P7903 Douxin’s Top

Source: [Eight Immortals Toasting](/paste/78f1vlm0), this can be clicked. - Lü Dongbin — Drunkenly lifting a kettle, strength like a thousand jin; - Tieguai Li — Spinning elbow and knee strike, drunk yet still real; - **Han Zhongli — Stumbling steps, hugging the jar, Douxin Top**; - Lan Caihe — One-hand toast, breaking at the waist; - Zhang Guolao — Drunkenly tossing a cup, continuous kicking combo; - Cao Guojiu — Immortal’s toast, throat-locking hook; - Han Xiangzi — Grabbing the wrist and striking the chest, drunkenly playing the flute; - He Xiangu — Bending the waist to offer wine, drunkenly swaying steps.

Given a positive integer $n$, you need to construct a tree with $n$ nodes such that the centroid of the tree’s diameter is **not** the centroid of the tree. At the same time, this tree must satisfy: the diameter$^1$, the centroid$^2$, and the centroid of the diameter$^3$ are all unique. --- Notes: - Tree diameter$^1$: [https://oi-wiki.org/graph/tree-diameter/](https://oi-wiki.org/graph/tree-diameter/) - Tree centroid$^2$: [https://oi-wiki.org/graph/tree-centroid/](https://oi-wiki.org/graph/tree-centroid/) - Centroid of the diameter$^3$: treat the tree diameter (a chain) as a tree, and find its center (a single node).

The first line contains a positive integer $n$, the number of nodes in the tree.

Output a positive integer $n$ on the first line. Then output $n-1$ lines. Each line contains two positive integers $u, v$, representing an edge of the tree. If there is no solution, output `-1`. **This problem uses a Special Judge. Any valid solution will be accepted.**

#### Sample Explanation  In Sample #1, the centroid of the diameter is $7$, and the centroid of the tree is $1$. Thus, $1 \ne 7$. In Sample #2, $n = 2$. With only two nodes, the centroid is obviously impossible to be unique. #### Constraints **This problem uses bundled testdata.** | Subtask ID | Score | Special Property | | :----------: | :----------: | :----------: | | $1$ | $30$ | $n \le 10$ | | $2$ | $30$ | $n$ is odd | | $3$ | $30$ | $n$ is even | | $4$ | $10$ | None | For $100\%$ of the testdata: $1 \le n \le 10^4$. **The Special Judge source code is provided; see the attachment below.** Translated by ChatGPT 5