103353: [Atcoder]ABC335 D - Loong and Takahashi

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

问题描述

有一个$N$行$N$列的网格，其中$N$是一个最多为45的奇数。

让$(i,j)$表示从顶部数第$i$行，从左数第$j$列的单元格。

在这个网格中，你需要放置Takahashi和一个由$N^2-1$个部分组成的大龙，编号为1到$N^2-1$，以满足以下条件：

• Takahashi必须放在网格的中心，即在单元格$(\frac{N+1}{2},\frac{N+1}{2})$中。
• 除Takahashi所在的单元格外，每个单元格中必须放置一个大龙部分。
• 对于满足$2 \leq x \leq N^2-1$的每个整数$x$，大龙部分$x$必须放在包含部分$x-1$的单元格相邻的单元格中。
• 如果且仅当$|i-k|+|j-l|=1$，则说单元格$(i,j)$和$(k,l)$相邻。

打印一种满足条件的布局方式。保证存在至少一种满足条件的布局方式。

约束

• $3 \leq N \leq 45$
• $N$是奇数。

输入

输入从标准输入按以下格式给出：

$N$

输出

打印$N$行。
第$i$行应该包含$X_{i,1},\ldots,X_{i,N}$，由空格分隔，其中$X_{i,j}$为T时在单元格$(i,j)$中放置Takahashi，为$x$时在该单元格中放置部分$x$。

样例输入1

5

样例输出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

以下输出也满足所有条件，是正确的。

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

另一方面，以下输出由于给出的原因是错误的。

Takahashi不在中心。

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

包含部分$23$和$24$的单元格没有相邻。

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