # threatened, either move to next col, or backtrack
            Q_prev = Q_list.pop()
            # move to next col
            cur_col += 1
            # or backtrack
            while (cur_col>num_cols-1) and (cur_row>=0):
                if len(Q_list)==0:
                    cur_row = -1
                else:
                    Q_prev = Q_list.pop()
                    cur_col = Q_prev[1]+1
                    cur_row -= 1
                    
        else:
            # if n queens placed, save result and remove last added
            if len(Q_list)==num_queens:
#                 print("found", Q_list)
                # make copy
                Q_result.append(copy.deepcopy(Q_list))
                Q_prev = Q_list.pop()
                cur_col += 1
                # backtrack if needed
                while (cur_col>num_cols-1):
                    Q_prev = Q_list.pop()
                    cur_col = Q_prev[1]+1
                    cur_row -= 1
            else:
                # works, move down, reset col
                cur_row += 1
                cur_col = 0
    
    return Q_result



    











In [78]:



    


r = place_n_queens(num_queens=9)
print("Number of solutions:", len(r))
for s in r:
    print(s)



    


















    






('Number of solutions:', 352)
[(0, 0), (1, 2), (2, 5), (3, 7), (4, 1), (5, 3), (6, 8), (7, 6), (8, 4)]
[(0, 0), (1, 2), (2, 6), (3, 1), (4, 7), (5, 4), (6, 8), (7, 3), (8, 5)]
[(0, 0), (1, 2), (2, 7), (3, 5), (4, 8), (5, 1), (6, 4), (7, 6), (8, 3)]
[(0, 0), (1, 3), (2, 1), (3, 7), (4, 5), (5, 8), (6, 2), (7, 4), (8, 6)]
[(0, 0), (1, 3), (2, 5), (3, 2), (4, 8), (5, 1), (6, 7), (7, 4), (8, 6)]
[(0, 0), (1, 3), (2, 5), (3, 7), (4, 1), (5, 4), (6, 2), (7, 8), (8, 6)]
[(0, 0), (1, 3), (2, 6), (3, 2), (4, 7), (5, 1), (6, 4), (7, 8), (8, 5)]
[(0, 0), (1, 3), (2, 6), (3, 8), (4, 1), (5, 4), (6, 7), (7, 5), (8, 2)]
[(0, 0), (1, 3), (2, 7), (3, 2), (4, 8), (5, 6), (6, 4), (7, 1), (8, 5)]
[(0, 0), (1, 4), (2, 1), (3, 5), (4, 8), (5, 2), (6, 7), (7, 3), (8, 6)]
[(0, 0), (1, 4), (2, 6), (3, 1), (4, 5), (5, 2), (6, 8), (7, 3), (8, 7)]
[(0, 0), (1, 4), (2, 6), (3, 8), (4, 2), (5, 7), (6, 1), (7, 3), (8, 5)]
[(0, 0), (1, 4), (2, 6), (3, 8), (4, 3), (5, 1), (6, 7), (7, 5), (8, 2)]
[(0, 0), (1, 4), (2, 8), (3, 1), (4, 5), (5, 7), (6, 2), (7, 6), (8, 3)]
