Rotation Puzzle

AtCoder

IDabc336_f

Time5000ms

Memory256MB

Difficulty—

There is a grid with $H$ rows and $W$ columns. Initially, each integer from $1$ to $(H \times W)$ is written exactly once in the grid. Specifically, for $1 \leq i \leq H$ and $1 \leq j \leq W$, the cell at the $i$\-th row from the top and the $j$\-th column from the left contains $S_{i,j}$. Below, let $(i,j)$ denote the cell at the $i$\-th row from the top and the $j$\-th column from the left. Determine if it is possible to reach a state where, for all pairs of integers $(i,j)$ $(1 \leq i \leq H, 1 \leq j \leq W)$, the cell $(i,j)$ contains the integer $((i-1) \times W + j)$ by repeating the following operation **at most $20$ times** (possibly zero). If it is possible, print the minimum number of operations required. If it is impossible in $20$ or fewer operations (including the case where it is impossible no matter how many times the operation is repeated), print $-1$. > Operation: Choose a rectangle of size $(H-1) \times (W-1)$ from the grid and rotate it $180$ degrees. > More precisely, choose integers $x$ and $y$ $(0 \leq x, y \leq 1)$, > and for all pairs of integers $(i,j)$ satisfying $1 \leq i \leq H-1$ and $1 \leq j \leq W-1$, > simultaneously replace the integer written in cell $(i+x,j+y)$ with the number written in cell $(H-i+x,W-j+y)$. Note that it is only necessary for the integers written in the cells to satisfy the condition; the orientation in which they are written does not matter. ## Constraints * $3 \leq H,W \leq 8$ * $1 \leq S_{i,j} \leq H \times W$ * If $(i,j) \neq (i',j')$, then $S_{i,j} \neq S_{i',j'}$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $H$ $W$ $S_{1,1}$ $S_{1,2}$ $\ldots$ $S_{1,W}$ $S_{2,1}$ $S_{2,2}$ $\ldots$ $S_{2,W}$ $\vdots$ $S_{H,1}$ $S_{H,2}$ $\ldots$ $S_{H,W}$ [samples]

Samples

Input #1

Output #1

2

Operating in the following order will satisfy the conditions of the problem statement in $2$ operations.

*   Operate by choosing the rectangle in the top-left corner. That is, by choosing $x=0$ and $y=0$.
*   Operate by choosing the rectangle in the bottom-right corner. That is, by choosing $x=1$ and $y=1$.

On the other hand, it is impossible to satisfy the conditions with one or fewer operations, so print $2$.
![image](https://img.atcoder.jp/abc336/75a97e79fc11bfe9406ef4e3bef74f37.png)

Input #2

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

Output #2

\-1

It is impossible to satisfy the conditions with $20$ or fewer operations, so print $-1$.

Input #3

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

Output #3

Input #4

Output #4

API Response (JSON)

{
  "problem": {
    "name": "Rotation Puzzle",
    "description": {
      "content": "There is a grid with $H$ rows and $W$ columns. Initially, each integer from $1$ to $(H \\times W)$ is written exactly once in the grid.   Specifically, for $1 \\leq i \\leq H$ and $1 \\leq j \\leq W$, the ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 5000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc336_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a grid with $H$ rows and $W$ columns. Initially, each integer from $1$ to $(H \\times W)$ is written exactly once in the grid.  \nSpecifically, for $1 \\leq i \\leq H$ and $1 \\leq j \\leq W$, the ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments