Row to Column

AtCoder

IDmujin_pc_2017_b

Time2000ms

Memory256MB

Difficulty—

There is a square-shaped grid with $N$ vertical rows and $N$ horizontal columns. We will denote the square at the $i$\-th row from the top and the $j$\-th column from the left as $(i,\ j)$. Initially, each square is either white or black. The initial color of the grid is given to you as characters $a_{ij}$, arranged in a square shape. If the square $(i,\ j)$ is white, $a_{ij}$ is `.`. If it is black, $a_{ij}$ is `#`. You are developing a robot that repaints the grid. It can repeatedly perform the following operation: * Select two integers $i$, $j$ ($1 ≤ i,\ j ≤ N$). Memorize the colors of the squares $(i,\ 1)$, $(i,\ 2)$, $...$, $(i,\ N)$ as $c_1$, $c_2$, $...$, $c_N$, respectively. Then, repaint the squares $(1,\ j)$, $(2,\ j)$, $...$, $(N,\ j)$ with the colors $c_1$, $c_2$, $...$, $c_N$, respectively. Your objective is to turn all the squares black. Determine whether it is possible, and find the minimum necessary number of operations to achieve it if the answer is positive. ## Constraints * $2 ≤ N ≤ 500$ * $a_{ij}$ is either `.` or `#`. ## Input The input is given from Standard Input in the following format: $N$ $a_{11}$$...$$a_{1N}$ $:$ $a_{N1}$$...$$a_{NN}$ ## Partial Score * In a test set worth $300$ points, $N ≤ 3$. [samples]

Samples

Input #1

2
#.
.#

Output #1

3

For example, perform the operation as follows:

*   Select $i = 1$, $j = 2$.
*   Select $i = 1$, $j = 1$.
*   Select $i = 1$, $j = 2$.

The transition of the colors of the squares is shown in the figure below:

![image](https://atcoder.jp/img/mujin/6a0314bb2b1073694a7ef5a062e77b13.png)

Input #2

2
..
..

Output #2

\-1

Input #3

2
##
##

Output #3

Input #4

3
.#.
###
.#.

Output #4

Input #5

3
...
.#.
...

Output #5

API Response (JSON)

{
  "problem": {
    "name": "Row to Column",
    "description": {
      "content": "There is a square-shaped grid with $N$ vertical rows and $N$ horizontal columns. We will denote the square at the $i$\\-th row from the top and the $j$\\-th column from the left as $(i,\\ j)$. Initially,",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "mujin_pc_2017_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a square-shaped grid with $N$ vertical rows and $N$ horizontal columns. We will denote the square at the $i$\\-th row from the top and the $j$\\-th column from the left as $(i,\\ j)$.\nInitially,...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments