CF2162G.Beautiful Tree

A tree is a connected graph without cycles.

A tree is called beautiful if the sum of the products of the vertex labels for all its edges is a perfect square.

More formally, let $E$ be the set of edges in the tree. The tree is called beautiful if the value $$S = \sum_{\{u, v\} \in E} (u \cdot v)$$ is a perfect square. That is, there exists an integer $x$ such that $S = x^2$.

You are given an integer $n$. Your task is to construct a beautiful tree having $n$ vertices or report that such a tree does not exist.

树是无环的连通图。

若一棵树的所有边的两个端点标签乘积之和为完全平方数，则称该树为“优美的”。

更准确地说，设 $E$ 为该树的边集。若值

$$S = \sum_{\{u, v\} \in E} (u \cdot v)$$

为完全平方数，则称该树为优美的。即存在整数 $x$，使得 $S = x^2$。

给定一个整数 $n$，你的任务是构造一棵含 $n$ 个顶点的优美树；若不存在这样的树，则报告不存在。

The first line of input contains a single integer $t$ ($1 \le t \le 10^4$) — the number of testcases.

Each testcase contains a single integer $n$ ($2 \le n \le 2\cdot10^5$).

It is guaranteed that the sum of $n$ over all the testcases does not exceed $2\cdot10^5$.

输入的第一行包含一个整数 $t$（$1 \le t \le 10^4$）—— 表示测试用例的数量。

每个测试用例包含一个整数 $n$（$2 \le n \le 2\cdot10^5$）。

保证所有测试用例中 $n$ 的总和不超过 $2\cdot10^5$。

For each testcase, if there is no beautiful tree having $n$ vertices, print $-1$.

Otherwise, print $n-1$ lines denoting the edges of a beautiful tree having $n$ vertices. Each line should contain two space-separated integers $u,v$ ($1 \le u,v \le n$) representing an edge.

The vertices can be printed in any order within an edge, and the edges can be printed in any order.

对于每个测试用例，如果不存在包含 $n$ 个顶点的优美树，则输出 $-1$。

否则，输出 $n-1$ 行，每行表示一棵包含 $n$ 个顶点的优美树的一条边。每行应包含两个以空格分隔的整数 $u,v$（$1 \le u,v \le n$），表示一条边。

每条边中两个顶点的顺序可以任意，各条边的输出顺序也可以任意。

Test case 1: No beautiful tree exists with $2$ vertices. Hence, print $-1$.

Test case 2:

$S = (2\cdot3) + (1\cdot3) = 9 = (3)^2$

Test case 3:

$S = (2\cdot1) + (3\cdot1) + (4\cdot1) = 9 = (3)^2$

测试用例 1：不存在包含 $2$ 个顶点的优美树。因此，输出 $-1$。

测试用例 2：

$S = (2\cdot3) + (1\cdot3) = 9 = (3)^2$

测试用例 3：

$S = (2\cdot1) + (3\cdot1) + (4\cdot1) = 9 = (3)^2$