Score : $350$ points

问题描述

存在一个 $N$ 行和 $N$ 列的网格，其中 $N$ 是不超过 $45$ 的奇数。

令 $(i,j)$ 表示从上到下第 $i$ 行、从左到右第 $j$ 列的格子。

在这个网格中，你需要放置 Takahashi 和一只由编号为 $1$ 到 $N^2-1$ 的 $N^2-1$ 个部分组成的龙，使得满足以下条件：

• Takahashi 必须放置在网格的中心位置，即格子 $(\frac{N+1}{2},\frac{N+1}{2})$。
• 除了 Takahashi 所在的格子外，每个格子中恰好放置一个龙的部分。
• 对于满足 $2 \leq x \leq N^2-1$ 的所有整数 $x$，龙的部分 $x$ 必须放置在一个与包含部分 $x-1$ 的格子相邻（通过边）的格子中。
  • 格子 $(i,j)$ 和 $(k,l)$ 被认为是通过边相邻，当且仅当 $|i-k|+|j-l|=1$。

请打印一种满足条件的摆放方式。保证至少存在一种满足条件的排列方式。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There is a grid with $N$ rows and $N$ columns, where $N$ is an odd number at most $45$.

Let $(i,j)$ denote the cell at the $i$-th row from the top and $j$-th column from the left.

In this grid, you will place Takahashi and a dragon consisting of $N^2-1$ parts numbered $1$ to $N^2-1$ in such a way that satisfies the following conditions:

• Takahashi must be placed at the center of the grid, that is, in cell $(\frac{N+1}{2},\frac{N+1}{2})$.
• Except for the cell where Takahashi is, exactly one dragon part must be placed in each cell.
• For every integer $x$ satisfying $2 \leq x \leq N^2-1$, the dragon part $x$ must be placed in a cell adjacent by an edge to the cell containing part $x-1$.
  • Cells $(i,j)$ and $(k,l)$ are said to be adjacent by an edge if and only if $|i-k|+|j-l|=1$.

Print one way to arrange the parts to satisfy the conditions. It is guaranteed that there is at least one arrangement that satisfies the conditions.

Constraints

• $3 \leq N \leq 45$
• $N$ is odd.

Input

The input is given from Standard Input in the following format:

$N$

Output

Print $N$ lines.
The $i$-th line should contain $X_{i,1},\ldots,X_{i,N}$ separated by spaces, where $X_{i,j}$ is T when placing Takahashi in cell $(i,j)$ and $x$ when placing part $x$ there.

Sample Input 1

5

Sample Output 1

1 2 3 4 5
16 17 18 19 6
15 24 T 20 7
14 23 22 21 8
13 12 11 10 9

The following output also satisfies all the conditions and is correct.

9 10 11 14 15
8 7 12 13 16
5 6 T 18 17
4 3 24 19 20 
1 2 23 22 21

On the other hand, the following outputs are incorrect for the reasons given.

Takahashi is not at the center.

1 2 3 4 5
10 9 8 7 6
11 12 13 14 15
20 19 18 17 16
21 22 23 24 T

The cells containing parts $23$ and $24$ are not adjacent by an edge.

1 2 3 4 5
10 9 8 7 6
11 12 24 22 23
14 13 T 21 20
15 16 17 18 19

update @ 2024/3/10 01:23:45