[(0, 0), (1, 4), (2, 8), (3, 5), (4, 3), (5, 1), (6, 7), (7, 2), (8, 6)]
[(0, 0), (1, 5), (2, 1), (3, 8), (4, 6), (5, 3), (6, 7), (7, 2), (8, 4)]
[(0, 0), (1, 5), (2, 3), (3, 1), (4, 6), (5, 8), (6, 2), (7, 4), (8, 7)]
[(0, 0), (1, 5), (2, 3), (3, 1), (4, 7), (5, 2), (6, 8), (7, 6), (8, 4)]
[(0, 0), (1, 5), (2, 7), (3, 2), (4, 6), (5, 3), (6, 1), (7, 8), (8, 4)]
[(0, 0), (1, 5), (2, 7), (3, 4), (4, 1), (5, 3), (6, 8), (7, 6), (8, 2)]
[(0, 0), (1, 5), (2, 8), (3, 4), (4, 1), (5, 7), (6, 2), (7, 6), (8, 3)]
[(0, 0), (1, 6), (2, 3), (3, 5), (4, 8), (5, 1), (6, 4), (7, 2), (8, 7)]
[(0, 0), (1, 6), (2, 3), (3, 7), (4, 2), (5, 4), (6, 8), (7, 1), (8, 5)]
[(0, 0), (1, 6), (2, 3), (3, 7), (4, 2), (5, 8), (6, 5), (7, 1), (8, 4)]
[(0, 0), (1, 6), (2, 4), (3, 7), (4, 1), (5, 8), (6, 2), (7, 5), (8, 3)]
[(0, 0), (1, 7), (2, 3), (3, 1), (4, 6), (5, 8), (6, 5), (7, 2), (8, 4)]
[(0, 0), (1, 7), (2, 4), (3, 2), (4, 5), (5, 8), (6, 1), (7, 3), (8, 6)]
[(0, 0), (1, 7), (2, 4), (3, 2), (4, 8), (5, 6), (6, 1), (7, 3), (8, 5)]
[(0, 1), (1, 3), (2, 0), (3, 6), (4, 8), (5, 5), (6, 2), (7, 4), (8, 7)]
[(0, 1), (1, 3), (2, 6), (3, 0), (4, 2), (5, 8), (6, 5), (7, 7), (8, 4)]
[(0, 1), (1, 3), (2, 7), (3, 2), (4, 8), (5, 5), (6, 0), (7, 4), (8, 6)]
[(0, 1), (1, 3), (2, 8), (3, 6), (4, 2), (5, 0), (6, 5), (7, 7), (8, 4)]
[(0, 1), (1, 3), (2, 8), (3, 6), (4, 4), (5, 2), (6, 0), (7, 5), (8, 7)]
[(0, 1), (1, 4), (2, 6), (3, 0), (4, 2), (5, 7), (6, 5), (7, 3), (8, 8)]
[(0, 1), (1, 4), (2, 6), (3, 3), (4, 0), (5, 2), (6, 8), (7, 5), (8, 7)]
[(0, 1), (1, 4), (2, 6), (3, 8), (4, 2), (5, 5), (6, 3), (7, 0), (8, 7)]
[(0, 1), (1, 4), (2, 6), (3, 8), (4, 3), (5, 7), (6, 0), (7, 2), (8, 5)]
[(0, 1), (1, 4), (2, 7), (3, 0), (4, 2), (5, 5), (6, 8), (7, 6), (8, 3)]
[(0, 1), (1, 4), (2, 7), (3, 0), (4, 8), (5, 5), (6, 2), (7, 6), (8, 3)]
[(0, 1), (1, 4), (2, 7), (3, 5), (4, 8), (5, 2), (6, 0), (7, 3), (8, 6)]
[(0, 1), (1, 4), (2, 7), (3, 5), (4, 8), (5, 2), (6, 0), (7, 6), (8, 3)]
[(0, 1), (1, 4), (2, 8), (3, 3), (4, 0), (5, 7), (6, 5), (7, 2), (8, 6)]
[(0, 1), (1, 5), (2, 0), (3, 2), (4, 6), (5, 8), (6, 3), (7, 7), (8, 4)]
[(0, 1), (1, 5), (2, 0), (3, 6), (4, 3), (5, 7), (6, 2), (7, 4), (8, 8)]
[(0, 1), (1, 5), (2, 0), (3, 6), (4, 4), (5, 2), (6, 8), (7, 3), (8, 7)]
[(0, 1), (1, 5), (2, 0), (3, 8), (4, 4), (5, 7), (6, 3), (7, 6), (8, 2)]
[(0, 1), (1, 5), (2, 2), (3, 0), (4, 7), (5, 3), (6, 8), (7, 6), (8, 4)]
