Problem H
Hexagon coloring
You are given a hexagonal grid with $n$ rows, where $n$ is an odd integer. The rows are numbered $1$ to $n$ from top to bottom. The oddnumbered rows have exactly $n$ hexagons, and the evennumbered rows have exactly $n1$ hexagons. Let’s denote the $j$th hexagon in the $i$th row by $(i, j)$.
For example, the below figure shows a hexagonal grid with $n = 3$.
Let’s assign an integer between $1$ and $6$ to each hexagon. Let’s $a_{i, j}$ be the integer assigned to the hexagon $(i, j)$. The following figure shows one example assignment:
Let’s color some edges of some hexagons. A coloring is valid iff it satisfies the following conditions:

For every pair of valid indices $i$ and $j$, either $a_{i, j} = 1$, or $a_{i, j}$ is equal to the number of colored edges of the hexagon $(i, j)$.

The colored edges form one or more loops. Each loop must not selfintersect. Two different loops must not share any vertices or edges.
The following figure shows a valid coloring:
The following two figures show two invalid colorings. The one on the left does not satisfy the $1$st condition, and the one on the right does not satisfy the $2$nd condition.
How many valid colorings are there?
Input
The first line of the input contains a single integer $n$ ($n$ is an odd integer between $1$ and $7$). The next $n$ lines contain the numbers $a_{i, j}$ $(1 \le a_{i, j} \le 6)$. The $i$th line contains exactly $n$ integers if $i$ is odd, and $n1$ integers otherwise.
Output
Print a single integer — the number of valid colorings.
Explanation of the sample input
The first sample was shown in the above figures.
The second example is shown below:
Sample Input 1  Sample Output 1 

3 1 2 1 2 2 1 1 1 
1 
Sample Input 2  Sample Output 2 

7 1 4 5 1 0 1 1 1 3 2 0 0 1 1 4 1 1 0 1 1 1 3 4 2 2 4 0 2 3 1 4 4 2 1 4 4 3 3 2 1 1 1 1 4 2 1 
1 
Sample Input 3  Sample Output 3 

3 1 2 1 2 2 1 1 1 
4 