Unbalanced Squares

AtCoder

IDarc142_b

Time2000ms

Memory256MB

Difficulty—

We have an $N \times N$ grid. Let $(i,j)$ denote the square at the $i$\-th row from the top and $j$\-th column from the left in this grid. Find one way to write an integer on every square to satisfy the conditions below. * Each integer between $1$ and $N^2$ is written exactly once. * For every pair of integers $i,j\, (1 \leq i,j \leq N)$, the square $(i,j)$ satisfies the following. * Among the squares horizontally, vertically, or diagonally adjacent to $(i,j)$ (there are at most eight of them), let $a$ and $b$ be the number of squares with integers larger than and smaller than that of $(i,j)$, respectively. Then, $a \neq b$ holds. Under the Constraints of this problem, it can be proved that such a way to write integers always exists. ## Constraints * $2 \leq N \leq 500$ * $N$ is an integer. ## Input Input is given from Standard Input in the following format: $N$ [samples]

Samples

Input #1

Output #1

1 2
3 4

This output contains each integer between $1$ and $N^2\, (=4)$ exactly once, so the first condition is satisfied.  
Additionally, among the squares horizontally, vertically, or diagonally adjacent to the square $(1,1)$, three squares $(1,2)$, $(2,1)$, and $(2,2)$ have integers larger than that of $(1,1)$, and none has an integer smaller than that.  
Thus, for $(1,1)$, we have $a=3$ and $b=0$, so $a\neq b$ holds.  
Similarly, it can be verified that $a\neq b$ also holds for the other squares, so the second condition is satisfied.  
Therefore, this output is valid.

Input #2

Output #2

1 2 3
5 4 6
7 8 9

API Response (JSON)

{
  "problem": {
    "name": "Unbalanced Squares",
    "description": {
      "content": "We have an $N \\times N$ grid. Let $(i,j)$ denote the square at the $i$\\-th row from the top and $j$\\-th column from the left in this grid.   Find one way to write an integer on every square to satisfy",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc142_b"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have an $N \\times N$ grid. Let $(i,j)$ denote the square at the $i$\\-th row from the top and $j$\\-th column from the left in this grid.  \nFind one way to write an integer on every square to satisfy...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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