Problem:
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15 Puzzle

The goal of the 15 puzzle problem is to complete pieces on $4 \times 4$ cells where one of the cells is empty space.

In this problem, the space is represented by 0 and pieces are represented by integers from 1 to 15 as shown below.

1 2 3 4
6 7 8 0
5 10 11 12
9 13 14 15

You can move a piece toward the empty space at one step. Your goal is to make the pieces the following configuration in the shortest move (fewest steps).

1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 0

Write a program which reads an initial state of the puzzle and prints the fewest steps to solve the puzzle.

Input

The $4 \times 4$ integers denoting the pieces or space are given.

Output

Print the fewest steps in a line.

Constraints

The given puzzle is solvable in at most 45 steps.

Sample Input

1 2 3 4
6 7 8 0
5 10 11 12
9 13 14 15

Sample Output

8

Solution in Python3 (judge verdict: Accepted)

from heapq import heappop, heappush
manhattan = tuple(tuple(abs((i % 4) - (j % 4)) + abs((i // 4) - (j // 4)) for j in range(16)) for i in range(16))
movables = ((1, 4), (0, 2, 5), (1, 3, 6), (2, 7), (0, 5, 8), (1, 4, 6, 9), (2, 5, 7, 10), (3, 6, 11),
            (4, 9, 12), (5, 8, 10, 13), (6, 9, 11, 14), (7, 10, 15), (8, 13), (9, 12, 14), (10, 13, 15), (11, 14))
swap_mul = tuple(tuple((1 << mf) - (1 << mt) for mt in range(0, 64, 4)) for mf in range(0, 64, 4))
destination = 0xfedcba9876543210
i = 0
board_init = 0
blank_init = 0
for _ in range(4):
    for n in map(int, input().split()):
        if n:
            n -= 1
        else:
            n = 15
            blank_init = i
        board_init += n * 16 ** i
        i += 1
estimation_init = sum(manhattan[i][((board_init >> (4 * i)) & 15)] for i in range(16) if i != blank_init)
queue = [(estimation_init, board_init, blank_init)]
visited = set()
while True:
    estimation, board, blank = heappop(queue)
    if board in visited:
        continue
    elif board == destination:
        print(estimation)
        break
    visited.add(board)
    for new_blank in movables[blank]:
        num = (board >> (4 * new_blank)) & 15
        new_board = board + swap_mul[new_blank][blank] * (15 - num)
        if new_board in visited:
            continue
        new_estimation = estimation + 1 - manhattan[new_blank][num] + manhattan[blank][num]
        if new_estimation > 45:
            continue
        heappush(queue, (new_estimation, new_board, new_blank))