Arithmetic Billiards Racks

output

row 4:         [5]
row 3:       [4, 9]
row 2:     [7, 11, 2]
row 1:   [8, 1, 12, 10]
row 0: [6, 14, 15, 3, 13]
Remaining: []
row 4:         [5]
row 3:       [9, 4]
row 2:     [2, 11, 7]
row 1:   [10, 12, 1, 8]
row 0: [13, 3, 15, 14, 6]
Remaining: []
{2: 196, 3: 1574, 4: 1734, 5: 2}

rack_finder.py

new_rack = None
some_racks = None
def compute_subracks(new_rack):
    some_racks = []
    for ball in new_rack.remaining_balls:
        try:
            sub_rack = copy.deepcopy(new_rack)
            sub_rack.add_ball(ball)
            some_racks.append(sub_rack)
        except ValueError:
            pass
    return some_racks
   
'''omg this should be recursive Im sorry'''    
def print_rack_tree(root_rack, puzzle_size):
    possibility_counts = {3:0,4:0,5:0,6:0,7:0}
    for r in compute_subracks(root_rack):
        possibility_counts[3]+=1
        if puzzle_size>=4:
            for s in compute_subracks(r):
                possibility_counts[4]+=1
                if puzzle_size>=5:
                    for k in compute_subracks(s):
                        possibility_counts[5]+=1
                        if puzzle_size>=6:
                            for m in compute_subracks(k):
                                possibility_counts[6]+=1
                                if puzzle_size>=7:
                                    for n in compute_subracks(m):
                                        possibility_counts[7]+=1
                                        n.print_rack()
                                else:
                                    m.print_rack()
                        else:
                            k.print_rack()
                else:
                    s.print_rack()
        else:
            r.print_rack()
    return possibility_counts

def solve_puzzle(puzzle_size, operation=add):
    if puzzle_size<3 or puzzle_size>7:
        raise ValueError("Change your puzzle size")
    rack_list = []
    number_of_balls = ((puzzle_size*(puzzle_size+1))/2)+1
    for i in range(1,number_of_balls):
        for j in [x for x in range(1,number_of_balls) if x != i]:  
            try:
                rack_list.append(rack(i, j, operation, puzzle_size))
            except ValueError:
                pass
    global_counts = {2:0,3:0,4:0,5:0,6:0,7:0}
    for r in rack_list:
        global_counts[2]+=1
        counts = print_rack_tree(r,puzzle_size)
        for val in counts:
            global_counts[val]+=counts[val]
    return global_counts

print solve_puzzle(5)

racks.py

import copy

def delta(first,second):
    return abs(first - second)

def add(first,second):
    return first+second

class row(object):
    def __init__(self, length, first):
        self.length = length
        self.elements = [first]
    def append(self, element):
        self.elements.append(element)
        
class rack(object):
    def __init__(self, leftmost, secondleftmost, operation=delta, rack_size=5):
        if (leftmost != secondleftmost and leftmost != 2*secondleftmost and 2*leftmost != secondleftmost):
            self.operation=operation
            self.rack_size=rack_size
            self.rows = [row(rack_size, leftmost)]
            self.rows[0].append(secondleftmost)
            self.remaining_balls = list(range(1,((rack_size*(rack_size+1))/2)+1))
            self.remaining_balls.remove(leftmost)
            self.remaining_balls.remove(secondleftmost)
            self.fill_last()
        else:
            raise ValueError("Incorrect Starting Configuration")
    def fill_last(self):
        current_last_length = self.rack_size+1-len(self.rows)
        diff = self.operation(self.rows[-1].elements[0], self.rows[-1].elements[1])
        if diff in self.remaining_balls:
            self.rows.append(row(current_last_length,diff))
            self.remaining_balls.remove(diff)
        else:
            raise ValueError("Impossible Rack")
    def add_ball(self, value):
        if value in self.remaining_balls:
            self.rows[0].append(value)
            self.remaining_balls.remove(value)
            for i in range(1,len(self.rows[0].elements)-1):
                diff = self.operation(self.rows[i-1].elements[-1],self.rows[i-1].elements[-2])
                if diff in self.remaining_balls:
                    self.rows[i].append(diff)
                    self.remaining_balls.remove(diff)
                else:
                    raise ValueError("Impossible Rack")
            self.fill_last()
            return self.remaining_balls
        else:
            raise ValueError("Impossible Rack")
    def print_rack(self):
        if len(self.rows[-1].elements) == 1:
            for i, row in enumerate(self.rows[::-1]):
                print "row %s:" % (len(self.rows)-i-1) + "  "*(len(self.rows)-i-1), row.elements
            print "Remaining:", self.remaining_balls
        else:
            raise RuntimeError("Attempting to print unfinished rack")
    def print_last_row(self):
        print self.rows[0].elements

all 5-racks with 21 balls

row 4:         [4]
row 3:       [5, 9]
row 2:     [7, 12, 3]
row 1:   [13, 6, 18, 15]
row 0: [1, 14, 20, 2, 17]
Remaining: [8, 10, 11, 16, 19, 21]
row 4:         [5]
row 3:       [4, 9]
row 2:     [7, 11, 2]
row 1:   [13, 6, 17, 15]
row 0: [1, 14, 20, 3, 18]
Remaining: [8, 10, 12, 16, 19, 21]
row 4:         [4]
row 3:       [5, 9]
row 2:     [6, 11, 2]
row 1:   [13, 7, 18, 16]
row 0: [1, 14, 21, 3, 19]
Remaining: [8, 10, 12, 15, 17, 20]
row 4:         [6]
row 3:       [2, 8]
row 2:     [9, 11, 3]
row 1:   [14, 5, 16, 13]
row 0: [1, 15, 20, 4, 17]
Remaining: [7, 10, 12, 18, 19, 21]
row 4:         [6]
row 3:       [10, 4]
row 2:     [13, 3, 7]
row 1:   [15, 2, 5, 12]
row 0: [1, 16, 14, 9, 21]
Remaining: [8, 11, 17, 18, 19, 20]
row 4:         [7]
row 3:       [2, 9]
row 2:     [10, 12, 3]
row 1:   [15, 5, 17, 14]
row 0: [1, 16, 21, 4, 18]
Remaining: [6, 8, 11, 13, 19, 20]
row 4:         [3]
row 3:       [9, 6]
row 2:     [11, 2, 8]
row 1:   [16, 5, 7, 15]
row 0: [1, 17, 12, 19, 4]
Remaining: [10, 13, 14, 18, 20, 21]
row 4:         [7]
row 3:       [10, 3]
row 2:     [12, 2, 5]
row 1:   [16, 4, 6, 11]
row 0: [1, 17, 13, 19, 8]
Remaining: [9, 14, 15, 18, 20, 21]
row 4:         [2]
row 3:       [4, 6]
row 2:     [9, 5, 11]
row 1:   [17, 8, 3, 14]
row 0: [1, 18, 10, 7, 21]
Remaining: [12, 13, 15, 16, 19, 20]
row 4:         [8]
row 3:       [10, 2]
row 2:     [13, 3, 5]
row 1:   [17, 4, 7, 12]
row 0: [1, 18, 14, 21, 9]
Remaining: [6, 11, 15, 16, 19, 20]
row 4:         [8]
row 3:       [12, 4]
row 2:     [14, 2, 6]
row 1:   [17, 3, 5, 11]
row 0: [1, 18, 15, 10, 21]
Remaining: [7, 9, 13, 16, 19, 20]
row 4:         [8]
row 3:       [12, 4]
row 2:     [14, 2, 6]
row 1:   [17, 3, 5, 11]
row 0: [1, 18, 15, 20, 9]
Remaining: [7, 10, 13, 16, 19, 21]
row 4:         [5]
row 3:       [3, 8]
row 2:     [15, 12, 4]
row 1:   [17, 2, 14, 10]
row 0: [1, 18, 20, 6, 16]
Remaining: [7, 9, 11, 13, 19, 21]
row 4:         [3]
row 3:       [9, 6]
row 2:     [11, 2, 8]
row 1:   [18, 7, 5, 13]
row 0: [1, 19, 12, 17, 4]
Remaining: [10, 14, 15, 16, 20, 21]
row 4:         [5]
row 3:       [8, 3]
row 2:     [12, 4, 7]
row 1:   [18, 6, 2, 9]
row 0: [1, 19, 13, 11, 20]
Remaining: [10, 14, 15, 16, 17, 21]
row 4:         [7]
row 3:       [10, 3]
row 2:     [12, 2, 5]
row 1:   [18, 6, 4, 9]
row 0: [1, 19, 13, 17, 8]
Remaining: [11, 14, 15, 16, 20, 21]
row 4:         [3]
row 3:       [10, 7]
row 2:     [12, 2, 9]
row 1:   [18, 6, 8, 17]
row 0: [1, 19, 13, 21, 4]
Remaining: [5, 11, 14, 15, 16, 20]
row 4:         [6]
row 3:       [10, 4]
row 2:     [13, 3, 7]
row 1:   [18, 5, 2, 9]
row 0: [1, 19, 14, 12, 21]
Remaining: [8, 11, 15, 16, 17, 20]
row 4:         [7]
row 3:       [11, 4]
row 2:     [13, 2, 6]
row 1:   [18, 5, 3, 9]
row 0: [1, 19, 14, 17, 8]
Remaining: [10, 12, 15, 16, 20, 21]
row 4:         [3]
row 3:       [11, 8]
row 2:     [13, 2, 10]
row 1:   [18, 5, 7, 17]
row 0: [1, 19, 14, 21, 4]
Remaining: [6, 9, 12, 15, 16, 20]
row 4:         [9]
row 3:       [11, 2]
row 2:     [14, 3, 5]
row 1:   [18, 4, 7, 12]
row 0: [1, 19, 15, 8, 20]
Remaining: [6, 10, 13, 16, 17, 21]
row 4:         [9]
row 3:       [12, 3]
row 2:     [14, 2, 5]
row 1:   [18, 4, 6, 11]
row 0: [1, 19, 15, 21, 10]
Remaining: [7, 8, 13, 16, 17, 20]
row 4:         [9]
row 3:       [13, 4]
row 2:     [15, 2, 6]
row 1:   [18, 3, 5, 11]
row 0: [1, 19, 16, 21, 10]
Remaining: [7, 8, 12, 14, 17, 20]
row 4:         [4]
row 3:       [7, 3]
row 2:     [16, 9, 6]
row 1:   [18, 2, 11, 5]
row 0: [1, 19, 21, 10, 15]
Remaining: [8, 12, 13, 14, 17, 20]
row 4:         [8]
row 3:       [10, 2]
row 2:     [15, 5, 3]
row 1:   [19, 4, 9, 6]
row 0: [1, 20, 16, 7, 13]
Remaining: [11, 12, 14, 17, 18, 21]
row 4:         [5]
row 3:       [12, 7]
row 2:     [3, 15, 8]
row 1:   [20, 17, 2, 10]
row 0: [1, 21, 4, 6, 16]
Remaining: [9, 11, 13, 14, 18, 19]
row 4:         [3]
row 3:       [10, 7]
row 2:     [12, 2, 9]
row 1:   [20, 8, 6, 15]
row 0: [1, 21, 13, 19, 4]
Remaining: [5, 11, 14, 16, 17, 18]
row 4:         [3]
row 3:       [11, 8]
row 2:     [13, 2, 10]
row 1:   [20, 7, 5, 15]
row 0: [1, 21, 14, 19, 4]
Remaining: [6, 9, 12, 16, 17, 18]
row 4:         [7]
row 3:       [9, 2]
row 2:     [14, 5, 3]
row 1:   [20, 6, 11, 8]
row 0: [1, 21, 15, 4, 12]
Remaining: [10, 13, 16, 17, 18, 19]
row 4:         [8]
row 3:       [11, 3]
row 2:     [18, 7, 4]
row 1:   [20, 2, 9, 5]
row 0: [1, 21, 19, 10, 15]
Remaining: [6, 12, 13, 14, 16, 17]
row 4:         [4]
row 3:       [6, 10]
row 2:     [7, 13, 3]
row 1:   [12, 5, 18, 15]
row 0: [2, 14, 19, 1, 16]
Remaining: [8, 9, 11, 17, 20, 21]
row 4:         [6]
row 3:       [4, 10]
row 2:     [7, 11, 1]
row 1:   [12, 5, 16, 15]
row 0: [2, 14, 19, 3, 18]
Remaining: [8, 9, 13, 17, 20, 21]
row 4:         [6]
row 3:       [4, 10]
row 2:     [7, 11, 1]
row 1:   [12, 5, 16, 17]
row 0: [2, 14, 19, 3, 20]
Remaining: [8, 9, 13, 15, 18, 21]
row 4:         [6]
row 3:       [3, 9]
row 2:     [7, 10, 1]
row 1:   [12, 5, 15, 16]
row 0: [2, 14, 19, 4, 20]
Remaining: [8, 11, 13, 17, 18, 21]
row 4:         [4]
row 3:       [6, 10]
row 2:     [5, 11, 1]
row 1:   [12, 7, 18, 17]
row 0: [2, 14, 21, 3, 20]
Remaining: [8, 9, 13, 15, 16, 19]
row 4:         [6]
row 3:       [1, 7]
row 2:     [9, 10, 3]
row 1:   [13, 4, 14, 11]
row 0: [2, 15, 19, 5, 16]
Remaining: [8, 12, 17, 18, 20, 21]
row 4:         [7]
row 3:       [4, 11]
row 2:     [8, 12, 1]
row 1:   [13, 5, 17, 16]
row 0: [2, 15, 20, 3, 19]
Remaining: [6, 9, 10, 14, 18, 21]
row 4:         [7]
row 3:       [4, 11]
row 2:     [8, 12, 1]
row 1:   [13, 5, 17, 18]
row 0: [2, 15, 20, 3, 21]
Remaining: [6, 9, 10, 14, 16, 19]
row 4:         [7]
row 3:       [3, 10]
row 2:     [8, 11, 1]
row 1:   [13, 5, 16, 17]
row 0: [2, 15, 20, 4, 21]
Remaining: [6, 9, 12, 14, 18, 19]
row 4:         [6]
row 3:       [9, 3]
row 2:     [13, 4, 7]
row 1:   [14, 1, 5, 12]
row 0: [2, 16, 15, 20, 8]
Remaining: [10, 11, 17, 18, 19, 21]
row 4:         [5]
row 3:       [4, 9]
row 2:     [11, 15, 6]
row 1:   [14, 3, 18, 12]
row 0: [2, 16, 19, 1, 13]
Remaining: [7, 8, 10, 17, 20, 21]
row 4:         [3]
row 3:       [5, 8]
row 2:     [10, 15, 7]
row 1:   [14, 4, 19, 12]
row 0: [2, 16, 20, 1, 13]
Remaining: [6, 9, 11, 17, 18, 21]
row 4:         [7]
row 3:       [1, 8]
row 2:     [10, 11, 3]
row 1:   [14, 4, 15, 12]
row 0: [2, 16, 20, 5, 17]
Remaining: [6, 9, 13, 18, 19, 21]
row 4:         [8]
row 3:       [4, 12]
row 2:     [9, 13, 1]
row 1:   [14, 5, 18, 17]
row 0: [2, 16, 21, 3, 20]
Remaining: [6, 7, 10, 11, 15, 19]
row 4:         [3]
row 3:       [9, 6]
row 2:     [10, 1, 7]
row 1:   [15, 5, 4, 11]
row 0: [2, 17, 12, 8, 19]
Remaining: [13, 14, 16, 18, 20, 21]
row 4:         [5]
row 3:       [11, 6]
row 2:     [14, 3, 9]
row 1:   [15, 1, 4, 13]
row 0: [2, 17, 16, 20, 7]
Remaining: [8, 10, 12, 18, 19, 21]
row 4:         [8]
row 3:       [1, 9]
row 2:     [11, 12, 3]
row 1:   [15, 4, 16, 13]
row 0: [2, 17, 21, 5, 18]
Remaining: [6, 7, 10, 14, 19, 20]
row 4:         [1]
row 3:       [5, 6]
row 2:     [9, 4, 10]
row 1:   [16, 7, 3, 13]
row 0: [2, 18, 11, 8, 21]
Remaining: [12, 14, 15, 17, 19, 20]
row 4:         [3]
row 3:       [10, 7]
row 2:     [11, 1, 8]
row 1:   [16, 5, 4, 12]
row 0: [2, 18, 13, 9, 21]
Remaining: [6, 14, 15, 17, 19, 20]
row 4:         [7]
row 3:       [12, 5]
row 2:     [13, 1, 6]
row 1:   [16, 3, 4, 10]
row 0: [2, 18, 15, 11, 21]
Remaining: [8, 9, 14, 17, 19, 20]
row 4:         [7]
row 3:       [12, 5]
row 2:     [13, 1, 6]
row 1:   [16, 3, 4, 10]
row 0: [2, 18, 15, 19, 9]
Remaining: [8, 11, 14, 17, 20, 21]
row 4:         [4]
row 3:       [3, 7]
row 2:     [15, 12, 5]
row 1:   [16, 1, 13, 8]
row 0: [2, 18, 19, 6, 14]
Remaining: [9, 10, 11, 17, 20, 21]
row 4:         [7]
row 3:       [4, 11]
row 2:     [13, 17, 6]
row 1:   [16, 3, 20, 14]
row 0: [2, 18, 21, 1, 15]
Remaining: [5, 8, 9, 10, 12, 19]
row 4:         [7]
row 3:       [1, 8]
row 2:     [13, 14, 6]
row 1:   [16, 3, 17, 11]
row 0: [2, 18, 21, 4, 15]
Remaining: [5, 9, 10, 12, 19, 20]
row 4:         [4]
row 3:       [1, 5]
row 2:     [13, 12, 7]
row 1:   [16, 3, 15, 8]
row 0: [2, 18, 21, 6, 14]
Remaining: [9, 10, 11, 17, 19, 20]
row 4:         [7]
row 3:       [1, 8]
row 2:     [13, 12, 4]
row 1:   [16, 3, 15, 11]
row 0: [2, 18, 21, 6, 17]
Remaining: [5, 9, 10, 14, 19, 20]
row 4:         [6]
row 3:       [10, 4]
row 2:     [1, 11, 7]
row 1:   [17, 16, 5, 12]
row 0: [2, 19, 3, 8, 20]
Remaining: [9, 13, 14, 15, 18, 21]
row 4:         [5]
row 3:       [9, 4]
row 2:     [16, 7, 3]
row 1:   [17, 1, 8, 11]
row 0: [2, 19, 18, 10, 21]
Remaining: [6, 12, 13, 14, 15, 20]
row 4:         [6]
row 3:       [9, 3]
row 2:     [10, 1, 4]
row 1:   [18, 8, 7, 11]
row 0: [2, 20, 12, 5, 16]
Remaining: [13, 14, 15, 17, 19, 21]
row 4:         [8]
row 3:       [11, 3]
row 2:     [12, 1, 4]
row 1:   [18, 6, 5, 9]
row 0: [2, 20, 14, 19, 10]
Remaining: [7, 13, 15, 16, 17, 21]
row 4:         [3]
row 3:       [11, 8]
row 2:     [12, 1, 9]
row 1:   [18, 6, 7, 16]
row 0: [2, 20, 14, 21, 5]
Remaining: [4, 10, 13, 15, 17, 19]
row 4:         [9]
row 3:       [10, 1]
row 2:     [13, 3, 4]
row 1:   [18, 5, 8, 12]
row 0: [2, 20, 15, 7, 19]
Remaining: [6, 11, 14, 16, 17, 21]
row 4:         [9]
row 3:       [12, 3]
row 2:     [13, 1, 4]
row 1:   [18, 5, 6, 10]
row 0: [2, 20, 15, 21, 11]
Remaining: [7, 8, 14, 16, 17, 19]
row 4:         [5]
row 3:       [11, 6]
row 2:     [14, 3, 9]
row 1:   [18, 4, 1, 10]
row 0: [2, 20, 16, 17, 7]
Remaining: [8, 12, 13, 15, 19, 21]
row 4:         [5]
row 3:       [13, 8]
row 2:     [14, 1, 9]
row 1:   [18, 4, 3, 12]
row 0: [2, 20, 16, 19, 7]
Remaining: [6, 10, 11, 15, 17, 21]
row 4:         [8]
row 3:       [13, 5]
row 2:     [14, 1, 6]
row 1:   [18, 4, 3, 9]
row 0: [2, 20, 16, 19, 10]
Remaining: [7, 11, 12, 15, 17, 21]
row 4:         [6]
row 3:       [10, 4]
row 2:     [15, 5, 1]
row 1:   [18, 3, 8, 7]
row 0: [2, 20, 17, 9, 16]
Remaining: [11, 12, 13, 14, 19, 21]
row 4:         [8]
row 3:       [14, 6]
row 2:     [15, 1, 7]
row 1:   [18, 3, 4, 11]
row 0: [2, 20, 17, 21, 10]
Remaining: [5, 9, 12, 13, 16, 19]
row 4:         [9]
row 3:       [14, 5]
row 2:     [15, 1, 6]
row 1:   [18, 3, 4, 10]
row 0: [2, 20, 17, 21, 11]
Remaining: [7, 8, 12, 13, 16, 19]
row 4:         [5]
row 3:       [3, 8]
row 2:     [17, 14, 6]
row 1:   [18, 1, 15, 9]
row 0: [2, 20, 19, 4, 13]
Remaining: [7, 10, 11, 12, 16, 21]
row 4:         [6]
row 3:       [4, 10]
row 2:     [17, 13, 3]
row 1:   [18, 1, 14, 11]
row 0: [2, 20, 19, 5, 16]
Remaining: [7, 8, 9, 12, 15, 21]
row 4:         [3]
row 3:       [5, 8]
row 2:     [17, 12, 4]
row 1:   [18, 1, 13, 9]
row 0: [2, 20, 19, 6, 15]
Remaining: [7, 10, 11, 14, 16, 21]
row 4:         [4]
row 3:       [5, 9]
row 2:     [17, 12, 3]
row 1:   [18, 1, 13, 10]
row 0: [2, 20, 19, 6, 16]
Remaining: [7, 8, 11, 14, 15, 21]
row 4:         [3]
row 3:       [7, 4]
row 2:     [17, 10, 6]
row 1:   [18, 1, 11, 5]
row 0: [2, 20, 19, 8, 13]
Remaining: [9, 12, 14, 15, 16, 21]
row 4:         [6]
row 3:       [10, 4]
row 2:     [17, 7, 3]
row 1:   [18, 1, 8, 5]
row 0: [2, 20, 19, 11, 16]
Remaining: [9, 12, 13, 14, 15, 21]
row 4:         [5]
row 3:       [8, 3]
row 2:     [17, 9, 6]
row 1:   [18, 1, 10, 4]
row 0: [2, 20, 21, 11, 7]
Remaining: [12, 13, 14, 15, 16, 19]
row 4:         [5]
row 3:       [8, 3]
row 2:     [17, 9, 6]
row 1:   [18, 1, 10, 4]
row 0: [2, 20, 21, 11, 15]
Remaining: [7, 12, 13, 14, 16, 19]
row 4:         [5]
row 3:       [13, 8]
row 2:     [1, 14, 6]
row 1:   [19, 18, 4, 10]
row 0: [2, 21, 3, 7, 17]
Remaining: [9, 11, 12, 15, 16, 20]
row 4:         [7]
row 3:       [10, 3]
row 2:     [4, 14, 11]
row 1:   [19, 15, 1, 12]
row 0: [2, 21, 6, 5, 17]
Remaining: [8, 9, 13, 16, 18, 20]
row 4:         [3]
row 3:       [11, 8]
row 2:     [12, 1, 9]
row 1:   [19, 7, 6, 15]
row 0: [2, 21, 14, 20, 5]
Remaining: [4, 10, 13, 16, 17, 18]
row 4:         [8]
row 3:       [9, 1]
row 2:     [13, 4, 3]
row 1:   [19, 6, 10, 7]
row 0: [2, 21, 15, 5, 12]
Remaining: [11, 14, 16, 17, 18, 20]
row 4:         [5]
row 3:       [14, 9]
row 2:     [15, 1, 10]
row 1:   [19, 4, 3, 13]
row 0: [2, 21, 17, 20, 7]
Remaining: [6, 8, 11, 12, 16, 18]
row 4:         [5]
row 3:       [3, 8]
row 2:     [18, 15, 7]
row 1:   [19, 1, 16, 9]
row 0: [2, 21, 20, 4, 13]
Remaining: [6, 10, 11, 12, 14, 17]
row 4:         [7]
row 3:       [4, 11]
row 2:     [18, 14, 3]
row 1:   [19, 1, 15, 12]
row 0: [2, 21, 20, 5, 17]
Remaining: [6, 8, 9, 10, 13, 16]
row 4:         [5]
row 3:       [8, 3]
row 2:     [18, 10, 7]
row 1:   [19, 1, 11, 4]
row 0: [2, 21, 20, 9, 13]
Remaining: [6, 12, 14, 15, 16, 17]
row 4:         [7]
row 3:       [10, 3]
row 2:     [18, 8, 5]
row 1:   [19, 1, 9, 4]
row 0: [2, 21, 20, 11, 15]
Remaining: [6, 12, 13, 14, 16, 17]
row 4:         [6]
row 3:       [2, 8]
row 2:     [7, 9, 1]
row 1:   [11, 4, 13, 12]
row 0: [3, 14, 18, 5, 17]
Remaining: [10, 15, 16, 19, 20, 21]
row 4:         [7]
row 3:       [2, 9]
row 2:     [8, 10, 1]
row 1:   [12, 4, 14, 13]
row 0: [3, 15, 19, 5, 18]
Remaining: [6, 11, 16, 17, 20, 21]
row 4:         [1]
row 3:       [5, 6]
row 2:     [9, 14, 8]
row 1:   [13, 4, 18, 10]
row 0: [3, 16, 20, 2, 12]
Remaining: [7, 11, 15, 17, 19, 21]
row 4:         [8]
row 3:       [2, 10]
row 2:     [9, 11, 1]
row 1:   [13, 4, 15, 14]
row 0: [3, 16, 20, 5, 19]
Remaining: [6, 7, 12, 17, 18, 21]
row 4:         [7]
row 3:       [1, 8]
row 2:     [9, 10, 2]
row 1:   [13, 4, 14, 12]
row 0: [3, 16, 20, 6, 18]
Remaining: [5, 11, 15, 17, 19, 21]
row 4:         [7]
row 3:       [2, 9]
row 2:     [8, 10, 1]
row 1:   [13, 5, 15, 14]
row 0: [3, 16, 21, 6, 20]
Remaining: [4, 11, 12, 17, 18, 19]
row 4:         [6]
row 3:       [1, 7]
row 2:     [5, 4, 11]
row 1:   [14, 9, 13, 2]
row 0: [3, 17, 8, 21, 19]
Remaining: [10, 12, 15, 16, 18, 20]
row 4:         [5]
row 3:       [2, 7]
row 2:     [6, 4, 11]
row 1:   [14, 8, 12, 1]
row 0: [3, 17, 9, 21, 20]
Remaining: [10, 13, 15, 16, 18, 19]
row 4:         [7]
row 3:       [8, 1]
row 2:     [12, 4, 5]
row 1:   [14, 2, 6, 11]
row 0: [3, 17, 15, 9, 20]
Remaining: [10, 13, 16, 18, 19, 21]
row 4:         [7]
row 3:       [8, 1]
row 2:     [12, 4, 5]
row 1:   [14, 2, 6, 11]
row 0: [3, 17, 15, 21, 10]
Remaining: [9, 13, 16, 18, 19, 20]
row 4:         [7]
row 3:       [9, 2]
row 2:     [13, 4, 6]
row 1:   [14, 1, 5, 11]
row 0: [3, 17, 16, 21, 10]
Remaining: [8, 12, 15, 18, 19, 20]
row 4:         [9]
row 3:       [2, 11]
row 2:     [10, 12, 1]
row 1:   [14, 4, 16, 15]
row 0: [3, 17, 21, 5, 20]
Remaining: [6, 7, 8, 13, 18, 19]
row 4:         [8]
row 3:       [1, 9]
row 2:     [10, 11, 2]
row 1:   [14, 4, 15, 13]
row 0: [3, 17, 21, 6, 19]
Remaining: [5, 7, 12, 16, 18, 20]
row 4:         [6]
row 3:       [1, 7]
row 2:     [10, 9, 2]
row 1:   [14, 4, 13, 11]
row 0: [3, 17, 21, 8, 19]
Remaining: [5, 12, 15, 16, 18, 20]
row 4:         [5]
row 3:       [2, 7]
row 2:     [10, 8, 1]
row 1:   [14, 4, 12, 11]
row 0: [3, 17, 21, 9, 20]
Remaining: [6, 13, 15, 16, 18, 19]
row 4:         [8]
row 3:       [9, 1]
row 2:     [13, 4, 5]
row 1:   [15, 2, 6, 11]
row 0: [3, 18, 16, 10, 21]
Remaining: [7, 12, 14, 17, 19, 20]
row 4:         [5]
row 3:       [4, 9]
row 2:     [13, 17, 8]
row 1:   [15, 2, 19, 11]
row 0: [3, 18, 20, 1, 12]
Remaining: [6, 7, 10, 14, 16, 21]
row 4:         [7]
row 3:       [1, 8]
row 2:     [13, 12, 4]
row 1:   [15, 2, 14, 10]
row 0: [3, 18, 20, 6, 16]
Remaining: [5, 9, 11, 17, 19, 21]
row 4:         [7]
row 3:       [11, 4]
row 2:     [2, 13, 9]
row 1:   [16, 18, 5, 14]
row 0: [3, 19, 1, 6, 20]
Remaining: [8, 10, 12, 15, 17, 21]
row 4:         [4]
row 3:       [9, 5]
row 2:     [2, 11, 6]
row 1:   [16, 18, 7, 13]
row 0: [3, 19, 1, 8, 21]
Remaining: [10, 12, 14, 15, 17, 20]
row 4:         [1]
row 3:       [9, 8]
row 2:     [11, 2, 10]
row 1:   [16, 5, 7, 17]
row 0: [3, 19, 14, 21, 4]
Remaining: [6, 12, 13, 15, 18, 20]
row 4:         [8]
row 3:       [9, 1]
row 2:     [14, 5, 4]
row 1:   [16, 2, 7, 11]
row 0: [3, 19, 17, 10, 21]
Remaining: [6, 12, 13, 15, 18, 20]
row 4:         [5]
row 3:       [2, 7]
row 2:     [15, 13, 6]
row 1:   [16, 1, 14, 8]
row 0: [3, 19, 18, 4, 12]
Remaining: [9, 10, 11, 17, 20, 21]
row 4:         [5]
row 3:       [4, 9]
row 2:     [15, 11, 2]
row 1:   [16, 1, 12, 14]
row 0: [3, 19, 18, 6, 20]
Remaining: [7, 8, 10, 13, 17, 21]
row 4:         [5]
row 3:       [4, 9]
row 2:     [15, 11, 2]
row 1:   [16, 1, 12, 10]
row 0: [3, 19, 20, 8, 18]
Remaining: [6, 7, 13, 14, 17, 21]
row 4:         [6]
row 3:       [4, 10]
row 2:     [14, 18, 8]
row 1:   [16, 2, 20, 12]
row 0: [3, 19, 21, 1, 13]
Remaining: [5, 7, 9, 11, 15, 17]
row 4:         [5]
row 3:       [1, 6]
row 2:     [14, 15, 9]
row 1:   [16, 2, 17, 8]
row 0: [3, 19, 21, 4, 12]
Remaining: [7, 10, 11, 13, 18, 20]
row 4:         [6]
row 3:       [1, 7]
row 2:     [14, 15, 8]
row 1:   [16, 2, 17, 9]
row 0: [3, 19, 21, 4, 13]
Remaining: [5, 10, 11, 12, 18, 20]
row 4:         [8]
row 3:       [1, 9]
row 2:     [14, 13, 4]
row 1:   [16, 2, 15, 11]
row 0: [3, 19, 21, 6, 17]
Remaining: [5, 7, 10, 12, 18, 20]
row 4:         [8]
row 3:       [12, 4]
row 2:     [2, 14, 10]
row 1:   [17, 19, 5, 15]
row 0: [3, 20, 1, 6, 21]
Remaining: [7, 9, 11, 13, 16, 18]
row 4:         [8]
row 3:       [12, 4]
row 2:     [1, 13, 9]
row 1:   [17, 18, 5, 14]
row 0: [3, 20, 2, 7, 21]
Remaining: [6, 10, 11, 15, 16, 19]
row 4:         [4]
row 3:       [10, 6]
row 2:     [1, 11, 5]
row 1:   [17, 18, 7, 12]
row 0: [3, 20, 2, 9, 21]
Remaining: [8, 13, 14, 15, 16, 19]
row 4:         [4]
row 3:       [6, 2]
row 2:     [5, 11, 13]
row 1:   [17, 12, 1, 14]
row 0: [3, 20, 8, 7, 21]
Remaining: [9, 10, 15, 16, 18, 19]
row 4:         [6]
row 3:       [4, 10]
row 2:     [5, 1, 11]
row 1:   [17, 12, 13, 2]
row 0: [3, 20, 8, 21, 19]
Remaining: [7, 9, 14, 15, 16, 18]
row 4:         [4]
row 3:       [9, 5]
row 2:     [10, 1, 6]
row 1:   [17, 7, 8, 2]
row 0: [3, 20, 13, 21, 19]
Remaining: [11, 12, 14, 15, 16, 18]
row 4:         [2]
row 3:       [10, 8]
row 2:     [11, 1, 9]
row 1:   [17, 6, 7, 16]
row 0: [3, 20, 14, 21, 5]
Remaining: [4, 12, 13, 15, 18, 19]
row 4:         [6]
row 3:       [4, 10]
row 2:     [15, 11, 1]
row 1:   [17, 2, 13, 14]
row 0: [3, 20, 18, 5, 19]
Remaining: [7, 8, 9, 12, 16, 21]
row 4:         [4]
row 3:       [5, 9]
row 2:     [15, 10, 1]
row 1:   [17, 2, 12, 13]
row 0: [3, 20, 18, 6, 19]
Remaining: [7, 8, 11, 14, 16, 21]
row 4:         [6]
row 3:       [10, 4]
row 2:     [15, 5, 1]
row 1:   [17, 2, 7, 8]
row 0: [3, 20, 18, 11, 19]
Remaining: [9, 12, 13, 14, 16, 21]
row 4:         [4]
row 3:       [5, 9]
row 2:     [16, 11, 2]
row 1:   [17, 1, 12, 14]
row 0: [3, 20, 19, 7, 21]
Remaining: [6, 8, 10, 13, 15, 18]
row 4:         [4]
row 3:       [9, 5]
row 2:     [16, 7, 2]
row 1:   [17, 1, 8, 10]
row 0: [3, 20, 19, 11, 21]
Remaining: [6, 12, 13, 14, 15, 18]
row 4:         [6]
row 3:       [4, 10]
row 2:     [16, 12, 2]
row 1:   [17, 1, 13, 11]
row 0: [3, 20, 21, 8, 19]
Remaining: [5, 7, 9, 14, 15, 18]
row 4:         [4]
row 3:       [9, 5]
row 2:     [16, 7, 2]
row 1:   [17, 1, 8, 6]
row 0: [3, 20, 21, 13, 19]
Remaining: [10, 11, 12, 14, 15, 18]
row 4:         [7]
row 3:       [13, 6]
row 2:     [2, 15, 9]
row 1:   [18, 16, 1, 10]
row 0: [3, 21, 5, 4, 14]
Remaining: [8, 11, 12, 17, 19, 20]
row 4:         [5]
row 3:       [2, 7]
row 2:     [6, 4, 11]
row 1:   [18, 12, 8, 19]
row 0: [3, 21, 9, 1, 20]
Remaining: [10, 13, 14, 15, 16, 17]
row 4:         [1]
row 3:       [9, 8]
row 2:     [11, 2, 10]
row 1:   [18, 7, 5, 15]
row 0: [3, 21, 14, 19, 4]
Remaining: [6, 12, 13, 16, 17, 20]
row 4:         [2]
row 3:       [10, 8]
row 2:     [11, 1, 9]
row 1:   [18, 7, 6, 15]
row 0: [3, 21, 14, 20, 5]
Remaining: [4, 12, 13, 16, 17, 19]
row 4:         [7]
row 3:       [4, 11]
row 2:     [16, 12, 1]
row 1:   [18, 2, 14, 15]
row 0: [3, 21, 19, 5, 20]
Remaining: [6, 8, 9, 10, 13, 17]
row 4:         [6]
row 3:       [2, 8]
row 2:     [17, 15, 7]
row 1:   [18, 1, 16, 9]
row 0: [3, 21, 20, 4, 13]
Remaining: [5, 10, 11, 12, 14, 19]
row 4:         [7]
row 3:       [2, 9]
row 2:     [17, 15, 6]
row 1:   [18, 1, 16, 10]
row 0: [3, 21, 20, 4, 14]
Remaining: [5, 8, 11, 12, 13, 19]
row 4:         [2]
row 3:       [6, 4]
row 2:     [17, 11, 7]
row 1:   [18, 1, 12, 5]
row 0: [3, 21, 20, 8, 13]
Remaining: [9, 10, 14, 15, 16, 19]
row 4:         [10]
row 3:       [12, 2]
row 2:     [17, 5, 7]
row 1:   [3, 20, 15, 8]
row 0: [4, 1, 21, 6, 14]
Remaining: [9, 11, 13, 16, 18, 19]
row 4:         [7]
row 3:       [15, 8]
row 2:     [17, 2, 10]
row 1:   [1, 18, 16, 6]
row 0: [4, 3, 21, 5, 11]
Remaining: [9, 12, 13, 14, 19, 20]
row 4:         [10]
row 3:       [12, 2]
row 2:     [17, 5, 7]
row 1:   [1, 18, 13, 6]
row 0: [4, 3, 21, 8, 14]
Remaining: [9, 11, 15, 16, 19, 20]
row 4:         [2]
row 3:       [6, 8]
row 2:     [7, 13, 5]
row 1:   [10, 3, 16, 11]
row 0: [4, 14, 17, 1, 12]
Remaining: [9, 15, 18, 19, 20, 21]
row 4:         [5]
row 3:       [6, 11]
row 2:     [7, 13, 2]
row 1:   [10, 3, 16, 18]
row 0: [4, 14, 17, 1, 19]
Remaining: [8, 9, 12, 15, 20, 21]
row 4:         [6]
row 3:       [5, 11]
row 2:     [7, 12, 1]
row 1:   [10, 3, 15, 16]
row 0: [4, 14, 17, 2, 18]
Remaining: [8, 9, 13, 19, 20, 21]
row 4:         [6]
row 3:       [2, 8]
row 2:     [7, 9, 1]
row 1:   [10, 3, 12, 11]
row 0: [4, 14, 17, 5, 16]
Remaining: [13, 15, 18, 19, 20, 21]
row 4:         [6]
row 3:       [2, 8]
row 2:     [7, 9, 1]
row 1:   [10, 3, 12, 13]
row 0: [4, 14, 17, 5, 18]
Remaining: [11, 15, 16, 19, 20, 21]
row 4:         [2]
row 3:       [6, 8]
row 2:     [3, 9, 1]
row 1:   [10, 7, 16, 15]
row 0: [4, 14, 21, 5, 20]
Remaining: [11, 12, 13, 17, 18, 19]
row 4:         [7]
row 3:       [3, 10]
row 2:     [5, 2, 12]
row 1:   [11, 6, 8, 20]
row 0: [4, 15, 9, 1, 21]
Remaining: [13, 14, 16, 17, 18, 19]
row 4:         [7]
row 3:       [5, 12]
row 2:     [8, 13, 1]
row 1:   [11, 3, 16, 17]
row 0: [4, 15, 18, 2, 19]
Remaining: [6, 9, 10, 14, 20, 21]
row 4:         [7]
row 3:       [2, 9]
row 2:     [8, 10, 1]
row 1:   [11, 3, 13, 12]
row 0: [4, 15, 18, 5, 17]
Remaining: [6, 14, 16, 19, 20, 21]
row 4:         [7]
row 3:       [2, 9]
row 2:     [8, 10, 1]
row 1:   [11, 3, 13, 14]
row 0: [4, 15, 18, 5, 19]
Remaining: [6, 12, 16, 17, 20, 21]
row 4:         [3]
row 3:       [9, 12]
row 2:     [5, 14, 2]
row 1:   [11, 6, 20, 18]
row 0: [4, 15, 21, 1, 19]
Remaining: [7, 8, 10, 13, 16, 17]
row 4:         [6]
row 3:       [2, 8]
row 2:     [3, 5, 13]
row 1:   [12, 15, 20, 7]
row 0: [4, 16, 1, 21, 14]
Remaining: [9, 10, 11, 17, 18, 19]
row 4:         [6]
row 3:       [2, 8]
row 2:     [3, 5, 13]
row 1:   [12, 9, 14, 1]
row 0: [4, 16, 7, 21, 20]
Remaining: [10, 11, 15, 17, 18, 19]
row 4:         [5]
row 3:       [1, 6]
row 2:     [10, 9, 3]
row 1:   [12, 2, 11, 8]
row 0: [4, 16, 18, 7, 15]
Remaining: [13, 14, 17, 19, 20, 21]
row 4:         [5]
row 3:       [1, 6]
row 2:     [10, 9, 3]
row 1:   [12, 2, 11, 14]
row 0: [4, 16, 18, 7, 21]
Remaining: [8, 13, 15, 17, 19, 20]
row 4:         [7]
row 3:       [6, 13]
row 2:     [9, 15, 2]
row 1:   [12, 3, 18, 20]
row 0: [4, 16, 19, 1, 21]
Remaining: [5, 8, 10, 11, 14, 17]
row 4:         [8]
row 3:       [5, 13]
row 2:     [9, 14, 1]
row 1:   [12, 3, 17, 18]
row 0: [4, 16, 19, 2, 20]
Remaining: [6, 7, 10, 11, 15, 21]
row 4:         [8]
row 3:       [2, 10]
row 2:     [9, 11, 1]
row 1:   [12, 3, 14, 13]
row 0: [4, 16, 19, 5, 18]
Remaining: [6, 7, 15, 17, 20, 21]
row 4:         [8]
row 3:       [2, 10]
row 2:     [9, 11, 1]
row 1:   [12, 3, 14, 15]
row 0: [4, 16, 19, 5, 20]
Remaining: [6, 7, 13, 17, 18, 21]
row 4:         [7]
row 3:       [1, 8]
row 2:     [9, 10, 2]
row 1:   [12, 3, 13, 11]
row 0: [4, 16, 19, 6, 17]
Remaining: [5, 14, 15, 18, 20, 21]
row 4:         [7]
row 3:       [1, 8]
row 2:     [9, 10, 2]
row 1:   [12, 3, 13, 15]
row 0: [4, 16, 19, 6, 21]
Remaining: [5, 11, 14, 17, 18, 20]
row 4:         [3]
row 3:       [6, 9]
row 2:     [8, 2, 11]
row 1:   [13, 5, 7, 18]
row 0: [4, 17, 12, 19, 1]
Remaining: [10, 14, 15, 16, 20, 21]
row 4:         [5]
row 3:       [3, 8]
row 2:     [12, 15, 7]
row 1:   [13, 1, 16, 9]
row 0: [4, 17, 18, 2, 11]
Remaining: [6, 10, 14, 19, 20, 21]
row 4:         [9]
row 3:       [5, 14]
row 2:     [10, 15, 1]
row 1:   [13, 3, 18, 19]
row 0: [4, 17, 20, 2, 21]
Remaining: [6, 7, 8, 11, 12, 16]
row 4:         [9]
row 3:       [2, 11]
row 2:     [10, 12, 1]
row 1:   [13, 3, 15, 14]
row 0: [4, 17, 20, 5, 19]
Remaining: [6, 7, 8, 16, 18, 21]
row 4:         [9]
row 3:       [2, 11]
row 2:     [10, 12, 1]
row 1:   [13, 3, 15, 16]
row 0: [4, 17, 20, 5, 21]
Remaining: [6, 7, 8, 14, 18, 19]
row 4:         [8]
row 3:       [1, 9]
row 2:     [10, 11, 2]
row 1:   [13, 3, 14, 12]
row 0: [4, 17, 20, 6, 18]
Remaining: [5, 7, 15, 16, 19, 21]
row 4:         [5]
row 3:       [2, 7]
row 2:     [10, 8, 1]
row 1:   [13, 3, 11, 12]
row 0: [4, 17, 20, 9, 21]
Remaining: [6, 14, 15, 16, 18, 19]
row 4:         [7]
row 3:       [9, 2]
row 2:     [3, 12, 10]
row 1:   [14, 17, 5, 15]
row 0: [4, 18, 1, 6, 21]
Remaining: [8, 11, 13, 16, 19, 20]
row 4:         [1]
row 3:       [7, 6]
row 2:     [9, 2, 8]
row 1:   [14, 5, 3, 11]
row 0: [4, 18, 13, 10, 21]
Remaining: [12, 15, 16, 17, 19, 20]
row 4:         [5]
row 3:       [11, 6]
row 2:     [12, 1, 7]
row 1:   [14, 2, 3, 10]
row 0: [4, 18, 16, 19, 9]
Remaining: [8, 13, 15, 17, 20, 21]
row 4:         [6]
row 3:       [2, 8]
row 2:     [13, 11, 3]
row 1:   [14, 1, 12, 15]
row 0: [4, 18, 17, 5, 20]
Remaining: [7, 9, 10, 16, 19, 21]
row 4:         [5]
row 3:       [3, 8]
row 2:     [13, 10, 2]
row 1:   [14, 1, 11, 9]
row 0: [4, 18, 17, 6, 15]
Remaining: [7, 12, 16, 19, 20, 21]
row 4:         [6]
row 3:       [3, 9]
row 2:     [13, 16, 7]
row 1:   [14, 1, 17, 10]
row 0: [4, 18, 19, 2, 12]
Remaining: [5, 8, 11, 15, 20, 21]
row 4:         [6]
row 3:       [2, 8]
row 2:     [13, 15, 7]
row 1:   [14, 1, 16, 9]
row 0: [4, 18, 19, 3, 12]
Remaining: [5, 10, 11, 17, 20, 21]
row 4:         [6]
row 3:       [2, 8]
row 2:     [13, 11, 3]
row 1:   [14, 1, 12, 9]
row 0: [4, 18, 19, 7, 16]
Remaining: [5, 10, 15, 17, 20, 21]
row 4:         [3]
row 3:       [5, 8]
row 2:     [12, 17, 9]
row 1:   [14, 2, 19, 10]
row 0: [4, 18, 20, 1, 11]
Remaining: [6, 7, 13, 15, 16, 21]
row 4:         [10]
row 3:       [2, 12]
row 2:     [11, 13, 1]
row 1:   [14, 3, 16, 15]
row 0: [4, 18, 21, 5, 20]
Remaining: [6, 7, 8, 9, 17, 19]
row 4:         [9]
row 3:       [1, 10]
row 2:     [11, 12, 2]
row 1:   [14, 3, 15, 13]
row 0: [4, 18, 21, 6, 19]
Remaining: [5, 7, 8, 16, 17, 20]
row 4:         [3]
row 3:       [6, 9]
row 2:     [8, 2, 11]
row 1:   [15, 7, 5, 16]
row 0: [4, 19, 12, 17, 1]
Remaining: [10, 13, 14, 18, 20, 21]
row 4:         [3]
row 3:       [7, 10]
row 2:     [9, 2, 12]
row 1:   [15, 6, 8, 20]
row 0: [4, 19, 13, 21, 1]
Remaining: [5, 11, 14, 16, 17, 18]
row 4:         [3]
row 3:       [8, 11]
row 2:     [10, 2, 13]
row 1:   [15, 5, 7, 20]
row 0: [4, 19, 14, 21, 1]
Remaining: [6, 9, 12, 16, 17, 18]
row 4:         [1]
row 3:       [8, 9]
row 2:     [10, 2, 11]
row 1:   [15, 5, 7, 18]
row 0: [4, 19, 14, 21, 3]
Remaining: [6, 12, 13, 16, 17, 20]
row 4:         [6]
row 3:       [3, 9]
row 2:     [13, 10, 1]
row 1:   [15, 2, 12, 11]
row 0: [4, 19, 17, 5, 16]
Remaining: [7, 8, 14, 18, 20, 21]
row 4:         [5]
row 3:       [12, 7]
row 2:     [13, 1, 8]
row 1:   [15, 2, 3, 11]
row 0: [4, 19, 17, 20, 9]
Remaining: [6, 10, 14, 16, 18, 21]
row 4:         [7]
row 3:       [12, 5]
row 2:     [13, 1, 6]
row 1:   [15, 2, 3, 9]
row 0: [4, 19, 17, 20, 11]
Remaining: [8, 10, 14, 16, 18, 21]
row 4:         [7]
row 3:       [2, 9]
row 2:     [14, 12, 3]
row 1:   [15, 1, 13, 16]
row 0: [4, 19, 18, 5, 21]
Remaining: [6, 8, 10, 11, 17, 20]
row 4:         [6]
row 3:       [3, 9]
row 2:     [14, 17, 8]
row 1:   [15, 1, 18, 10]
row 0: [4, 19, 20, 2, 12]
Remaining: [5, 7, 11, 13, 16, 21]
row 4:         [5]
row 3:       [2, 7]
row 2:     [14, 16, 9]
row 1:   [15, 1, 17, 8]
row 0: [4, 19, 20, 3, 11]
Remaining: [6, 10, 12, 13, 18, 21]
row 4:         [6]
row 3:       [3, 9]
row 2:     [14, 11, 2]
row 1:   [15, 1, 12, 10]
row 0: [4, 19, 20, 8, 18]
Remaining: [5, 7, 13, 16, 17, 21]
row 4:         [8]
row 3:       [1, 9]
row 2:     [13, 12, 3]
row 1:   [15, 2, 14, 11]
row 0: [4, 19, 21, 7, 18]
Remaining: [5, 6, 10, 16, 17, 20]
row 4:         [2]
row 3:       [10, 8]
row 2:     [3, 13, 5]
row 1:   [16, 19, 6, 11]
row 0: [4, 20, 1, 7, 18]
Remaining: [9, 12, 14, 15, 17, 21]
row 4:         [9]
row 3:       [11, 2]
row 2:     [1, 12, 10]
row 1:   [16, 15, 3, 13]
row 0: [4, 20, 5, 8, 21]
Remaining: [6, 7, 14, 17, 18, 19]
row 4:         [7]
row 3:       [9, 2]
row 2:     [10, 1, 3]
row 1:   [16, 6, 5, 8]
row 0: [4, 20, 14, 19, 11]
Remaining: [12, 13, 15, 17, 18, 21]
row 4:         [8]
row 3:       [10, 2]
row 2:     [11, 1, 3]
row 1:   [16, 5, 6, 9]
row 0: [4, 20, 15, 21, 12]
Remaining: [7, 13, 14, 17, 18, 19]
row 4:         [2]
row 3:       [5, 7]
row 2:     [13, 8, 1]
row 1:   [16, 3, 11, 12]
row 0: [4, 20, 17, 6, 18]
Remaining: [9, 10, 14, 15, 19, 21]
row 4:         [2]
row 3:       [7, 5]
row 2:     [13, 6, 1]
row 1:   [16, 3, 9, 10]
row 0: [4, 20, 17, 8, 18]
Remaining: [11, 12, 14, 15, 19, 21]
row 4:         [6]
row 3:       [11, 5]
row 2:     [13, 2, 7]
row 1:   [16, 3, 1, 8]
row 0: [4, 20, 17, 18, 10]
Remaining: [9, 12, 14, 15, 19, 21]
row 4:         [5]
row 3:       [12, 7]
row 2:     [13, 1, 8]
row 1:   [16, 3, 2, 10]
row 0: [4, 20, 17, 19, 9]
Remaining: [6, 11, 14, 15, 18, 21]
row 4:         [7]
row 3:       [12, 5]
row 2:     [13, 1, 6]
row 1:   [16, 3, 2, 8]
row 0: [4, 20, 17, 19, 11]
Remaining: [9, 10, 14, 15, 18, 21]
row 4:         [5]
row 3:       [1, 6]
row 2:     [14, 13, 7]
row 1:   [16, 2, 15, 8]
row 0: [4, 20, 18, 3, 11]
Remaining: [9, 10, 12, 17, 19, 21]
row 4:         [7]
row 3:       [3, 10]
row 2:     [14, 11, 1]
row 1:   [16, 2, 13, 12]
row 0: [4, 20, 18, 5, 17]
Remaining: [6, 8, 9, 15, 19, 21]
row 4:         [1]
row 3:       [5, 6]
row 2:     [14, 9, 3]
row 1:   [16, 2, 11, 8]
row 0: [4, 20, 18, 7, 15]
Remaining: [10, 12, 13, 17, 19, 21]
row 4:         [3]
row 3:       [5, 8]
row 2:     [14, 9, 1]
row 1:   [16, 2, 11, 10]
row 0: [4, 20, 18, 7, 17]
Remaining: [6, 12, 13, 15, 19, 21]
row 4:         [3]
row 3:       [5, 8]
row 2:     [14, 9, 1]
row 1:   [16, 2, 11, 12]
row 0: [4, 20, 18, 7, 19]
Remaining: [6, 10, 13, 15, 17, 21]
row 4:         [6]
row 3:       [13, 7]
row 2:     [14, 1, 8]
row 1:   [16, 2, 3, 11]
row 0: [4, 20, 18, 21, 10]
Remaining: [5, 9, 12, 15, 17, 19]
row 4:         [8]
row 3:       [13, 5]
row 2:     [14, 1, 6]
row 1:   [16, 2, 3, 9]
row 0: [4, 20, 18, 21, 12]
Remaining: [7, 10, 11, 15, 17, 19]
row 4:         [7]
row 3:       [3, 10]
row 2:     [15, 12, 2]
row 1:   [16, 1, 13, 11]
row 0: [4, 20, 19, 6, 17]
Remaining: [5, 8, 9, 14, 18, 21]
row 4:         [5]
row 3:       [7, 2]
row 2:     [15, 8, 6]
row 1:   [16, 1, 9, 3]
row 0: [4, 20, 19, 10, 13]
Remaining: [11, 12, 14, 17, 18, 21]
row 4:         [7]
row 3:       [10, 3]
row 2:     [15, 5, 2]
row 1:   [16, 1, 6, 8]
row 0: [4, 20, 19, 13, 21]
Remaining: [9, 11, 12, 14, 17, 18]
row 4:         [5]
row 3:       [3, 8]
row 2:     [15, 18, 10]
row 1:   [16, 1, 19, 9]
row 0: [4, 20, 21, 2, 11]
Remaining: [6, 7, 12, 13, 14, 17]
row 4:         [7]
row 3:       [3, 10]
row 2:     [15, 18, 8]
row 1:   [16, 1, 19, 11]
