P9890 [ICPC 2018 Qingdao R] Tournament

DreamGrid, the king of Gridland, is making a knight tournament. There are $n$ knights, numbered from 1 to $n$, participating in the tournament. The rules of the tournament are listed as follows: - The tournament consists of $k$ rounds. Each round consists of several duels. Each duel happens between exactly two knights. - Each knight must participate in exactly one duel during each round. - For each pair of knights, there can be at most one duel between them during all the $k$ rounds. - Let $1 \le i, j \le k$, $i \ne j$, and $1 \le a, b, c, d \le n$, $a, b, c, d$ be four distinct integers. If - Knight $a$ fights against knight $b$ during round $i$, and - Knight $c$ fights against knight $d$ during round $i$, and - Knight $a$ fights against knight $c$ during round $j$, - then knight $b$ must fight against knight $d$ during round $j$. As DreamGrid's general, you are asked to write a program to arrange all the duels in all the $k$ rounds, so that the resulting arrangement satisfies the rules above.

There are multiple test cases. The first line of the input is an integer $T$, indicating the number of test cases. For each test case: The first and only line contains two integers $n$ and $k$ ($1 \le n, k \le 1000$), indicating the number of knights participating in the tournament and the number of rounds. It's guaranteed that neither the sum of $n$ nor the sum of $k$ in all test cases will exceed $5000$.

For each test case: - If it's possible to make a valid arrangement, output $k$ lines. On the $i$-th line, output $n$ integers $c_{i, 1}, c_{i, 2}, \dots, c_{i, n}$ separated by one space, indicating that in the $i$-th round, knight $j$ will fight against knight $c_{i, j}$ for all $1 \le j \le n$. If there are multiple valid answers, output the lexicographically smallest answer. Consider two answers $A$ and $B$, let's denote $a_{i, j}$ as the $j$-th integer on the $i$-th line in answer $A$, and $b_{i, j}$ as the $j$-th integer on the $i$-th line in answer $B$. Answer $A$ is lexicographically smaller than answer $B$, if there exists two integers $p$ ($1 \le p \le k$) and $q$ ($1 \le q \le n$), such that - for all $1 \le i < p$ and $1 \le j \le n$, $a_{i, j} = b_{i, j}$, and - for all $1 \le j < q$, $a_{p, j} = b_{p, j}$, and finally $a_{p, q} < b_{p, q}$. -If it's impossible to make a valid arrangement, output ``Impossible`` (without quotes) in one line. Please, DO NOT output extra spaces at the end of each line, or your answer may be considered incorrect!