F - XOR on Grid Path
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F - XOR on Grid Path
Score : $500$ points
Problem Statement
There is a grid with $N$ rows and $N$ columns. We denote by $(i, j)$ the square at the $i$-th $(1 \leq i \leq N)$ row from the top and $j$-th $(1 \leq j \leq N)$ column from the left.
Square $(i, j)$ has a non-negative integer $a_{i, j}$ written on it.
When you are at square $(i, j)$, you can move to either square $(i+1, j)$ or $(i, j+1)$. Here, you are not allowed to go outside the grid.
Find the number of ways to travel from square $(1, 1)$ to square $(N, N)$ such that the exclusive logical sum of the integers written on the squares visited (including $(1, 1)$ and $(N, N)$) is $0$.
What is the exclusive logical sum?
The exclusive logical sum $a \oplus b$ of two integers $a$ and $b$ is defined as follows.- The $2^k$'s place ($k \geq 0$) in the binary notation of $a \oplus b$ is $1$ if exactly one of the $2^k$'s places in the binary notation of $a$ and $b$ is $1$; otherwise, it is $0$.
In general, the exclusive logical sum of $k$ integers $p_1, \dots, p_k$ is defined as $(\cdots ((p_1 \oplus p_2) \oplus p_3) \oplus \cdots \oplus p_k)$. We can prove that it is independent of the order of $p_1, \dots, p_k$.
Constraints
- $2 \leq N \leq 20$
- $0 \leq a_{i, j} \lt 2^{30} \, (1 \leq i, j \leq N)$
- All values in the input are integers.
Input
The input is given from Standard Input in the following format:
Output
Print the answer.
3
1 5 2
7 0 5
4 2 3
2
The following two ways satisfy the condition:
- $(1, 1) \rightarrow (1, 2) \rightarrow (1, 3) \rightarrow (2, 3) \rightarrow (3, 3)$;
- $(1, 1) \rightarrow (2, 1) \rightarrow (2, 2) \rightarrow (2, 3) \rightarrow (3, 3)$.
2
1 2
2 1
0
10
1 0 1 0 0 1 0 0 0 1
0 0 0 1 0 1 0 1 1 0
1 0 0 0 1 0 1 0 0 0
0 1 0 0 0 1 1 0 0 1
0 0 1 1 0 1 1 0 1 0
1 0 0 0 1 0 0 1 1 0
1 1 1 0 0 0 1 1 0 0
0 1 1 0 0 1 1 0 1 0
1 0 1 1 0 0 0 0 0 0
1 0 1 1 0 0 1 1 1 0
24307