[(0, 1), (1, 5), (2, 8), (3, 2), (4, 4), (5, 7), (6, 3), (7, 0), (8, 6)]
[(0, 1), (1, 6), (2, 4), (3, 0), (4, 8), (5, 3), (6, 5), (7, 7), (8, 2)]
[(0, 1), (1, 6), (2, 4), (3, 7), (4, 0), (5, 3), (6, 5), (7, 2), (8, 8)]
[(0, 1), (1, 6), (2, 8), (3, 5), (4, 2), (5, 0), (6, 3), (7, 7), (8, 4)]
[(0, 1), (1, 7), (2, 0), (3, 3), (4, 6), (5, 8), (6, 5), (7, 2), (8, 4)]
[(0, 1), (1, 7), (2, 4), (3, 2), (4, 8), (5, 5), (6, 3), (7, 0), (8, 6)]
[(0, 1), (1, 7), (2, 5), (3, 8), (4, 2), (5, 0), (6, 3), (7, 6), (8, 4)]
[(0, 1), (1, 8), (2, 4), (3, 2), (4, 7), (5, 3), (6, 6), (7, 0), (8, 5)]
[(0, 1), (1, 8), (2, 5), (3, 2), (4, 4), (5, 7), (6, 0), (7, 3), (8, 6)]
[(0, 1), (1, 8), (2, 5), (3, 2), (4, 6), (5, 3), (6, 0), (7, 7), (8, 4)]
[(0, 1), (1, 8), (2, 5), (3, 3), (4, 6), (5, 0), (6, 2), (7, 4), (8, 7)]
[(0, 2), (1, 0), (2, 3), (3, 6), (4, 8), (5, 1), (6, 4), (7, 7), (8, 5)]
[(0, 2), (1, 0), (2, 5), (3, 7), (4, 4), (5, 1), (6, 3), (7, 8), (8, 6)]
[(0, 2), (1, 0), (2, 6), (3, 1), (4, 7), (5, 5), (6, 3), (7, 8), (8, 4)]
[(0, 2), (1, 0), (2, 6), (3, 4), (4, 7), (5, 1), (6, 3), (7, 5), (8, 8)]
[(0, 2), (1, 0), (2, 7), (3, 3), (4, 8), (5, 6), (6, 4), (7, 1), (8, 5)]
[(0, 2), (1, 0), (2, 8), (3, 6), (4, 4), (5, 1), (6, 7), (7, 5), (8, 3)]
[(0, 2), (1, 4), (2, 1), (3, 7), (4, 0), (5, 3), (6, 6), (7, 8), (8, 5)]
[(0, 2), (1, 4), (2, 1), (3, 7), (4, 0), (5, 6), (6, 3), (7, 5), (8, 8)]
[(0, 2), (1, 4), (2, 6), (3, 0), (4, 3), (5, 1), (6, 7), (7, 5), (8, 8)]
[(0, 2), (1, 4), (2, 7), (3, 1), (4, 8), (5, 5), (6, 0), (7, 6), (8, 3)]
[(0, 2), (1, 4), (2, 7), (3, 1), (4, 8), (5, 6), (6, 0), (7, 3), (8, 5)]
[(0, 2), (1, 4), (2, 8), (3, 1), (4, 3), (5, 6), (6, 0), (7, 7), (8, 5)]
[(0, 2), (1, 4), (2, 8), (3, 3), (4, 0), (5, 6), (6, 1), (7, 5), (8, 7)]
[(0, 2), (1, 5), (2, 1), (3, 6), (4, 0), (5, 3), (6, 7), (7, 4), (8, 8)]
[(0, 2), (1, 5), (2, 1), (3, 8), (4, 4), (5, 0), (6, 7), (7, 3), (8, 6)]
[(0, 2), (1, 5), (2, 7), (3, 0), (4, 3), (5, 6), (6, 4), (7, 1), (8, 8)]
[(0, 2), (1, 5), (2, 7), (3, 0), (4, 4), (5, 8), (6, 1), (7, 3), (8, 6)]
[(0, 2), (1, 5), (2, 7), (3, 1), (4, 3), (5, 8), (6, 6), (7, 4), (8, 0)]
[(0, 2), (1, 5), (2, 7), (3, 4), (4, 0), (5, 8), (6, 6), (7, 1), (8, 3)]
[(0, 2), (1, 5), (2, 7), (3, 4), (4, 1), (5, 8), (6, 6), (7, 3), (8, 0)]
[(0, 2), (1, 5), (2, 8), (3, 0), (4, 7), (5, 3), (6, 1), (7, 6), (8, 4)]
