Problem M
You be The Judge, Again
You are a judge, again! The contest you’re judging includes the following problem:
“You have one L-shaped triomino of each of $\frac{4^ n-1}{3}$ different colors. Tile a $2^ n$ by $2^ n$ grid using each of these triominos such that there is exactly one blank square and all other squares are covered by exactly one square of such a triomino. All triominos must be used.”
Your team is to write a checker for this problem. Validation of the input values and format has already taken place. You will be given a purported tiling of a $2^ n$ by $2^ n$ grid, where each square in the grid is either 0 or a positive integer from $1$ to $\frac{4^ n-1}{3}$ representing one of the colors. Determine if it is, indeed, a covering of the grid with $\frac{4^ n-1}{3}$ unique triominos and a single empty space.
L-shaped triominos look like this:
|
$\blacksquare $ |
$\blacksquare $ |
$\blacksquare $ |
$\blacksquare $ |
$\blacksquare $ |
$\blacksquare $ |
|||||
|
$\blacksquare $ |
$\blacksquare $ |
$\blacksquare $ |
$\blacksquare $ |
$\blacksquare $ |
$\blacksquare $ |
Input
The first line of input contains a single integer $n$ ($1 \le n \le 10$), which is the $n$ of the description.
Each of the next $2^ n$ lines contains $2^ n$ integers $x$ ($0 \le x \le \frac{4^ n-1}{3}$), where 0 represents an empty space, and any positive number is a unique identifier of a triomino.
Output
Output a single integer, which is $1$ if the given grid is covered with $\frac{4^ n-1}{3}$ unique triominos and a single empty space. Otherwise, output $0$.
| Sample Input 1 | Sample Output 1 |
|---|---|
2 1 1 2 2 1 3 3 2 4 4 3 5 4 0 5 5 |
1 |
| Sample Input 2 | Sample Output 2 |
|---|---|
1 1 1 1 1 |
0 |
