{"problem":{"name":"Coloring Torus","description":{"content":"For an $n \\times n$ grid, let $(r, c)$ denote the square at the $(r+1)$\\-th row from the top and the $(c+1)$\\-th column from the left. A _good_ coloring of this grid using $K$ colors is a coloring tha","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"agc030_c"},"statements":[{"statement_type":"Markdown","content":"For an $n \\times n$ grid, let $(r, c)$ denote the square at the $(r+1)$\\-th row from the top and the $(c+1)$\\-th column from the left. A _good_ coloring of this grid using $K$ colors is a coloring that satisfies the following:\n\n*   Each square is painted in one of the $K$ colors.\n*   Each of the $K$ colors is used for some squares.\n*   Let us number the $K$ colors $1, 2, ..., K$. For any colors $i$ and $j$ ($1 \\leq i \\leq K, 1 \\leq j \\leq K$), every square in Color $i$ has the same number of adjacent squares in Color $j$. Here, the squares adjacent to square $(r, c)$ are $((r-1)\\; mod\\; n, c), ((r+1)\\; mod\\; n, c), (r, (c-1)\\; mod\\; n)$ and $(r, (c+1)\\; mod\\; n)$ (if the same square appears multiple times among these four, the square is counted that number of times).\n\nGiven $K$, choose **$n$ between $1$ and $500$** (inclusive) freely and construct a good coloring of an $n \\times n$ grid using $K$ colors. It can be proved that this is always possible under the constraints of this problem,\n\n## Constraints\n\n*   $1 \\leq K \\leq 1000$\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$K$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"agc030_c","tags":[],"sample_group":[["2","3\n1 1 1\n1 1 1\n2 2 2\n\n*   Every square in Color $1$ has three adjacent squares in Color $1$ and one adjacent square in Color $2$.\n*   Every square in Color $2$ has two adjacent squares in Color $1$ and two adjacent squares in Color $2$.\n\nOutput such as the following will be judged incorrect:\n\n2\n1 2\n2 2\n\n3\n1 1 1\n1 1 1\n1 1 1"],["9","3\n1 2 3\n4 5 6\n7 8 9"]],"created_at":"2026-03-03 11:01:14"}}