[(0, 2), (1, 5), (2, 8), (3, 1), (4, 4), (5, 6), (6, 3), (7, 0), (8, 7)]
[(0, 2), (1, 5), (2, 8), (3, 1), (4, 7), (5, 0), (6, 3), (7, 6), (8, 4)]
[(0, 2), (1, 5), (2, 8), (3, 4), (4, 7), (5, 0), (6, 3), (7, 1), (8, 6)]
[(0, 2), (1, 5), (2, 8), (3, 6), (4, 0), (5, 3), (6, 1), (7, 4), (8, 7)]
[(0, 2), (1, 5), (2, 8), (3, 6), (4, 1), (5, 3), (6, 7), (7, 0), (8, 4)]
[(0, 2), (1, 5), (2, 8), (3, 6), (4, 3), (5, 0), (6, 7), (7, 1), (8, 4)]
[(0, 2), (1, 6), (2, 1), (3, 3), (4, 7), (5, 0), (6, 4), (7, 8), (8, 5)]
[(0, 2), (1, 6), (2, 1), (3, 7), (4, 4), (5, 8), (6, 0), (7, 5), (8, 3)]
[(0, 2), (1, 6), (2, 1), (3, 7), (4, 5), (5, 3), (6, 0), (7, 4), (8, 8)]
[(0, 2), (1, 6), (2, 3), (3, 1), (4, 8), (5, 4), (6, 0), (7, 7), (8, 5)]
[(0, 2), (1, 6), (2, 3), (3, 1), (4, 8), (5, 5), (6, 0), (7, 4), (8, 7)]
[(0, 2), (1, 6), (2, 3), (3, 7), (4, 4), (5, 8), (6, 0), (7, 5), (8, 1)]
[(0, 2), (1, 6), (2, 8), (3, 0), (4, 4), (5, 1), (6, 7), (7, 5), (8, 3)]
[(0, 2), (1, 6), (2, 8), (3, 3), (4, 1), (5, 4), (6, 7), (7, 5), (8, 0)]
[(0, 2), (1, 7), (2, 1), (3, 3), (4, 8), (5, 6), (6, 4), (7, 0), (8, 5)]
[(0, 2), (1, 7), (2, 3), (3, 6), (4, 8), (5, 1), (6, 4), (7, 0), (8, 5)]
[(0, 2), (1, 7), (2, 5), (3, 0), (4, 8), (5, 1), (6, 4), (7, 6), (8, 3)]
[(0, 2), (1, 7), (2, 5), (3, 3), (4, 8), (5, 0), (6, 4), (7, 6), (8, 1)]
[(0, 2), (1, 7), (2, 5), (3, 8), (4, 1), (5, 4), (6, 0), (7, 3), (8, 6)]
[(0, 2), (1, 8), (2, 1), (3, 4), (4, 7), (5, 0), (6, 6), (7, 3), (8, 5)]
[(0, 2), (1, 8), (2, 3), (3, 0), (4, 7), (5, 5), (6, 1), (7, 6), (8, 4)]
[(0, 2), (1, 8), (2, 3), (3, 1), (4, 7), (5, 5), (6, 0), (7, 6), (8, 4)]
[(0, 2), (1, 8), (2, 3), (3, 7), (4, 4), (5, 1), (6, 5), (7, 0), (8, 6)]
[(0, 2), (1, 8), (2, 5), (3, 1), (4, 4), (5, 6), (6, 0), (7, 3), (8, 7)]
[(0, 2), (1, 8), (2, 5), (3, 3), (4, 0), (5, 6), (6, 4), (7, 1), (8, 7)]
[(0, 2), (1, 8), (2, 5), (3, 7), (4, 1), (5, 3), (6, 0), (7, 6), (8, 4)]
[(0, 3), (1, 0), (2, 2), (3, 5), (4, 8), (5, 1), (6, 7), (7, 4), (8, 6)]
[(0, 3), (1, 0), (2, 4), (3, 1), (4, 8), (5, 6), (6, 2), (7, 7), (8, 5)]
[(0, 3), (1, 0), (2, 4), (3, 7), (4, 1), (5, 6), (6, 2), (7, 5), (8, 8)]
[(0, 3), (1, 0), (2, 4), (3, 8), (4, 1), (5, 5), (6, 7), (7, 2), (8, 6)]
[(0, 3), (1, 0), (2, 6), (3, 8), (4, 1), (5, 5), (6, 7), (7, 2), (8, 4)]
[(0, 3), (1, 0), (2, 8), (3, 5), (4, 2), (5, 6), (6, 1), (7, 7), (8, 4)]
[(0, 3), (1, 1), (2, 4), (3, 7), (4, 0), (5, 2), (6, 5), (7, 8), (8, 6)]
