Minimum Ties
Contest: Educational Codeforces Round 104 (Rated for Div. 2)
Difficulty rating: 1500
Tags: brute force, constructive algorithms, dfs and similar, graphs, greedy, implementation, math

A big football championship will occur soon! $$$n$$$ teams will compete in it, and each pair of teams will play exactly one game against each other.

There are two possible outcomes of a game:

- the game may result in a tie, then both teams get $$$1$$$ point;
- one team might win in a game, then the winning team gets $$$3$$$ points and the losing team gets $$$0$$$ points.

The score of a team is the number of points it gained during all games that it played.

You are interested in a hypothetical situation when all teams get the same score at the end of the championship. A simple example of that situation is when all games result in ties, but you want to minimize the number of ties as well.

Your task is to describe a situation (choose the result of each game) so that all teams get the same score, and the number of ties is the minimum possible.

Input format:
The first line contains one integer $$$t$$$ ($$$1 \le t \le 100$$$) — the number of test cases.

Then the test cases follow. Each test case is described by one line containing one integer $$$n$$$ ($$$2 \le n \le 100$$$) — the number of teams.

Output format:
For each test case, print $$$\frac{n(n - 1)}{2}$$$ integers describing the results of the games in the following order: the first integer should correspond to the match between team $$$1$$$ and team $$$2$$$, the second — between team $$$1$$$ and team $$$3$$$, then $$$1$$$ and $$$4$$$, ..., $$$1$$$ and $$$n$$$, $$$2$$$ and $$$3$$$, $$$2$$$ and $$$4$$$, ..., $$$2$$$ and $$$n$$$, and so on, until the game between the team $$$n - 1$$$ and the team $$$n$$$.

The integer corresponding to the game between the team $$$x$$$ and the team $$$y$$$ should be $$$1$$$ if $$$x$$$ wins, $$$-1$$$ if $$$y$$$ wins, or $$$0$$$ if the game results in a tie.

All teams should get the same score, and the number of ties should be the minimum possible. If there are multiple optimal answers, print any of them. It can be shown that there always exists a way to make all teams have the same score.

Constraints:
Time limit: 1.0 s
Memory limit: 256.0 MB

Note:
In the first test case of the example, both teams get $$$1$$$ point since the game between them is a tie.

In the second test case of the example, team $$$1$$$ defeats team $$$2$$$ (team $$$1$$$ gets $$$3$$$ points), team $$$1$$$ loses to team $$$3$$$ (team $$$3$$$ gets $$$3$$$ points), and team $$$2$$$ wins against team $$$3$$$ (team $$$2$$$ gets $$$3$$$ points).

Example input:
2
2
3

Example output:
0 
1 -1 1

Editorial:
If $$$n$$$ is odd, then we can solve the problem without any ties: each team should win exactly $$$\lfloor\frac{n}{2}\rfloor$$$ matches and lose the same number of matches. Finding which matches each team wins and which matches each team loses can be done with some graph algorithms (like Eulerian cycles or circulations), or with a simple construction: place all teams in a circle in any order, and let the $$$i$$$-th team win against the next $$$\lfloor\frac{n}{2}\rfloor$$$ teams after it in the circle, and lose to all other teams.
Unfortunately, if $$$n$$$ is even, we need to use some ties since the total sum of scores over all teams is exactly $$$\frac{3n(n-1)}{2}$$$ when there are no ties, and this number is not divisible by $$$n$$$ when $$$n$$$ is even. Each tie reduces the total sum by $$$1$$$, and the minimum number of ties to make $$$\frac{3n(n-1)}{2} - t$$$ divisible by $$$n$$$ is $$$t = \frac{n}{2}$$$ (since $$$\frac{3n(n-1)}{2} \bmod n = \frac{n}{2}$$$). So, if we find an answer with exactly $$$\frac{n}{2}$$$ ties, it is optimal. And it's easy to find one: once again, place all teams in a circle in any order; make the $$$i$$$-th team win against $$$\frac{n - 2}{2}$$$ next teams in the circle, lose against $$$\frac{n - 2}{2}$$$ previous teams in the circle, and tie with the opposite team in the circle.