row 0: [4, 20, 21, 2, 13]
Remaining: [5, 6, 9, 12, 14, 17]
row 4:         [8]
row 3:       [3, 11]
row 2:     [15, 18, 7]
row 1:   [16, 1, 19, 12]
row 0: [4, 20, 21, 2, 14]
Remaining: [5, 6, 9, 10, 13, 17]
row 4:         [5]
row 3:       [2, 7]
row 2:     [15, 17, 10]
row 1:   [16, 1, 18, 8]
row 0: [4, 20, 21, 3, 11]
Remaining: [6, 9, 12, 13, 14, 19]
row 4:         [7]
row 3:       [2, 9]
row 2:     [15, 17, 8]
row 1:   [16, 1, 18, 10]
row 0: [4, 20, 21, 3, 13]
Remaining: [5, 6, 11, 12, 14, 19]
row 4:         [8]
row 3:       [2, 10]
row 2:     [15, 17, 7]
row 1:   [16, 1, 18, 11]
row 0: [4, 20, 21, 3, 14]
Remaining: [5, 6, 9, 12, 13, 19]
row 4:         [8]
row 3:       [2, 10]
row 2:     [15, 13, 3]
row 1:   [16, 1, 14, 11]
row 0: [4, 20, 21, 7, 18]
Remaining: [5, 6, 9, 12, 17, 19]
row 4:         [2]
row 3:       [3, 5]
row 2:     [15, 12, 7]
row 1:   [16, 1, 13, 6]
row 0: [4, 20, 21, 8, 14]
Remaining: [9, 10, 11, 17, 18, 19]
row 4:         [7]
row 3:       [3, 10]
row 2:     [15, 12, 2]
row 1:   [16, 1, 13, 11]
row 0: [4, 20, 21, 8, 19]
Remaining: [5, 6, 9, 14, 17, 18]
row 4:         [2]
row 3:       [7, 5]
row 2:     [15, 8, 3]
row 1:   [16, 1, 9, 6]
row 0: [4, 20, 21, 12, 18]
Remaining: [10, 11, 13, 14, 17, 19]
row 4:         [8]
row 3:       [9, 1]
row 2:     [3, 12, 11]
row 1:   [17, 14, 2, 13]
row 0: [4, 21, 7, 5, 18]
Remaining: [6, 10, 15, 16, 19, 20]
row 4:         [8]
row 3:       [9, 1]
row 2:     [3, 12, 13]
row 1:   [17, 14, 2, 15]
row 0: [4, 21, 7, 5, 20]
Remaining: [6, 10, 11, 16, 18, 19]
row 4:         [8]
row 3:       [10, 2]
row 2:     [3, 13, 11]
row 1:   [17, 14, 1, 12]
row 0: [4, 21, 7, 6, 18]
Remaining: [5, 9, 15, 16, 19, 20]
row 4:         [3]
row 3:       [7, 10]
row 2:     [9, 2, 12]
row 1:   [17, 8, 6, 18]
row 0: [4, 21, 13, 19, 1]
Remaining: [5, 11, 14, 15, 16, 20]
row 4:         [3]
row 3:       [8, 11]
row 2:     [10, 2, 13]
row 1:   [17, 7, 5, 18]
row 0: [4, 21, 14, 19, 1]
Remaining: [6, 9, 12, 15, 16, 20]
row 4:         [1]
row 3:       [8, 9]
row 2:     [10, 2, 11]
row 1:   [17, 7, 5, 16]
row 0: [4, 21, 14, 19, 3]
Remaining: [6, 12, 13, 15, 18, 20]
row 4:         [5]
row 3:       [1, 6]
row 2:     [14, 13, 7]
row 1:   [17, 3, 16, 9]
row 0: [4, 21, 18, 2, 11]
Remaining: [8, 10, 12, 15, 19, 20]
row 4:         [7]
row 3:       [1, 8]
row 2:     [15, 14, 6]
row 1:   [17, 2, 16, 10]
row 0: [4, 21, 19, 3, 13]
Remaining: [5, 9, 11, 12, 18, 20]
row 4:         [8]
row 3:       [3, 11]
row 2:     [15, 12, 1]
row 1:   [17, 2, 14, 13]
row 0: [4, 21, 19, 5, 18]
Remaining: [6, 7, 9, 10, 16, 20]
row 4:         [8]
row 3:       [3, 11]
row 2:     [16, 13, 2]
row 1:   [17, 1, 14, 12]
row 0: [4, 21, 20, 6, 18]
Remaining: [5, 7, 9, 10, 15, 19]
row 4:         [6]
row 3:       [13, 7]
row 2:     [15, 2, 9]
row 1:   [4, 19, 17, 8]
row 0: [5, 1, 20, 3, 11]
Remaining: [10, 12, 14, 16, 18, 21]
row 4:         [7]
row 3:       [13, 6]
row 2:     [15, 2, 8]
row 1:   [4, 19, 17, 9]
row 0: [5, 1, 20, 3, 12]
Remaining: [10, 11, 14, 16, 18, 21]
row 4:         [6]
row 3:       [4, 10]
row 2:     [15, 11, 1]
row 1:   [3, 18, 7, 8]
row 0: [5, 2, 20, 13, 21]
Remaining: [9, 12, 14, 16, 17, 19]
row 4:         [8]
row 3:       [1, 9]
row 2:     [16, 15, 6]
row 1:   [3, 19, 4, 10]
row 0: [5, 2, 21, 17, 7]
Remaining: [11, 12, 13, 14, 18, 20]
row 4:         [6]
row 3:       [14, 8]
row 2:     [15, 1, 9]
row 1:   [2, 17, 16, 7]
row 0: [5, 3, 20, 4, 11]
Remaining: [10, 12, 13, 18, 19, 21]
row 4:         [8]
row 3:       [14, 6]
row 2:     [15, 1, 7]
row 1:   [2, 17, 16, 9]
row 0: [5, 3, 20, 4, 13]
Remaining: [10, 11, 12, 18, 19, 21]
row 4:         [6]
row 3:       [15, 9]
row 2:     [16, 1, 10]
row 1:   [2, 18, 17, 7]
row 0: [5, 3, 21, 4, 11]
Remaining: [8, 12, 13, 14, 19, 20]
row 4:         [9]
row 3:       [15, 6]
row 2:     [16, 1, 7]
row 1:   [2, 18, 17, 10]
row 0: [5, 3, 21, 4, 14]
Remaining: [8, 11, 12, 13, 19, 20]
row 4:         [1]
row 3:       [9, 8]
row 2:     [12, 3, 11]
row 1:   [2, 14, 17, 6]
row 0: [5, 7, 21, 4, 10]
Remaining: [13, 15, 16, 18, 19, 20]
row 4:         [3]
row 3:       [9, 6]
row 2:     [1, 10, 4]
row 1:   [7, 8, 18, 14]
row 0: [5, 12, 20, 2, 16]
Remaining: [11, 13, 15, 17, 19, 21]
row 4:         [7]
row 3:       [2, 9]
row 2:     [4, 6, 15]
row 1:   [8, 12, 18, 3]
row 0: [5, 13, 1, 19, 16]
Remaining: [10, 11, 14, 17, 20, 21]
row 4:         [6]
row 3:       [4, 10]
row 2:     [3, 7, 17]
row 1:   [8, 11, 18, 1]
row 0: [5, 13, 2, 20, 19]
Remaining: [9, 12, 14, 15, 16, 21]
row 4:         [6]
row 3:       [4, 10]
row 2:     [3, 7, 17]
row 1:   [8, 11, 18, 1]
row 0: [5, 13, 2, 20, 21]
Remaining: [9, 12, 14, 15, 16, 19]
row 4:         [8]
row 3:       [2, 10]
row 2:     [4, 6, 16]
row 1:   [9, 13, 19, 3]
row 0: [5, 14, 1, 20, 17]
Remaining: [7, 11, 12, 15, 18, 21]
row 4:         [8]
row 3:       [3, 11]
row 2:     [4, 7, 18]
row 1:   [9, 13, 20, 2]
row 0: [5, 14, 1, 21, 19]
Remaining: [6, 10, 12, 15, 16, 17]
row 4:         [4]
row 3:       [6, 10]
row 2:     [7, 13, 3]
row 1:   [9, 2, 15, 18]
row 0: [5, 14, 16, 1, 19]
Remaining: [8, 11, 12, 17, 20, 21]
row 4:         [6]
row 3:       [4, 10]
row 2:     [7, 11, 1]
row 1:   [9, 2, 13, 12]
row 0: [5, 14, 16, 3, 15]
Remaining: [8, 17, 18, 19, 20, 21]
row 4:         [4]
row 3:       [7, 11]
row 2:     [6, 13, 2]
row 1:   [9, 3, 16, 18]
row 0: [5, 14, 17, 1, 19]
Remaining: [8, 10, 12, 15, 20, 21]
row 4:         [9]
row 3:       [2, 11]
row 2:     [4, 6, 17]
row 1:   [10, 14, 20, 3]
row 0: [5, 15, 1, 21, 18]
Remaining: [7, 8, 12, 13, 16, 19]
row 4:         [7]
row 3:       [4, 11]
row 2:     [2, 6, 17]
row 1:   [10, 12, 18, 1]
row 0: [5, 15, 3, 21, 20]
Remaining: [8, 9, 13, 14, 16, 19]
row 4:         [8]
row 3:       [3, 11]
row 2:     [4, 1, 12]
row 1:   [10, 6, 7, 19]
row 0: [5, 15, 9, 2, 21]
Remaining: [13, 14, 16, 17, 18, 20]
row 4:         [7]
row 3:       [4, 11]
row 2:     [8, 12, 1]
row 1:   [10, 2, 14, 13]
row 0: [5, 15, 17, 3, 16]
Remaining: [6, 9, 18, 19, 20, 21]
row 4:         [3]
row 3:       [9, 12]
row 2:     [4, 13, 1]
row 1:   [10, 6, 19, 18]
row 0: [5, 15, 21, 2, 20]
Remaining: [7, 8, 11, 14, 16, 17]
row 4:         [8]
row 3:       [4, 12]
row 2:     [9, 13, 1]
row 1:   [11, 2, 15, 14]
row 0: [5, 16, 18, 3, 17]
Remaining: [6, 7, 10, 19, 20, 21]
row 4:         [2]
row 3:       [7, 9]
row 2:     [8, 15, 6]
row 1:   [11, 3, 18, 12]
row 0: [5, 16, 19, 1, 13]
Remaining: [4, 10, 14, 17, 20, 21]
row 4:         [6]
row 3:       [7, 13]
row 2:     [8, 15, 2]
row 1:   [11, 3, 18, 20]
row 0: [5, 16, 19, 1, 21]
Remaining: [4, 9, 10, 12, 14, 17]
row 4:         [7]
row 3:       [6, 13]
row 2:     [8, 14, 1]
row 1:   [11, 3, 17, 18]
row 0: [5, 16, 19, 2, 20]
Remaining: [4, 9, 10, 12, 15, 21]
row 4:         [7]
row 3:       [2, 9]
row 2:     [8, 10, 1]
row 1:   [11, 3, 13, 12]
row 0: [5, 16, 19, 6, 18]
Remaining: [4, 14, 15, 17, 20, 21]
row 4:         [7]
row 3:       [2, 9]
row 2:     [8, 10, 1]
row 1:   [11, 3, 13, 14]
row 0: [5, 16, 19, 6, 20]
Remaining: [4, 12, 15, 17, 18, 21]
row 4:         [4]
row 3:       [6, 10]
row 2:     [8, 2, 12]
row 1:   [11, 3, 1, 13]
row 0: [5, 16, 19, 20, 7]
Remaining: [9, 14, 15, 17, 18, 21]
row 4:         [7]
row 3:       [8, 1]
row 2:     [3, 11, 10]
row 1:   [12, 15, 4, 14]
row 0: [5, 17, 2, 6, 20]
Remaining: [9, 13, 16, 18, 19, 21]
row 4:         [6]
row 3:       [1, 7]
row 2:     [3, 4, 11]
row 1:   [12, 15, 19, 8]
row 0: [5, 17, 2, 21, 13]
Remaining: [9, 10, 14, 16, 18, 20]
row 4:         [6]
row 3:       [1, 7]
row 2:     [3, 4, 11]
row 1:   [12, 9, 13, 2]
row 0: [5, 17, 8, 21, 19]
Remaining: [10, 14, 15, 16, 18, 20]
row 4:         [2]
row 3:       [8, 6]
row 2:     [9, 1, 7]
row 1:   [12, 3, 4, 11]
row 0: [5, 17, 14, 10, 21]
Remaining: [13, 15, 16, 18, 19, 20]
row 4:         [2]
row 3:       [8, 6]
row 2:     [11, 3, 9]
row 1:   [12, 1, 4, 13]
row 0: [5, 17, 16, 20, 7]
Remaining: [10, 14, 15, 18, 19, 21]
row 4:         [4]
row 3:       [2, 6]
row 2:     [11, 9, 3]
row 1:   [12, 1, 10, 7]
row 0: [5, 17, 18, 8, 15]
Remaining: [13, 14, 16, 19, 20, 21]
row 4:         [4]
row 3:       [2, 6]
row 2:     [11, 9, 3]
row 1:   [12, 1, 10, 13]
row 0: [5, 17, 18, 8, 21]
Remaining: [7, 14, 15, 16, 19, 20]
row 4:         [9]
row 3:       [4, 13]
row 2:     [10, 14, 1]
row 1:   [12, 2, 16, 15]
row 0: [5, 17, 19, 3, 18]
Remaining: [6, 7, 8, 11, 20, 21]
row 4:         [8]
row 3:       [6, 14]
row 2:     [9, 15, 1]
row 1:   [12, 3, 18, 19]
row 0: [5, 17, 20, 2, 21]
Remaining: [4, 7, 10, 11, 13, 16]
row 4:         [8]
row 3:       [2, 10]
row 2:     [9, 11, 1]
row 1:   [12, 3, 14, 13]
row 0: [5, 17, 20, 6, 19]
Remaining: [4, 7, 15, 16, 18, 21]
row 4:         [8]
row 3:       [2, 10]
row 2:     [9, 11, 1]
row 1:   [12, 3, 14, 15]
row 0: [5, 17, 20, 6, 21]
Remaining: [4, 7, 13, 16, 18, 19]
row 4:         [7]
row 3:       [3, 10]
row 2:     [8, 11, 1]
row 1:   [12, 4, 15, 14]
row 0: [5, 17, 21, 6, 20]
Remaining: [2, 9, 13, 16, 18, 19]
row 4:         [8]
row 3:       [9, 1]
row 2:     [3, 12, 11]
row 1:   [13, 16, 4, 15]
row 0: [5, 18, 2, 6, 21]
Remaining: [7, 10, 14, 17, 19, 20]
row 4:         [8]
row 3:       [9, 1]
row 2:     [2, 11, 10]
row 1:   [13, 15, 4, 14]
row 0: [5, 18, 3, 7, 21]
Remaining: [6, 12, 16, 17, 19, 20]
row 4:         [8]
row 3:       [10, 2]
row 2:     [1, 11, 9]
row 1:   [13, 14, 3, 12]
row 0: [5, 18, 4, 7, 19]
Remaining: [6, 15, 16, 17, 20, 21]
row 4:         [4]
row 3:       [6, 2]
row 2:     [3, 9, 11]
row 1:   [13, 10, 1, 12]
row 0: [5, 18, 8, 7, 19]
Remaining: [14, 15, 16, 17, 20, 21]
row 4:         [2]
row 3:       [8, 6]
row 2:     [9, 1, 7]
row 1:   [13, 4, 3, 10]
row 0: [5, 18, 14, 11, 21]
Remaining: [12, 15, 16, 17, 19, 20]
row 4:         [6]
row 3:       [1, 7]
row 2:     [11, 10, 3]
row 1:   [13, 2, 12, 15]
row 0: [5, 18, 16, 4, 19]
Remaining: [8, 9, 14, 17, 20, 21]
row 4:         [4]
row 3:       [3, 7]
row 2:     [11, 8, 1]
row 1:   [13, 2, 10, 9]
row 0: [5, 18, 16, 6, 15]
Remaining: [12, 14, 17, 19, 20, 21]
row 4:         [4]
row 3:       [10, 6]
row 2:     [12, 2, 8]
row 1:   [13, 1, 3, 11]
row 0: [5, 18, 17, 20, 9]
Remaining: [7, 14, 15, 16, 19, 21]
row 4:         [3]
row 3:       [4, 7]
row 2:     [12, 16, 9]
row 1:   [13, 1, 17, 8]
row 0: [5, 18, 19, 2, 10]
Remaining: [6, 11, 14, 15, 20, 21]
row 4:         [10]
row 3:       [4, 14]
row 2:     [11, 15, 1]
row 1:   [13, 2, 17, 16]
row 0: [5, 18, 20, 3, 19]
Remaining: [6, 7, 8, 9, 12, 21]
row 4:         [7]
row 3:       [1, 8]