[(0, 3), (1, 1), (2, 6), (3, 2), (4, 0), (5, 7), (6, 4), (7, 8), (8, 5)]
[(0, 3), (1, 1), (2, 6), (3, 8), (4, 0), (5, 4), (6, 7), (7, 5), (8, 2)]
[(0, 3), (1, 1), (2, 6), (3, 8), (4, 0), (5, 7), (6, 4), (7, 2), (8, 5)]
[(0, 3), (1, 1), (2, 7), (3, 2), (4, 8), (5, 6), (6, 4), (7, 0), (8, 5)]
[(0, 3), (1, 1), (2, 8), (3, 2), (4, 5), (5, 7), (6, 0), (7, 4), (8, 6)]
[(0, 3), (1, 1), (2, 8), (3, 4), (4, 0), (5, 7), (6, 5), (7, 2), (8, 6)]
[(0, 3), (1, 5), (2, 0), (3, 4), (4, 1), (5, 7), (6, 2), (7, 6), (8, 8)]
[(0, 3), (1, 5), (2, 0), (3, 8), (4, 4), (5, 7), (6, 1), (7, 6), (8, 2)]
[(0, 3), (1, 5), (2, 0), (3, 8), (4, 6), (5, 2), (6, 7), (7, 1), (8, 4)]
[(0, 3), (1, 5), (2, 2), (3, 8), (4, 1), (5, 4), (6, 7), (7, 0), (8, 6)]
[(0, 3), (1, 5), (2, 2), (3, 8), (4, 1), (5, 7), (6, 4), (7, 6), (8, 0)]
[(0, 3), (1, 5), (2, 2), (3, 8), (4, 6), (5, 0), (6, 7), (7, 1), (8, 4)]
[(0, 3), (1, 5), (2, 7), (3, 1), (4, 4), (5, 0), (6, 8), (7, 6), (8, 2)]
[(0, 3), (1, 5), (2, 7), (3, 1), (4, 4), (5, 6), (6, 8), (7, 0), (8, 2)]
[(0, 3), (1, 5), (2, 7), (3, 1), (4, 6), (5, 0), (6, 2), (7, 4), (8, 8)]
[(0, 3), (1, 5), (2, 7), (3, 2), (4, 0), (5, 6), (6, 4), (7, 1), (8, 8)]
[(0, 3), (1, 5), (2, 8), (3, 2), (4, 0), (5, 7), (6, 1), (7, 4), (8, 6)]
[(0, 3), (1, 6), (2, 0), (3, 2), (4, 8), (5, 5), (6, 7), (7, 4), (8, 1)]
[(0, 3), (1, 6), (2, 0), (3, 5), (4, 8), (5, 1), (6, 7), (7, 4), (8, 2)]
[(0, 3), (1, 6), (2, 0), (3, 7), (4, 4), (5, 1), (6, 8), (7, 2), (8, 5)]
[(0, 3), (1, 6), (2, 2), (3, 5), (4, 8), (5, 0), (6, 7), (7, 4), (8, 1)]
[(0, 3), (1, 6), (2, 2), (3, 7), (4, 1), (5, 4), (6, 8), (7, 5), (8, 0)]
[(0, 3), (1, 6), (2, 2), (3, 7), (4, 5), (5, 0), (6, 8), (7, 1), (8, 4)]
[(0, 3), (1, 6), (2, 2), (3, 7), (4, 5), (5, 1), (6, 8), (7, 4), (8, 0)]
[(0, 3), (1, 6), (2, 4), (3, 1), (4, 8), (5, 0), (6, 2), (7, 7), (8, 5)]
[(0, 3), (1, 6), (2, 4), (3, 1), (4, 8), (5, 0), (6, 5), (7, 7), (8, 2)]
[(0, 3), (1, 6), (2, 4), (3, 1), (4, 8), (5, 5), (6, 7), (7, 2), (8, 0)]
[(0, 3), (1, 6), (2, 8), (3, 1), (4, 4), (5, 7), (6, 0), (7, 2), (8, 5)]
[(0, 3), (1, 6), (2, 8), (3, 1), (4, 5), (5, 0), (6, 2), (7, 4), (8, 7)]
[(0, 3), (1, 6), (2, 8), (3, 5), (4, 2), (5, 0), (6, 7), (7, 4), (8, 1)]
[(0, 3), (1, 7), (2, 0), (3, 4), (4, 6), (5, 1), (6, 5), (7, 2), (8, 8)]
[(0, 3), (1, 7), (2, 4), (3, 2), (4, 0), (5, 5), (6, 1), (7, 8), (8, 6)]
