Cruel Bingo

Time Limit : 8 sec, Memory Limit : 65536 KB

Problem J: Cruel Bingo

Bingo is a party game played by many players and one game master. Each player is given a bingo card containing N2 different numbers in a N × N grid (typically N = 5). The master draws numbers from a lottery one by one during the game. Each time a number is drawn, a player marks a square with that number if it exists. The playerfs goal is to have N marked squares in a single vertical, horizontal, or diagonal line and then call gBingo!h The first player calling gBingo!h wins the game.

In ultimately unfortunate cases, a card can have exactly N unmarked squares, or N(N-1) marked squares, but not a bingo pattern. Your task in this problem is to write a program counting how many such patterns are possible from a given initial pattern, which contains zero or more marked squares.


The input is given in the following format:

x1 y1
xK yK

The input begins with a line containing two numbers N (1 ≤ N ≤ 32) and K (0 ≤ K ≤ 8), which represent the size of the bingo card and the number of marked squares in the initial pattern respectively. Then K lines follow, each containing two numbers xi and yi to indicate the square at (xi, yi) is marked. The coordinate values are zero-based (i.e. 0 ≤ xi, yiN - 1). No pair of marked squares coincides.


Count the number of possible non-bingo patterns with exactly N unmarked squares that can be made from the given initial pattern, and print the number in modulo 10007 in a line (since it is supposed to be huge). Rotated and mirrored patterns should be regarded as different and counted each.

Sample Input 1

4 2
0 2
3 1

Output for the Sample Input 1


Sample Input 2

4 2
2 3
3 2

Output for the Sample Input 2


Sample Input 3

10 3
0 0
4 4
1 4

Output for the Sample Input 3


Source: ACM-ICPC Japan Alumni Group Summer Camp 2010 , Day 4, Tokyo, Japan, 2010-09-20