ICPC Asia Can Tho Regional Contest 2020 - Open

Start

2020-12-10 17:00 AKST

ICPC Asia Can Tho Regional Contest 2020 - Open

End

2020-12-10 22:00 AKST
The end is near!
Contest is over.
Not yet started.
Contest is starting in -194 days 21:42:23

Time elapsed

5:00:00

Time remaining

0:00:00

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 odd-numbered rows have exactly $n$ hexagons, and the even-numbered rows have exactly $n-1$ 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 $n = 3$.

\includegraphics[width=0.3\textwidth ]{img/1_numbered_grid.png}

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:

\includegraphics[width=0.3\textwidth ]{img/1.png}

Let’s color some edges of some hexagons. A coloring is valid iff it satisfies the following conditions:

  1. 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 $(i, j)$.

  2. The colored edges form one or more loops. Each loop must not self-intersect. Two different loops must not share any vertices or edges.

The following figure shows a valid coloring:

\includegraphics[width=0.3\textwidth ]{img/1_good.png}

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.

\includegraphics[width=.7\textwidth ]{img/1_bad_combine.png}

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 $n-1$ 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:

\includegraphics[width=0.6\textwidth ]{img/2.png}
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