[(0, 3), (1, 7), (2, 4), (3, 2), (4, 0), (5, 6), (6, 1), (7, 5), (8, 8)]
[(0, 3), (1, 8), (2, 2), (3, 5), (4, 1), (5, 6), (6, 4), (7, 0), (8, 7)]
[(0, 3), (1, 8), (2, 4), (3, 2), (4, 0), (5, 5), (6, 7), (7, 1), (8, 6)]
[(0, 3), (1, 8), (2, 4), (3, 2), (4, 0), (5, 6), (6, 1), (7, 7), (8, 5)]
[(0, 3), (1, 8), (2, 4), (3, 7), (4, 0), (5, 2), (6, 5), (7, 1), (8, 6)]
[(0, 4), (1, 0), (2, 5), (3, 3), (4, 1), (5, 7), (6, 2), (7, 8), (8, 6)]
[(0, 4), (1, 0), (2, 7), (3, 3), (4, 1), (5, 6), (6, 8), (7, 5), (8, 2)]
[(0, 4), (1, 0), (2, 7), (3, 5), (4, 2), (5, 6), (6, 1), (7, 3), (8, 8)]
[(0, 4), (1, 1), (2, 3), (3, 0), (4, 6), (5, 8), (6, 2), (7, 5), (8, 7)]
[(0, 4), (1, 1), (2, 3), (3, 8), (4, 6), (5, 2), (6, 0), (7, 5), (8, 7)]
[(0, 4), (1, 1), (2, 5), (3, 0), (4, 2), (5, 6), (6, 8), (7, 3), (8, 7)]
[(0, 4), (1, 1), (2, 5), (3, 8), (4, 2), (5, 7), (6, 3), (7, 6), (8, 0)]
[(0, 4), (1, 1), (2, 5), (3, 8), (4, 6), (5, 3), (6, 0), (7, 2), (8, 7)]
[(0, 4), (1, 1), (2, 7), (3, 0), (4, 3), (5, 6), (6, 8), (7, 5), (8, 2)]
[(0, 4), (1, 1), (2, 7), (3, 0), (4, 6), (5, 8), (6, 2), (7, 5), (8, 3)]
[(0, 4), (1, 1), (2, 7), (3, 2), (4, 6), (5, 3), (6, 0), (7, 8), (8, 5)]
[(0, 4), (1, 1), (2, 7), (3, 2), (4, 6), (5, 8), (6, 0), (7, 5), (8, 3)]
[(0, 4), (1, 1), (2, 8), (3, 0), (4, 5), (5, 7), (6, 2), (7, 6), (8, 3)]
[(0, 4), (1, 1), (2, 8), (3, 5), (4, 2), (5, 6), (6, 3), (7, 0), (8, 7)]
[(0, 4), (1, 2), (2, 0), (3, 5), (4, 1), (5, 8), (6, 6), (7, 3), (8, 7)]
[(0, 4), (1, 2), (2, 0), (3, 5), (4, 7), (5, 1), (6, 3), (7, 6), (8, 8)]
[(0, 4), (1, 2), (2, 0), (3, 6), (4, 1), (5, 7), (6, 5), (7, 3), (8, 8)]
[(0, 4), (1, 2), (2, 5), (3, 8), (4, 1), (5, 7), (6, 0), (7, 3), (8, 6)]
[(0, 4), (1, 2), (2, 5), (3, 8), (4, 6), (5, 0), (6, 3), (7, 1), (8, 7)]
[(0, 4), (1, 2), (2, 5), (3, 8), (4, 6), (5, 1), (6, 3), (7, 7), (8, 0)]
[(0, 4), (1, 2), (2, 5), (3, 8), (4, 6), (5, 3), (6, 0), (7, 7), (8, 1)]
[(0, 4), (1, 2), (2, 7), (3, 3), (4, 1), (5, 8), (6, 5), (7, 0), (8, 6)]
[(0, 4), (1, 2), (2, 7), (3, 3), (4, 6), (5, 8), (6, 1), (7, 5), (8, 0)]
[(0, 4), (1, 2), (2, 7), (3, 5), (4, 1), (5, 8), (6, 0), (7, 3), (8, 6)]
[(0, 4), (1, 2), (2, 7), (3, 5), (4, 1), (5, 8), (6, 6), (7, 0), (8, 3)]
[(0, 4), (1, 2), (2, 8), (3, 3), (4, 1), (5, 7), (6, 5), (7, 0), (8, 6)]
[(0, 4), (1, 2), (2, 8), (3, 5), (4, 7), (5, 1), (6, 3), (7, 0), (8, 6)]
[(0, 4), (1, 6), (2, 0), (3, 3), (4, 1), (5, 7), (6, 5), (7, 8), (8, 2)]
[(0, 4), (1, 6), (2, 0), (3, 5), (4, 7), (5, 1), (6, 3), (7, 8), (8, 2)]
[(0, 4), (1, 6), (2, 1), (3, 3), (4, 7), (5, 0), (6, 2), (7, 8), (8, 5)]
[(0, 4), (1, 6), (2, 1), (3, 3), (4, 7), (5, 0), (6, 8), (7, 5), (8, 2)]
[(0, 4), (1, 6), (2, 1), (3, 5), (4, 2), (5, 0), (6, 7), (7, 3), (8, 8)]
[(0, 4), (1, 6), (2, 1), (3, 5), (4, 7), (5, 0), (6, 3), (7, 8), (8, 2)]
[(0, 4), (1, 6), (2, 3), (3, 0), (4, 2), (5, 5), (6, 8), (7, 1), (8, 7)]
[(0, 4), (1, 6), (2, 3), (3, 0), (4, 2), (5, 7), (6, 5), (7, 1), (8, 8)]
[(0, 4), (1, 6), (2, 3), (3, 0), (4, 2), (5, 8), (6, 5), (7, 7), (8, 1)]
[(0, 4), (1, 6), (2, 3), (3, 0), (4, 7), (5, 1), (6, 8), (7, 5), (8, 2)]
[(0, 4), (1, 6), (2, 8), (3, 2), (4, 7), (5, 1), (6, 3), (7, 5), (8, 0)]
[(0, 4), (1, 6), (2, 8), (3, 3), (4, 1), (5, 7), (6, 5), (7, 2), (8, 0)]
[(0, 4), (1, 6), (2, 8), (3, 3), (4, 7), (5, 0), (6, 2), (7, 5), (8, 1)]
[(0, 4), (1, 7), (2, 0), (3, 3), (4, 6), (5, 2), (6, 5), (7, 8), (8, 1)]
[(0, 4), (1, 7), (2, 0), (3, 8), (4, 3), (5, 1), (6, 6), (7, 2), (8, 5)]
[(0, 4), (1, 7), (2, 1), (3, 6), (4, 2), (5, 0), (6, 8), (7, 3), (8, 5)]
[(0, 4), (1, 7), (2, 1), (3, 6), (4, 2), (5, 5), (6, 8), (7, 0), (8, 3)]
[(0, 4), (1, 7), (2, 1), (3, 8), (4, 2), (5, 0), (6, 6), (7, 3), (8, 5)]
[(0, 4), (1, 7), (2, 1), (3, 8), (4, 5), (5, 2), (6, 0), (7, 3), (8, 6)]
[(0, 4), (1, 7), (2, 3), (3, 0), (4, 2), (5, 5), (6, 8), (7, 6), (8, 1)]
[(0, 4), (1, 7), (2, 3), (3, 0), (4, 6), (5, 1), (6, 5), (7, 2), (8, 8)]
[(0, 4), (1, 7), (2, 3), (3, 8), (4, 6), (5, 2), (6, 0), (7, 5), (8, 1)]
[(0, 4), (1, 7), (2, 5), (3, 0), (4, 2), (5, 6), (6, 8), (7, 3), (8, 1)]
[(0, 4), (1, 7), (2, 5), (3, 8), (4, 2), (5, 0), (6, 6), (7, 3), (8, 1)]
[(0, 4), (1, 8), (2, 1), (3, 3), (4, 6), (5, 2), (6, 7), (7, 5), (8, 0)]
[(0, 4), (1, 8), (2, 1), (3, 5), (4, 7), (5, 2), (6, 0), (7, 3), (8, 6)]
[(0, 4), (1, 8), (2, 3), (3, 5), (4, 7), (5, 1), (6, 6), (7, 0), (8, 2)]
[(0, 5), (1, 0), (2, 4), (3, 1), (4, 8), (5, 6), (6, 3), (7, 7), (8, 2)]
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[(0, 5), (1, 0), (2, 4), (3, 6), (4, 8), (5, 3), (6, 1), (7, 7), (8, 2)]
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[(0, 5), (1, 2), (2, 4), (3, 7), (4, 0), (5, 8), (6, 6), (7, 1), (8, 3)]
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[(0, 7), (1, 5), (2, 0), (3, 2), (4, 6), (5, 8), (6, 3), (7, 1), (8, 4)]