row 2:     [11, 12, 4]
row 1:   [13, 2, 14, 10]
row 0: [5, 18, 20, 6, 16]
Remaining: [3, 9, 15, 17, 19, 21]
row 4:         [6]
row 3:       [1, 7]
row 2:     [11, 10, 3]
row 1:   [13, 2, 12, 9]
row 0: [5, 18, 20, 8, 17]
Remaining: [4, 14, 15, 16, 19, 21]
row 4:         [4]
row 3:       [7, 11]
row 2:     [10, 17, 6]
row 1:   [13, 3, 20, 14]
row 0: [5, 18, 21, 1, 15]
Remaining: [2, 8, 9, 12, 16, 19]
row 4:         [9]
row 3:       [2, 11]
row 2:     [10, 12, 1]
row 1:   [13, 3, 15, 14]
row 0: [5, 18, 21, 6, 20]
Remaining: [4, 7, 8, 16, 17, 19]
row 4:         [8]
row 3:       [1, 9]
row 2:     [10, 11, 2]
row 1:   [13, 3, 14, 12]
row 0: [5, 18, 21, 7, 19]
Remaining: [4, 6, 15, 16, 17, 20]
row 4:         [9]
row 3:       [11, 2]
row 2:     [1, 12, 10]
row 1:   [14, 15, 3, 13]
row 0: [5, 19, 4, 7, 20]
Remaining: [6, 8, 16, 17, 18, 21]
row 4:         [6]
row 3:       [1, 7]
row 2:     [3, 2, 9]
row 1:   [14, 11, 13, 4]
row 0: [5, 19, 8, 21, 17]
Remaining: [10, 12, 15, 16, 18, 20]
row 4:         [3]
row 3:       [4, 7]
row 2:     [6, 2, 9]
row 1:   [14, 8, 10, 1]
row 0: [5, 19, 11, 21, 20]
Remaining: [12, 13, 15, 16, 17, 18]
row 4:         [7]
row 3:       [8, 1]
row 2:     [10, 2, 3]
row 1:   [14, 4, 6, 9]
row 0: [5, 19, 15, 21, 12]
Remaining: [11, 13, 16, 17, 18, 20]
row 4:         [6]
row 3:       [2, 8]
row 2:     [11, 9, 1]
row 1:   [14, 3, 12, 13]
row 0: [5, 19, 16, 4, 17]
Remaining: [7, 10, 15, 18, 20, 21]
row 4:         [2]
row 3:       [10, 8]
row 2:     [11, 1, 9]
row 1:   [14, 3, 4, 13]
row 0: [5, 19, 16, 20, 7]
Remaining: [6, 12, 15, 17, 18, 21]
row 4:         [7]
row 3:       [1, 8]
row 2:     [12, 11, 3]
row 1:   [14, 2, 13, 16]
row 0: [5, 19, 17, 4, 20]
Remaining: [6, 9, 10, 15, 18, 21]
row 4:         [7]
row 3:       [8, 1]
row 2:     [12, 4, 3]
row 1:   [14, 2, 6, 9]
row 0: [5, 19, 17, 11, 20]
Remaining: [10, 13, 15, 16, 18, 21]
row 4:         [2]
row 3:       [4, 6]
row 2:     [13, 9, 3]
row 1:   [14, 1, 10, 7]
row 0: [5, 19, 18, 8, 15]
Remaining: [11, 12, 16, 17, 20, 21]
row 4:         [3]
row 3:       [4, 7]
row 2:     [13, 9, 2]
row 1:   [14, 1, 10, 12]
row 0: [5, 19, 18, 8, 20]
Remaining: [6, 11, 15, 16, 17, 21]
row 4:         [2]
row 3:       [6, 4]
row 2:     [13, 7, 3]
row 1:   [14, 1, 8, 11]
row 0: [5, 19, 18, 10, 21]
Remaining: [9, 12, 15, 16, 17, 20]
row 4:         [7]
row 3:       [11, 4]
row 2:     [13, 2, 6]
row 1:   [14, 1, 3, 9]
row 0: [5, 19, 18, 21, 12]
Remaining: [8, 10, 15, 16, 17, 20]
row 4:         [7]
row 3:       [11, 4]
row 2:     [13, 2, 6]
row 1:   [14, 1, 3, 9]
row 0: [5, 19, 20, 17, 8]
Remaining: [10, 12, 15, 16, 18, 21]
row 4:         [11]
row 3:       [4, 15]
row 2:     [12, 16, 1]
row 1:   [14, 2, 18, 17]
row 0: [5, 19, 21, 3, 20]
Remaining: [6, 7, 8, 9, 10, 13]
row 4:         [8]
row 3:       [1, 9]
row 2:     [12, 13, 4]
row 1:   [14, 2, 15, 11]
row 0: [5, 19, 21, 6, 17]
Remaining: [3, 7, 10, 16, 18, 20]
row 4:         [6]
row 3:       [1, 7]
row 2:     [12, 11, 4]
row 1:   [14, 2, 13, 9]
row 0: [5, 19, 21, 8, 17]
Remaining: [3, 10, 15, 16, 18, 20]
row 4:         [3]
row 3:       [4, 7]
row 2:     [12, 8, 1]
row 1:   [14, 2, 10, 9]
row 0: [5, 19, 21, 11, 20]
Remaining: [6, 13, 15, 16, 17, 18]
row 4:         [10]
row 3:       [12, 2]
row 2:     [1, 13, 11]
row 1:   [15, 16, 3, 14]
row 0: [5, 20, 4, 7, 21]
Remaining: [6, 8, 9, 17, 18, 19]
row 4:         [2]
row 3:       [8, 6]
row 2:     [1, 9, 3]
row 1:   [15, 16, 7, 10]
row 0: [5, 20, 4, 11, 21]
Remaining: [12, 13, 14, 17, 18, 19]
row 4:         [3]
row 3:       [8, 11]
row 2:     [9, 1, 12]
row 1:   [15, 6, 7, 19]
row 0: [5, 20, 14, 21, 2]
Remaining: [4, 10, 13, 16, 17, 18]
row 4:         [2]
row 3:       [8, 10]
row 2:     [9, 1, 11]
row 1:   [15, 6, 7, 18]
row 0: [5, 20, 14, 21, 3]
Remaining: [4, 12, 13, 16, 17, 19]
row 4:         [6]
row 3:       [1, 7]
row 2:     [11, 10, 3]
row 1:   [15, 4, 14, 17]
row 0: [5, 20, 16, 2, 19]
Remaining: [8, 9, 12, 13, 18, 21]
row 4:         [6]
row 3:       [2, 8]
row 2:     [11, 9, 1]
row 1:   [15, 4, 13, 14]
row 0: [5, 20, 16, 3, 17]
Remaining: [7, 10, 12, 18, 19, 21]
row 4:         [2]
row 3:       [8, 6]
row 2:     [11, 3, 9]
row 1:   [15, 4, 1, 10]
row 0: [5, 20, 16, 17, 7]
Remaining: [12, 13, 14, 18, 19, 21]
row 4:         [2]
row 3:       [10, 8]
row 2:     [11, 1, 9]
row 1:   [15, 4, 3, 12]
row 0: [5, 20, 16, 19, 7]
Remaining: [6, 13, 14, 17, 18, 21]
row 4:         [7]
row 3:       [2, 9]
row 2:     [12, 10, 1]
row 1:   [15, 3, 13, 14]
row 0: [5, 20, 17, 4, 18]
Remaining: [6, 8, 11, 16, 19, 21]
row 4:         [4]
row 3:       [10, 6]
row 2:     [12, 2, 8]
row 1:   [15, 3, 1, 9]
row 0: [5, 20, 17, 16, 7]
Remaining: [11, 13, 14, 18, 19, 21]
row 4:         [4]
row 3:       [11, 7]
row 2:     [12, 1, 8]
row 1:   [15, 3, 2, 10]
row 0: [5, 20, 17, 19, 9]
Remaining: [6, 13, 14, 16, 18, 21]
row 4:         [2]
row 3:       [11, 9]
row 2:     [12, 1, 10]
row 1:   [15, 3, 4, 14]
row 0: [5, 20, 17, 21, 7]
Remaining: [6, 8, 13, 16, 18, 19]
row 4:         [8]
row 3:       [1, 9]
row 2:     [13, 12, 3]
row 1:   [15, 2, 14, 17]
row 0: [5, 20, 18, 4, 21]
Remaining: [6, 7, 10, 11, 16, 19]
row 4:         [7]
row 3:       [2, 9]
row 2:     [14, 12, 3]
row 1:   [15, 1, 13, 10]
row 0: [5, 20, 19, 6, 16]
Remaining: [4, 8, 11, 17, 18, 21]
row 4:         [6]
row 3:       [3, 9]
row 2:     [14, 11, 2]
row 1:   [15, 1, 12, 10]
row 0: [5, 20, 19, 7, 17]
Remaining: [4, 8, 13, 16, 18, 21]
row 4:         [8]
row 3:       [12, 4]
row 2:     [14, 2, 6]
row 1:   [15, 1, 3, 9]
row 0: [5, 20, 19, 16, 7]
Remaining: [10, 11, 13, 17, 18, 21]
row 4:         [3]
row 3:       [4, 7]
row 2:     [14, 18, 11]
row 1:   [15, 1, 19, 8]
row 0: [5, 20, 21, 2, 10]
Remaining: [6, 9, 12, 13, 16, 17]
row 4:         [6]
row 3:       [4, 10]
row 2:     [14, 18, 8]
row 1:   [15, 1, 19, 11]
row 0: [5, 20, 21, 2, 13]
Remaining: [3, 7, 9, 12, 16, 17]
row 4:         [7]
row 3:       [2, 9]
row 2:     [14, 12, 3]
row 1:   [15, 1, 13, 10]
row 0: [5, 20, 21, 8, 18]
Remaining: [4, 6, 11, 16, 17, 19]
row 4:         [6]
row 3:       [14, 8]
row 2:     [4, 18, 10]
row 1:   [16, 20, 2, 12]
row 0: [5, 21, 1, 3, 15]
Remaining: [7, 9, 11, 13, 17, 19]
row 4:         [7]
row 3:       [15, 8]
row 2:     [2, 17, 9]
row 1:   [16, 18, 1, 10]
row 0: [5, 21, 3, 4, 14]
Remaining: [6, 11, 12, 13, 19, 20]
row 4:         [7]
row 3:       [10, 3]
row 2:     [1, 11, 8]
row 1:   [16, 15, 4, 12]
row 0: [5, 21, 6, 2, 14]
Remaining: [9, 13, 17, 18, 19, 20]
row 4:         [7]
row 3:       [10, 3]
row 2:     [1, 11, 14]
row 1:   [16, 15, 4, 18]
row 0: [5, 21, 6, 2, 20]
Remaining: [8, 9, 12, 13, 17, 19]
row 4:         [8]
row 3:       [9, 1]
row 2:     [2, 11, 10]
row 1:   [16, 14, 3, 13]
row 0: [5, 21, 7, 4, 17]
Remaining: [6, 12, 15, 18, 19, 20]
row 4:         [8]
row 3:       [9, 1]
row 2:     [2, 11, 12]
row 1:   [16, 14, 3, 15]
row 0: [5, 21, 7, 4, 19]
Remaining: [6, 10, 13, 17, 18, 20]
row 4:         [3]
row 3:       [8, 11]
row 2:     [9, 1, 12]
row 1:   [16, 7, 6, 18]
row 0: [5, 21, 14, 20, 2]
Remaining: [4, 10, 13, 15, 17, 19]
row 4:         [2]
row 3:       [8, 10]
row 2:     [9, 1, 11]
row 1:   [16, 7, 6, 17]
row 0: [5, 21, 14, 20, 3]
Remaining: [4, 12, 13, 15, 18, 19]
row 4:         [7]
row 3:       [1, 8]
row 2:     [12, 11, 3]
row 1:   [16, 4, 15, 18]
row 0: [5, 21, 17, 2, 20]
Remaining: [6, 9, 10, 13, 14, 19]
row 4:         [7]
row 3:       [2, 9]
row 2:     [12, 10, 1]
row 1:   [16, 4, 14, 15]
row 0: [5, 21, 17, 3, 18]
Remaining: [6, 8, 11, 13, 19, 20]
row 4:         [2]
row 3:       [11, 9]
row 2:     [12, 1, 10]
row 1:   [16, 4, 3, 13]
row 0: [5, 21, 17, 20, 7]
Remaining: [6, 8, 14, 15, 18, 19]
row 4:         [8]
row 3:       [2, 10]
row 2:     [13, 11, 1]
row 1:   [16, 3, 14, 15]
row 0: [5, 21, 18, 4, 19]
Remaining: [6, 7, 9, 12, 17, 20]
row 4:         [7]
row 3:       [3, 10]
row 2:     [14, 11, 1]
row 1:   [16, 2, 13, 12]
row 0: [5, 21, 19, 6, 18]
Remaining: [4, 8, 9, 15, 17, 20]
row 4:         [8]
row 3:       [2, 10]
row 2:     [15, 13, 3]
row 1:   [16, 1, 14, 11]
row 0: [5, 21, 20, 6, 17]
Remaining: [4, 7, 9, 12, 18, 19]
row 4:         [4]
row 3:       [3, 7]
row 2:     [13, 16, 9]
row 1:   [5, 18, 2, 11]
row 0: [6, 1, 19, 21, 10]
Remaining: [8, 12, 14, 15, 17, 20]
row 4:         [4]
row 3:       [13, 9]
row 2:     [15, 2, 11]
row 1:   [5, 20, 18, 7]
row 0: [6, 1, 21, 3, 10]
Remaining: [8, 12, 14, 16, 17, 19]
row 4:         [8]
row 3:       [2, 10]
row 2:     [15, 13, 3]
row 1:   [5, 20, 7, 4]
row 0: [6, 1, 21, 14, 18]
Remaining: [9, 11, 12, 16, 17, 19]
row 4:         [8]
row 3:       [3, 11]
row 2:     [15, 18, 7]
row 1:   [5, 20, 2, 9]
row 0: [6, 1, 21, 19, 10]
Remaining: [4, 12, 13, 14, 16, 17]
row 4:         [9]
row 3:       [1, 10]
row 2:     [14, 13, 3]
row 1:   [4, 18, 5, 8]
row 0: [6, 2, 20, 15, 7]
Remaining: [11, 12, 16, 17, 19, 21]
row 4:         [5]
row 3:       [3, 8]
row 2:     [14, 17, 9]
row 1:   [4, 18, 1, 10]
row 0: [6, 2, 20, 21, 11]
Remaining: [7, 12, 13, 15, 16, 19]
row 4:         [5]
row 3:       [14, 9]
row 2:     [15, 1, 10]
row 1:   [4, 19, 18, 8]
row 0: [6, 2, 21, 3, 11]
Remaining: [7, 12, 13, 16, 17, 20]
row 4:         [5]
row 3:       [3, 8]
row 2:     [15, 18, 10]
row 1:   [4, 19, 1, 11]
row 0: [6, 2, 21, 20, 9]
Remaining: [7, 12, 13, 14, 16, 17]
row 4:         [7]
row 3:       [3, 10]
row 2:     [15, 18, 8]
row 1:   [4, 19, 1, 9]
row 0: [6, 2, 21, 20, 11]
Remaining: [5, 12, 13, 14, 16, 17]
row 4:         [5]
row 3:       [14, 9]
row 2:     [15, 1, 10]
row 1:   [2, 17, 18, 8]
row 0: [6, 4, 21, 3, 11]
Remaining: [7, 12, 13, 16, 19, 20]
row 4:         [4]
row 3:       [13, 9]
row 2:     [15, 2, 11]
row 1:   [1, 16, 18, 7]
row 0: [6, 5, 21, 3, 10]
Remaining: [8, 12, 14, 17, 19, 20]
row 4:         [4]
row 3:       [13, 9]
row 2:     [15, 2, 11]
row 1:   [1, 16, 14, 3]
row 0: [6, 5, 21, 7, 10]
Remaining: [8, 12, 17, 18, 19, 20]
row 4:         [8]
row 3:       [10, 2]
row 2:     [13, 3, 5]
row 1:   [1, 14, 17, 12]
row 0: [6, 7, 21, 4, 16]
Remaining: [9, 11, 15, 18, 19, 20]
row 4:         [7]
row 3:       [2, 9]
row 2:     [5, 3, 12]
row 1:   [8, 13, 16, 4]
row 0: [6, 14, 1, 17, 21]
Remaining: [10, 11, 15, 18, 19, 20]
row 4:         [4]
row 3:       [5, 9]
row 2:     [7, 12, 3]
row 1:   [8, 1, 13, 16]
row 0: [6, 14, 15, 2, 18]
Remaining: [10, 11, 17, 19, 20, 21]
row 4:         [5]
row 3:       [4, 9]
row 2:     [7, 11, 2]
row 1:   [8, 1, 12, 10]
row 0: [6, 14, 15, 3, 13]
