Score : $300$ points

问题描述

存在一个 $N\times N$ 的网格，其中包含水平行和垂直列，所有方格最初都被涂成白色。令 $(i,j)$ 表示第 $i$ 行第 $j$ 列的方格。

Takahashi 拥有两个整数 $A$ 和 $B$，它们都在 $1$ 到 $N$（包括两端点）之间。他将执行以下操作：

• 对于每一个满足 $\max(1-A,1-B)\leq k\leq \min(N-A,N-B)$ 的整数 $k$，将 $(A+k,B+k)$ 涂成黑色。
• 对于每一个满足 $\max(1-A,B-N)\leq k\leq \min(N-A,B-1)$ 的整数 $k$，将 $(A+k,B-k)$ 涂成黑色。

在这些操作之后，请找出网格中方格 $(i,j)$ 的颜色，条件是 $P\leq i\leq Q$ 且 $R\leq j\leq S$。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There is an $N\times N$ grid with horizontal rows and vertical columns, where all squares are initially painted white. Let $(i,j)$ denote the square at the $i$-th row and $j$-th column.

Takahashi has integers $A$ and $B$, which are between $1$ and $N$ (inclusive). He will do the following operations.

• For every integer $k$ such that $\max(1-A,1-B)\leq k\leq \min(N-A,N-B)$, paint $(A+k,B+k)$ black.
• For every integer $k$ such that $\max(1-A,B-N)\leq k\leq \min(N-A,B-1)$, paint $(A+k,B-k)$ black.

In the grid after these operations, find the color of each square $(i,j)$ such that $P\leq i\leq Q$ and $R\leq j\leq S$.

Constraints

• $1 \leq N \leq 10^{18}$
• $1 \leq A \leq N$
• $1 \leq B \leq N$
• $1 \leq P \leq Q \leq N$
• $1 \leq R \leq S \leq N$
• $(Q-P+1)\times(S-R+1)\leq 3\times 10^5$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $A$ $B$

$P$ $Q$ $R$ $S$

Output

Print $Q-P+1$ lines.
Each line should contain a string of length $S-R+1$ consisting of # and .. The $j$-th character of the string in the $i$-th line should be # to represent that $(P+i-1, R+j-1)$ is painted black, and . to represent that $(P+i-1, R+j-1)$ is white.

Sample Input 1

5 3 2
1 5 1 5

Sample Output 1

...#.
#.#..
.#...
#.#..
...#.

The first operation paints the four squares $(2,1)$, $(3,2)$, $(4,3)$, $(5,4)$ black, and the second paints the four squares $(4,1)$, $(3,2)$, $(2,3)$, $(1,4)$ black.
Thus, the above output should be printed, since $P=1$, $Q=5$, $R=1$, $S=5$.

Sample Input 2

5 3 3
4 5 2 5

Sample Output 2

#.#.
...#

The operations paint the nine squares $(1,1)$, $(1,5)$, $(2,2)$, $(2,4)$, $(3,3)$, $(4,2)$, $(4,4)$, $(5,1)$, $(5,5)$.
Thus, the above output should be printed, since $P=4$, $Q=5$, $R=2$, $S=5$.

Sample Input 3

1000000000000000000 999999999999999999 999999999999999999
999999999999999998 1000000000000000000 999999999999999998 1000000000000000000

Sample Output 3

#.#
.#.
#.#

Note that the input may not fit into a $32$-bit integer type.

update @ 2024/3/10 10:03:12