[(0, 7), (1, 5), (2, 1), (3, 6), (4, 0), (5, 3), (6, 8), (7, 4), (8, 2)]
[(0, 7), (1, 5), (2, 2), (3, 8), (4, 6), (5, 0), (6, 3), (7, 1), (8, 4)]
[(0, 7), (1, 5), (2, 8), (3, 2), (4, 0), (5, 3), (6, 6), (7, 4), (8, 1)]
[(0, 8), (1, 1), (2, 4), (3, 6), (4, 0), (5, 2), (6, 7), (7, 5), (8, 3)]
[(0, 8), (1, 1), (2, 4), (3, 6), (4, 3), (5, 0), (6, 7), (7, 5), (8, 2)]
[(0, 8), (1, 1), (2, 5), (3, 7), (4, 2), (5, 0), (6, 3), (7, 6), (8, 4)]
[(0, 8), (1, 2), (2, 4), (3, 1), (4, 7), (5, 0), (6, 6), (7, 3), (8, 5)]
[(0, 8), (1, 2), (2, 5), (3, 1), (4, 6), (5, 0), (6, 3), (7, 7), (8, 4)]
[(0, 8), (1, 2), (2, 5), (3, 1), (4, 6), (5, 4), (6, 0), (7, 7), (8, 3)]
[(0, 8), (1, 2), (2, 5), (3, 3), (4, 0), (5, 7), (6, 4), (7, 6), (8, 1)]
[(0, 8), (1, 3), (2, 0), (3, 4), (4, 7), (5, 1), (6, 6), (7, 2), (8, 5)]
[(0, 8), (1, 3), (2, 1), (3, 4), (4, 7), (5, 5), (6, 0), (7, 2), (8, 6)]
[(0, 8), (1, 3), (2, 1), (3, 6), (4, 2), (5, 5), (6, 7), (7, 0), (8, 4)]
[(0, 8), (1, 3), (2, 5), (3, 7), (4, 1), (5, 6), (6, 0), (7, 2), (8, 4)]
[(0, 8), (1, 3), (2, 5), (3, 7), (4, 2), (5, 0), (6, 6), (7, 4), (8, 1)]
[(0, 8), (1, 3), (2, 7), (3, 0), (4, 2), (5, 5), (6, 1), (7, 6), (8, 4)]
[(0, 8), (1, 4), (2, 0), (3, 3), (4, 5), (5, 7), (6, 1), (7, 6), (8, 2)]
[(0, 8), (1, 4), (2, 0), (3, 7), (4, 3), (5, 1), (6, 6), (7, 2), (8, 5)]
[(0, 8), (1, 4), (2, 2), (3, 0), (4, 5), (5, 7), (6, 1), (7, 3), (8, 6)]
[(0, 8), (1, 4), (2, 2), (3, 0), (4, 6), (5, 1), (6, 7), (7, 5), (8, 3)]
[(0, 8), (1, 4), (2, 2), (3, 7), (4, 3), (5, 6), (6, 0), (7, 5), (8, 1)]
[(0, 8), (1, 4), (2, 7), (3, 3), (4, 0), (5, 6), (6, 1), (7, 5), (8, 2)]
[(0, 8), (1, 5), (2, 1), (3, 6), (4, 0), (5, 2), (6, 4), (7, 7), (8, 3)]
[(0, 8), (1, 5), (2, 2), (3, 0), (4, 7), (5, 4), (6, 1), (7, 3), (8, 6)]
[(0, 8), (1, 5), (2, 2), (3, 6), (4, 1), (5, 7), (6, 4), (7, 0), (8, 3)]
[(0, 8), (1, 5), (2, 3), (3, 1), (4, 7), (5, 4), (6, 6), (7, 0), (8, 2)]
[(0, 8), (1, 5), (2, 3), (3, 6), (4, 0), (5, 7), (6, 1), (7, 4), (8, 2)]
[(0, 8), (1, 5), (2, 7), (3, 1), (4, 3), (5, 0), (6, 6), (7, 4), (8, 2)]
[(0, 8), (1, 6), (2, 1), (3, 3), (4, 0), (5, 7), (6, 4), (7, 2), (8, 5)]
[(0, 8), (1, 6), (2, 2), (3, 7), (4, 1), (5, 4), (6, 0), (7, 5), (8, 3)]
[(0, 8), (1, 6), (2, 3), (3, 1), (4, 7), (5, 5), (6, 0), (7, 2), (8, 4)]

















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