Remaining: [16, 17, 18, 19, 20, 21]
row 4:         [4]
row 3:       [7, 11]
row 2:     [5, 12, 1]
row 1:   [8, 3, 15, 16]
row 0: [6, 14, 17, 2, 18]
Remaining: [9, 10, 13, 19, 20, 21]
row 4:         [2]
row 3:       [7, 9]
row 2:     [3, 10, 1]
row 1:   [8, 5, 15, 16]
row 0: [6, 14, 19, 4, 20]
Remaining: [11, 12, 13, 17, 18, 21]
row 4:         [7]
row 3:       [3, 10]
row 2:     [5, 2, 12]
row 1:   [9, 14, 16, 4]
row 0: [6, 15, 1, 17, 13]
Remaining: [8, 11, 18, 19, 20, 21]
row 4:         [7]
row 3:       [3, 10]
row 2:     [5, 2, 12]
row 1:   [9, 14, 16, 4]
row 0: [6, 15, 1, 17, 21]
Remaining: [8, 11, 13, 18, 19, 20]
row 4:         [8]
row 3:       [3, 11]
row 2:     [5, 2, 13]
row 1:   [10, 15, 17, 4]
row 0: [6, 16, 1, 18, 14]
Remaining: [7, 9, 12, 19, 20, 21]
row 4:         [5]
row 3:       [3, 8]
row 2:     [9, 12, 4]
row 1:   [10, 1, 13, 17]
row 0: [6, 16, 15, 2, 19]
Remaining: [7, 11, 14, 18, 20, 21]
row 4:         [4]
row 3:       [3, 7]
row 2:     [9, 12, 5]
row 1:   [10, 1, 13, 18]
row 0: [6, 16, 15, 2, 20]
Remaining: [8, 11, 14, 17, 19, 21]
row 4:         [7]
row 3:       [4, 11]
row 2:     [9, 13, 2]
row 1:   [10, 1, 14, 12]
row 0: [6, 16, 17, 3, 15]
Remaining: [5, 8, 18, 19, 20, 21]
row 4:         [5]
row 3:       [7, 12]
row 2:     [8, 15, 3]
row 1:   [10, 2, 17, 20]
row 0: [6, 16, 18, 1, 21]
Remaining: [4, 9, 11, 13, 14, 19]
row 4:         [7]
row 3:       [5, 12]
row 2:     [8, 13, 1]
row 1:   [10, 2, 15, 14]
row 0: [6, 16, 18, 3, 17]
Remaining: [4, 9, 11, 19, 20, 21]
row 4:         [5]
row 3:       [8, 13]
row 2:     [7, 15, 2]
row 1:   [10, 3, 18, 20]
row 0: [6, 16, 19, 1, 21]
Remaining: [4, 9, 11, 12, 14, 17]
row 4:         [4]
row 3:       [5, 9]
row 2:     [7, 2, 11]
row 1:   [10, 3, 1, 12]
row 0: [6, 16, 19, 20, 8]
Remaining: [13, 14, 15, 17, 18, 21]
row 4:         [9]
row 3:       [3, 12]
row 2:     [5, 2, 14]
row 1:   [11, 16, 18, 4]
row 0: [6, 17, 1, 19, 15]
Remaining: [7, 8, 10, 13, 20, 21]
row 4:         [7]
row 3:       [8, 1]
row 2:     [4, 12, 13]
row 1:   [11, 15, 3, 16]
row 0: [6, 17, 2, 5, 21]
Remaining: [9, 10, 14, 18, 19, 20]
row 4:         [7]
row 3:       [10, 3]
row 2:     [2, 12, 15]
row 1:   [11, 13, 1, 16]
row 0: [6, 17, 4, 5, 21]
Remaining: [8, 9, 14, 18, 19, 20]
row 4:         [7]
row 3:       [3, 10]
row 2:     [1, 4, 14]
row 1:   [11, 12, 16, 2]
row 0: [6, 17, 5, 21, 19]
Remaining: [8, 9, 13, 15, 18, 20]
row 4:         [5]
row 3:       [3, 8]
row 2:     [1, 4, 12]
row 1:   [11, 10, 14, 2]
row 0: [6, 17, 7, 21, 19]
Remaining: [9, 13, 15, 16, 18, 20]
row 4:         [5]
row 3:       [3, 8]
row 2:     [9, 12, 4]
row 1:   [11, 2, 14, 18]
row 0: [6, 17, 15, 1, 19]
Remaining: [7, 10, 13, 16, 20, 21]
row 4:         [4]
row 3:       [3, 7]
row 2:     [9, 12, 5]
row 1:   [11, 2, 14, 19]
row 0: [6, 17, 15, 1, 20]
Remaining: [8, 10, 13, 16, 18, 21]
row 4:         [7]
row 3:       [5, 12]
row 2:     [10, 15, 3]
row 1:   [11, 1, 16, 19]
row 0: [6, 17, 18, 2, 21]
Remaining: [4, 8, 9, 13, 14, 20]
row 4:         [8]
row 3:       [4, 12]
row 2:     [10, 14, 2]
row 1:   [11, 1, 15, 13]
row 0: [6, 17, 18, 3, 16]
Remaining: [5, 7, 9, 19, 20, 21]
row 4:         [7]
row 3:       [2, 9]
row 2:     [10, 12, 3]
row 1:   [11, 1, 13, 16]
row 0: [6, 17, 18, 5, 21]
Remaining: [4, 8, 14, 15, 19, 20]
row 4:         [5]
row 3:       [7, 12]
row 2:     [9, 16, 4]
row 1:   [11, 2, 18, 14]
row 0: [6, 17, 19, 1, 15]
Remaining: [3, 8, 10, 13, 20, 21]
row 4:         [8]
row 3:       [5, 13]
row 2:     [9, 14, 1]
row 1:   [11, 2, 16, 15]
row 0: [6, 17, 19, 3, 18]
Remaining: [4, 7, 10, 12, 20, 21]
row 4:         [7]
row 3:       [5, 12]
row 2:     [8, 13, 1]
row 1:   [11, 3, 16, 15]
row 0: [6, 17, 20, 4, 19]
Remaining: [2, 9, 10, 14, 18, 21]
row 4:         [5]
row 3:       [9, 14]
row 2:     [7, 16, 2]
row 1:   [11, 4, 20, 18]
row 0: [6, 17, 21, 1, 19]
Remaining: [3, 8, 10, 12, 13, 15]
row 4:         [10]
row 3:       [3, 13]
row 2:     [5, 2, 15]
row 1:   [12, 17, 19, 4]
row 0: [6, 18, 1, 20, 16]
Remaining: [7, 8, 9, 11, 14, 21]
row 4:         [8]
row 3:       [2, 10]
row 2:     [5, 3, 13]
row 1:   [12, 17, 20, 7]
row 0: [6, 18, 1, 21, 14]
Remaining: [4, 9, 11, 15, 16, 19]
row 4:         [8]
row 3:       [9, 1]
row 2:     [2, 11, 10]
row 1:   [12, 14, 3, 13]
row 0: [6, 18, 4, 7, 20]
Remaining: [5, 15, 16, 17, 19, 21]
row 4:         [7]
row 3:       [3, 10]
row 2:     [11, 14, 4]
row 1:   [12, 1, 15, 19]
row 0: [6, 18, 17, 2, 21]
Remaining: [5, 8, 9, 13, 16, 20]
row 4:         [3]
row 3:       [2, 5]
row 2:     [11, 9, 4]
row 1:   [12, 1, 10, 14]
row 0: [6, 18, 17, 7, 21]
Remaining: [8, 13, 15, 16, 19, 20]
row 4:         [9]
row 3:       [4, 13]
row 2:     [11, 15, 2]
row 1:   [12, 1, 16, 14]
row 0: [6, 18, 19, 3, 17]
Remaining: [5, 7, 8, 10, 20, 21]
row 4:         [7]
row 3:       [2, 9]
row 2:     [11, 13, 4]
row 1:   [12, 1, 14, 10]
row 0: [6, 18, 19, 5, 15]
Remaining: [3, 8, 16, 17, 20, 21]
row 4:         [9]
row 3:       [5, 14]
row 2:     [10, 15, 1]
row 1:   [12, 2, 17, 16]
row 0: [6, 18, 20, 3, 19]
Remaining: [4, 7, 8, 11, 13, 21]
row 4:         [2]
row 3:       [8, 10]
row 2:     [9, 17, 7]
row 1:   [12, 3, 20, 13]
row 0: [6, 18, 21, 1, 14]
Remaining: [4, 5, 11, 15, 16, 19]
row 4:         [8]
row 3:       [5, 13]
row 2:     [9, 14, 1]
row 1:   [12, 3, 17, 16]
row 0: [6, 18, 21, 4, 20]
Remaining: [2, 7, 10, 11, 15, 19]
row 4:         [7]
row 3:       [4, 11]
row 2:     [9, 13, 2]
row 1:   [12, 3, 16, 14]
row 0: [6, 18, 21, 5, 19]
Remaining: [1, 8, 10, 15, 17, 20]
row 4:         [8]
row 3:       [2, 10]
row 2:     [9, 11, 1]
row 1:   [12, 3, 14, 13]
row 0: [6, 18, 21, 7, 20]
Remaining: [4, 5, 15, 16, 17, 19]
row 4:         [7]
row 3:       [11, 4]
row 2:     [5, 16, 12]
row 1:   [13, 18, 2, 14]
row 0: [6, 19, 1, 3, 17]
Remaining: [8, 9, 10, 15, 20, 21]
row 4:         [7]
row 3:       [3, 10]
row 2:     [5, 2, 12]
row 1:   [13, 18, 16, 4]
row 0: [6, 19, 1, 17, 21]
Remaining: [8, 9, 11, 14, 15, 20]
row 4:         [4]
row 3:       [3, 7]
row 2:     [5, 2, 9]
row 1:   [13, 18, 20, 11]
row 0: [6, 19, 1, 21, 10]
Remaining: [8, 12, 14, 15, 16, 17]
row 4:         [11]
row 3:       [3, 14]
row 2:     [5, 2, 16]
row 1:   [13, 18, 20, 4]
row 0: [6, 19, 1, 21, 17]
Remaining: [7, 8, 9, 10, 12, 15]
row 4:         [5]
row 3:       [3, 8]
row 2:     [4, 1, 9]
row 1:   [13, 17, 16, 7]
row 0: [6, 19, 2, 18, 11]
Remaining: [10, 12, 14, 15, 20, 21]
row 4:         [9]
row 3:       [10, 1]
row 2:     [2, 12, 11]
row 1:   [13, 15, 3, 14]
row 0: [6, 19, 4, 7, 21]
Remaining: [5, 8, 16, 17, 18, 20]
row 4:         [7]
row 3:       [11, 4]
row 2:     [1, 12, 16]
row 1:   [13, 14, 2, 18]
row 0: [6, 19, 5, 3, 21]
Remaining: [8, 9, 10, 15, 17, 20]
row 4:         [1]
row 3:       [5, 4]
row 2:     [3, 8, 12]
row 1:   [13, 10, 2, 14]
row 0: [6, 19, 9, 7, 21]
Remaining: [11, 15, 16, 17, 18, 20]
row 4:         [4]
row 3:       [3, 7]
row 2:     [5, 2, 9]
row 1:   [13, 8, 10, 1]
row 0: [6, 19, 11, 21, 20]
Remaining: [12, 14, 15, 16, 17, 18]
row 4:         [7]
row 3:       [3, 10]
row 2:     [11, 14, 4]
row 1:   [13, 2, 16, 20]
row 0: [6, 19, 17, 1, 21]
Remaining: [5, 8, 9, 12, 15, 18]
row 4:         [7]
row 3:       [1, 8]
row 2:     [11, 12, 4]
row 1:   [13, 2, 14, 18]
row 0: [6, 19, 17, 3, 21]
Remaining: [5, 9, 10, 15, 16, 20]
row 4:         [7]
row 3:       [8, 1]
row 2:     [11, 3, 4]
row 1:   [13, 2, 5, 9]
row 0: [6, 19, 17, 12, 21]
Remaining: [10, 14, 15, 16, 18, 20]
row 4:         [5]
row 3:       [3, 8]
row 2:     [12, 15, 7]
row 1:   [13, 1, 16, 9]
row 0: [6, 19, 18, 2, 11]
Remaining: [4, 10, 14, 17, 20, 21]
row 4:         [10]
row 3:       [4, 14]
row 2:     [12, 16, 2]
row 1:   [13, 1, 17, 15]
row 0: [6, 19, 20, 3, 18]
Remaining: [5, 7, 8, 9, 11, 21]
row 4:         [8]
row 3:       [2, 10]
row 2:     [12, 14, 4]
row 1:   [13, 1, 15, 11]
row 0: [6, 19, 20, 5, 16]
Remaining: [3, 7, 9, 17, 18, 21]
row 4:         [5]
row 3:       [2, 7]
row 2:     [12, 10, 3]
row 1:   [13, 1, 11, 8]
row 0: [6, 19, 20, 9, 17]
Remaining: [4, 14, 15, 16, 18, 21]
row 4:         [7]
row 3:       [5, 12]
row 2:     [11, 16, 4]
row 1:   [13, 2, 18, 14]
row 0: [6, 19, 21, 3, 17]
Remaining: [1, 8, 9, 10, 15, 20]
row 4:         [10]
row 3:       [5, 15]
row 2:     [11, 16, 1]
row 1:   [13, 2, 18, 17]
row 0: [6, 19, 21, 3, 20]
Remaining: [4, 7, 8, 9, 12, 14]
row 4:         [7]
row 3:       [3, 10]
row 2:     [11, 14, 4]
row 1:   [13, 2, 16, 12]
row 0: [6, 19, 21, 5, 17]
Remaining: [1, 8, 9, 15, 18, 20]
row 4:         [8]
row 3:       [12, 4]
row 2:     [5, 17, 13]
row 1:   [14, 19, 2, 15]
row 0: [6, 20, 1, 3, 18]
Remaining: [7, 9, 10, 11, 16, 21]
row 4:         [7]
row 3:       [2, 9]
row 2:     [5, 3, 12]
row 1:   [14, 19, 16, 4]
row 0: [6, 20, 1, 17, 13]
Remaining: [8, 10, 11, 15, 18, 21]
row 4:         [7]
row 3:       [2, 9]
row 2:     [5, 3, 12]
row 1:   [14, 19, 16, 4]
row 0: [6, 20, 1, 17, 21]
Remaining: [8, 10, 11, 13, 15, 18]
row 4:         [4]
row 3:       [3, 7]
row 2:     [5, 2, 9]
row 1:   [14, 19, 17, 8]
row 0: [6, 20, 1, 18, 10]
Remaining: [11, 12, 13, 15, 16, 21]
row 4:         [1]
row 3:       [9, 10]
row 2:     [4, 13, 3]
row 1:   [14, 18, 5, 8]
row 0: [6, 20, 2, 7, 15]
Remaining: [11, 12, 16, 17, 19, 21]
row 4:         [5]
row 3:       [3, 8]
row 2:     [4, 1, 9]
row 1:   [14, 18, 19, 10]
row 0: [6, 20, 2, 21, 11]
Remaining: [7, 12, 13, 15, 16, 17]
row 4:         [7]
row 3:       [3, 10]
row 2:     [4, 1, 11]
row 1:   [14, 18, 19, 8]
row 0: [6, 20, 2, 21, 13]
Remaining: [5, 9, 12, 15, 16, 17]
row 4:         [7]
row 3:       [11, 4]
row 2:     [1, 12, 16]
row 1:   [14, 15, 3, 19]
row 0: [6, 20, 5, 2, 21]
Remaining: [8, 9, 10, 13, 17, 18]
row 4:         [8]
row 3:       [12, 4]
row 2:     [1, 13, 9]
row 1:   [14, 15, 2, 11]
row 0: [6, 20, 5, 7, 18]
Remaining: [3, 10, 16, 17, 19, 21]
row 4:         [9]
row 3:       [11, 2]
row 2:     [1, 12, 10]
row 1:   [14, 15, 3, 13]
row 0: [6, 20, 5, 8, 21]
Remaining: [4, 7, 16, 17, 18, 19]
row 4:         [2]
row 3:       [10, 8]
row 2:     [1, 11, 3]
row 1:   [14, 15, 4, 7]
row 0: [6, 20, 5, 9, 16]
Remaining: [12, 13, 17, 18, 19, 21]
row 4:         [7]
row 3:       [10, 3]
row 2:     [1, 11, 8]
row 1:   [14, 15, 4, 12]
row 0: [6, 20, 5, 9, 21]
Remaining: [2, 13, 16, 17, 18, 19]
row 4:         [5]
row 3:       [7, 2]
row 2:     [3, 10, 12]
row 1:   [14, 11, 1, 13]
row 0: [6, 20, 9, 8, 21]
Remaining: [4, 15, 16, 17, 18, 19]
row 4:         [5]
row 3:       [2, 7]
row 2:     [3, 1, 8]
row 1:   [14, 11, 12, 4]
row 0: [6, 20, 9, 21, 17]
Remaining: [10, 13, 15, 16, 18, 19]
row 4:         [3]
row 3:       [4, 7]
row 2:     [5, 1, 8]
row 1:   [14, 9, 10, 2]
row 0: [6, 20, 11, 21, 19]
Remaining: [12, 13, 15, 16, 17, 18]
row 4:         [7]
row 3:       [9, 2]
row 2:     [10, 1, 3]
row 1:   [14, 4, 5, 8]
row 0: [6, 20, 16, 11, 19]
Remaining: [12, 13, 15, 17, 18, 21]
row 4:         [7]
row 3:       [9, 2]
row 2:     [10, 1, 3]
row 1:   [14, 4, 5, 8]
row 0: [6, 20, 16, 21, 13]
Remaining: [11, 12, 15, 17, 18, 19]
row 4:         [4]
row 3:       [1, 5]
row 2:     [11, 12, 7]
row 1:   [14, 3, 15, 8]
row 0: [6, 20, 17, 2, 10]
Remaining: [9, 13, 16, 18, 19, 21]
row 4:         [7]
row 3:       [1, 8]
row 2:     [11, 12, 4]
row 1:   [14, 3, 15, 19]
row 0: [6, 20, 17, 2, 21]
Remaining: [5, 9, 10, 13, 16, 18]
row 4:         [7]
row 3:       [1, 8]
row 2:     [11, 10, 2]
row 1:   [14, 3, 13, 15]
row 0: [6, 20, 17, 4, 19]
Remaining: [5, 9, 12, 16, 18, 21]
row 4:         [4]
row 3:       [9, 5]
row 2:     [11, 2, 7]
row 1:   [14, 3, 1, 8]
row 0: [6, 20, 17, 18, 10]
Remaining: [12, 13, 15, 16, 19, 21]
row 4:         [4]
row 3:       [3, 7]
row 2:     [12, 15, 8]
row 1:   [14, 2, 17, 9]
row 0: [6, 20, 18, 1, 10]
Remaining: [5, 11, 13, 16, 19, 21]
row 4:         [5]
row 3:       [3, 8]
row 2:     [12, 15, 7]
row 1:   [14, 2, 17, 10]
row 0: [6, 20, 18, 1, 11]
Remaining: [4, 9, 13, 16, 19, 21]
row 4:         [4]
row 3:       [1, 5]
row 2:     [12, 13, 8]
row 1:   [14, 2, 15, 7]
row 0: [6, 20, 18, 3, 10]
Remaining: [9, 11, 16, 17, 19, 21]
row 4:         [7]
row 3:       [1, 8]
row 2:     [12, 11, 3]
row 1:   [14, 2, 13, 10]
row 0: [6, 20, 18, 5, 15]
Remaining: [4, 9, 16, 17, 19, 21]
row 4:         [7]
row 3:       [1, 8]
row 2:     [12, 11, 3]
row 1:   [14, 2, 13, 16]
row 0: [6, 20, 18, 5, 21]
Remaining: [4, 9, 10, 15, 17, 19]
row 4:         [5]
row 3:       [3, 8]
row 2:     [12, 9, 1]
row 1:   [14, 2, 11, 10]
row 0: [6, 20, 18, 7, 17]
Remaining: [4, 13, 15, 16, 19, 21]
row 4:         [7]
row 3:       [11, 4]
row 2:     [12, 1, 5]
row 1:   [14, 2, 3, 8]
row 0: [6, 20, 18, 21, 13]
Remaining: [9, 10, 15, 16, 17, 19]
row 4:         [4]
row 3:       [3, 7]
row 2:     [13, 16, 9]
row 1:   [14, 1, 17, 8]
row 0: [6, 20, 19, 2, 10]
Remaining: [5, 11, 12, 15, 18, 21]
row 4:         [3]
row 3:       [5, 8]
row 2:     [13, 18, 10]
row 1:   [14, 1, 19, 9]
row 0: [6, 20, 21, 2, 11]
Remaining: [4, 7, 12, 15, 16, 17]
row 4:         [11]
row 3:       [4, 15]
row 2:     [13, 17, 2]
row 1:   [14, 1, 18, 16]
row 0: [6, 20, 21, 3, 19]
Remaining: [5, 7, 8, 9, 10, 12]
row 4:         [9]
row 3:       [2, 11]
row 2:     [13, 15, 4]
row 1:   [14, 1, 16, 12]
row 0: [6, 20, 21, 5, 17]
Remaining: [3, 7, 8, 10, 18, 19]
row 4:         [5]
row 3:       [2, 7]
row 2:     [13, 11, 4]
row 1:   [14, 1, 12, 8]
row 0: [6, 20, 21, 9, 17]
Remaining: [3, 10, 15, 16, 18, 19]
row 4:         [3]
row 3:       [4, 7]
row 2:     [13, 9, 2]
row 1:   [14, 1, 10, 8]
row 0: [6, 20, 21, 11, 19]
Remaining: [5, 12, 15, 16, 17, 18]
row 4:         [9]
row 3:       [13, 4]
row 2:     [5, 18, 14]
row 1:   [15, 20, 2, 16]
row 0: [6, 21, 1, 3, 19]
Remaining: [7, 8, 10, 11, 12, 17]
row 4:         [8]
row 3:       [2, 10]
row 2:     [5, 3, 13]
row 1:   [15, 20, 17, 4]
row 0: [6, 21, 1, 18, 14]
Remaining: [7, 9, 11, 12, 16, 19]
row 4:         [7]
row 3:       [1, 8]
row 2:     [4, 3, 11]
row 1:   [15, 19, 16, 5]
row 0: [6, 21, 2, 18, 13]
Remaining: [9, 10, 12, 14, 17, 20]
row 4:         [5]
row 3:       [14, 9]
row 2:     [2, 16, 7]
row 1:   [15, 17, 1, 8]
row 0: [6, 21, 4, 3, 11]
Remaining: [10, 12, 13, 18, 19, 20]
row 4:         [9]
row 3:       [13, 4]
row 2:     [1, 14, 10]
row 1:   [15, 16, 2, 12]
row 0: [6, 21, 5, 7, 19]
Remaining: [3, 8, 11, 17, 18, 20]
row 4:         [8]
row 3:       [10, 2]
row 2:     [1, 11, 9]
row 1:   [15, 14, 3, 12]
row 0: [6, 21, 7, 4, 16]
Remaining: [5, 13, 17, 18, 19, 20]
row 4:         [8]
row 3:       [10, 2]
row 2:     [1, 11, 13]
row 1:   [15, 14, 3, 16]
row 0: [6, 21, 7, 4, 20]
Remaining: [5, 9, 12, 17, 18, 19]
row 4:         [3]
row 3:       [4, 7]
row 2:     [5, 1, 8]
row 1:   [15, 10, 9, 17]
row 0: [6, 21, 11, 2, 19]
Remaining: [12, 13, 14, 16, 18, 20]
row 4:         [7]
row 3:       [1, 8]
row 2:     [11, 10, 2]
row 1:   [15, 4, 14, 16]
row 0: [6, 21, 17, 3, 19]
Remaining: [5, 9, 12, 13, 18, 20]
row 4:         [8]
row 3:       [1, 9]
row 2:     [12, 11, 2]
row 1:   [15, 3, 14, 16]
row 0: [6, 21, 18, 4, 20]
Remaining: [5, 7, 10, 13, 17, 19]
row 4:         [7]
row 3:       [2, 9]
row 2:     [12, 10, 1]
row 1:   [15, 3, 13, 14]
row 0: [6, 21, 18, 5, 19]
Remaining: [4, 8, 11, 16, 17, 20]
row 4:         [4]
row 3:       [1, 5]
row 2:     [13, 14, 9]
row 1:   [15, 2, 16, 7]
row 0: [6, 21, 19, 3, 10]
Remaining: [8, 11, 12, 17, 18, 20]
row 4:         [8]
row 3:       [1, 9]
row 2:     [13, 12, 3]
row 1:   [15, 2, 14, 11]
row 0: [6, 21, 19, 5, 16]
Remaining: [4, 7, 10, 17, 18, 20]
row 4:         [4]
row 3:       [5, 1]
row 2:     [13, 8, 7]
row 1:   [15, 2, 10, 3]
row 0: [6, 21, 19, 9, 12]
Remaining: [11, 14, 16, 17, 18, 20]
row 4:         [5]
row 3:       [2, 7]
row 2:     [14, 16, 9]
row 1:   [15, 1, 17, 8]
row 0: [6, 21, 20, 3, 11]
Remaining: [4, 10, 12, 13, 18, 19]
row 4:         [8]
row 3:       [4, 12]
row 2:     [11, 15, 3]
row 1:   [6, 17, 2, 5]
row 0: [7, 1, 18, 16, 21]
Remaining: [9, 10, 13, 14, 19, 20]
row 4:         [2]
row 3:       [10, 8]
row 2:     [13, 3, 11]
row 1:   [6, 19, 16, 5]
row 0: [7, 1, 20, 4, 9]
Remaining: [12, 14, 15, 17, 18, 21]
row 4:         [8]
row 3:       [10, 2]
row 2:     [13, 3, 5]
row 1:   [6, 19, 16, 11]
row 0: [7, 1, 20, 4, 15]
Remaining: [9, 12, 14, 17, 18, 21]
row 4:         [9]
row 3:       [11, 2]
row 2:     [14, 3, 5]
row 1:   [6, 20, 17, 12]
row 0: [7, 1, 21, 4, 16]
Remaining: [8, 10, 13, 15, 18, 19]
row 4:         [8]
row 3:       [9, 1]
row 2:     [13, 4, 3]
row 1:   [5, 18, 14, 11]
row 0: [7, 2, 20, 6, 17]
Remaining: [10, 12, 15, 16, 19, 21]
row 4:         [4]
row 3:       [13, 9]
row 2:     [14, 1, 10]
row 1:   [5, 19, 18, 8]
row 0: [7, 2, 21, 3, 11]
Remaining: [6, 12, 15, 16, 17, 20]
row 4:         [9]
row 3:       [10, 1]
row 2:     [14, 4, 3]
row 1:   [5, 19, 15, 12]
row 0: [7, 2, 21, 6, 18]
Remaining: [8, 11, 13, 16, 17, 20]
row 4:         [6]
row 3:       [4, 10]
row 2:     [14, 18, 8]
row 1:   [5, 19, 1, 9]
row 0: [7, 2, 21, 20, 11]
Remaining: [3, 12, 13, 15, 16, 17]
row 4:         [8]
row 3:       [2, 10]
row 2:     [13, 11, 1]
row 1:   [4, 17, 6, 5]
row 0: [7, 3, 20, 14, 9]
Remaining: [12, 15, 16, 18, 19, 21]
row 4:         [8]
row 3:       [2, 10]
row 2:     [13, 11, 1]
row 1:   [4, 17, 6, 5]
row 0: [7, 3, 20, 14, 19]
Remaining: [9, 12, 15, 16, 18, 21]
row 4:         [5]
row 3:       [13, 8]
row 2:     [14, 1, 9]
row 1:   [4, 18, 19, 10]
row 0: [7, 3, 21, 2, 12]
Remaining: [6, 11, 15, 16, 17, 20]
row 4:         [9]
row 3:       [2, 11]
row 2:     [14, 12, 1]
row 1:   [4, 18, 6, 5]
row 0: [7, 3, 21, 15, 10]
Remaining: [8, 13, 16, 17, 19, 20]
row 4:         [9]
row 3:       [2, 11]
row 2:     [14, 12, 1]
row 1:   [4, 18, 6, 5]
row 0: [7, 3, 21, 15, 20]
Remaining: [8, 10, 13, 16, 17, 19]
row 4:         [8]
row 3:       [2, 10]
row 2:     [13, 11, 1]
row 1:   [3, 16, 5, 6]
row 0: [7, 4, 20, 15, 9]
Remaining: [12, 14, 17, 18, 19, 21]
row 4:         [8]
row 3:       [2, 10]
row 2:     [13, 11, 1]
row 1:   [3, 16, 5, 6]
row 0: [7, 4, 20, 15, 21]
Remaining: [9, 12, 14, 17, 18, 19]
row 4:         [9]
row 3:       [2, 11]
row 2:     [14, 12, 1]
row 1:   [3, 17, 5, 6]
row 0: [7, 4, 21, 16, 10]
Remaining: [8, 13, 15, 18, 19, 20]
row 4:         [5]
row 3:       [1, 6]
row 2:     [14, 15, 9]
row 1:   [3, 17, 2, 11]
row 0: [7, 4, 21, 19, 8]
Remaining: [10, 12, 13, 16, 18, 20]
row 4:         [3]
row 3:       [11, 8]
row 2:     [12, 1, 9]
row 1:   [2, 14, 15, 6]
row 0: [7, 5, 19, 4, 10]
Remaining: [13, 16, 17, 18, 20, 21]
row 4:         [3]
row 3:       [11, 8]
row 2:     [12, 1, 9]
row 1:   [2, 14, 13, 4]
row 0: [7, 5, 19, 6, 10]
Remaining: [15, 16, 17, 18, 20, 21]
row 4:         [8]
row 3:       [11, 3]
row 2:     [12, 1, 4]
row 1:   [2, 14, 13, 9]
row 0: [7, 5, 19, 6, 15]
Remaining: [10, 16, 17, 18, 20, 21]
row 4:         [9]
row 3:       [12, 3]
row 2:     [13, 1, 4]
row 1:   [2, 15, 14, 10]
row 0: [7, 5, 20, 6, 16]
Remaining: [8, 11, 17, 18, 19, 21]
row 4:         [10]
row 3:       [13, 3]
row 2:     [14, 1, 4]
row 1:   [2, 16, 15, 11]
row 0: [7, 5, 21, 6, 17]
Remaining: [8, 9, 12, 18, 19, 20]
row 4:         [10]
row 3:       [11, 1]
row 2:     [14, 3, 4]
row 1:   [2, 16, 13, 9]
row 0: [7, 5, 21, 8, 17]
Remaining: [6, 12, 15, 18, 19, 20]
row 4:         [5]
row 3:       [3, 8]
row 2:     [6, 9, 17]
row 1:   [4, 10, 19, 2]
row 0: [7, 11, 1, 20, 18]
Remaining: [12, 13, 14, 15, 16, 21]
row 4:         [5]
row 3:       [3, 8]
row 2:     [6, 9, 1]
row 1:   [4, 10, 19, 18]
row 0: [7, 11, 21, 2, 20]
Remaining: [12, 13, 14, 15, 16, 17]
row 4:         [5]
row 3:       [3, 8]
row 2:     [6, 9, 17]
row 1:   [4, 10, 1, 18]
row 0: [7, 11, 21, 20, 2]
Remaining: [12, 13, 14, 15, 16, 19]
row 4:         [6]
row 3:       [4, 10]
row 2:     [5, 1, 11]
row 1:   [8, 13, 14, 3]
row 0: [7, 15, 2, 16, 19]
Remaining: [9, 12, 17, 18, 20, 21]
row 4:         [9]
row 3:       [1, 10]
row 2:     [5, 4, 14]
row 1:   [8, 13, 17, 3]
row 0: [7, 15, 2, 19, 16]
Remaining: [6, 11, 12, 18, 20, 21]
row 4:         [3]
row 3:       [1, 4]
row 2:     [5, 6, 10]
row 1:   [8, 13, 19, 9]
row 0: [7, 15, 2, 21, 12]
Remaining: [11, 14, 16, 17, 18, 20]
row 4:         [9]
row 3:       [1, 10]
row 2:     [5, 6, 16]
row 1:   [8, 13, 19, 3]
row 0: [7, 15, 2, 21, 18]
Remaining: [4, 11, 12, 14, 17, 20]
row 4:         [9]
row 3:       [2, 11]
row 2:     [4, 6, 17]
row 1:   [8, 12, 18, 1]
row 0: [7, 15, 3, 21, 20]
Remaining: [5, 10, 13, 14, 16, 19]
row 4:         [9]
row 3:       [10, 1]
row 2:     [3, 13, 14]
row 1:   [8, 5, 18, 4]
row 0: [7, 15, 20, 2, 6]
Remaining: [11, 12, 16, 17, 19, 21]
row 4:         [1]
row 3:       [10, 9]
row 2:     [3, 13, 4]
row 1:   [8, 5, 18, 14]
row 0: [7, 15, 20, 2, 16]
Remaining: [6, 11, 12, 17, 19, 21]
row 4:         [8]
row 3:       [4, 12]
row 2:     [6, 2, 14]
row 1:   [9, 15, 17, 3]
row 0: [7, 16, 1, 18, 21]
Remaining: [5, 10, 11, 13, 19, 20]
row 4:         [10]
row 3:       [1, 11]
row 2:     [5, 4, 15]
row 1:   [9, 14, 18, 3]
row 0: [7, 16, 2, 20, 17]
Remaining: [6, 8, 12, 13, 19, 21]