A15397 | Funny Permutation

A sequence of $n$ numbers is called permutation if it contains all numbers from $1$ to $n$ exactly once. For example, the sequences $[3, 1, 4, 2]$ , \[ $1$ \] and $[2,1]$ are permutations, but $[1,2,1]$ , $[0,1]$ and $[1,3,4]$ are not.

For a given number $n$ you need to make a permutation $p$ such that two requirements are satisfied at the same time:

- For each element $p_i$ , at least one of its neighbors has a value that differs from the value of $p_i$ by one. That is, for each element $p_i$ ( $1 \le i \le n$ ), at least one of its neighboring elements (standing to the left or right of $p_i$ ) must be $p_i + 1$ , or $p_i - 1$ .
- the permutation must have no fixed points. That is, for every $i$ ( $1 \le i \le n$ ), $p_i \neq i$ must be satisfied.

Let's call the permutation that satisfies these requirements funny.

For example, let $n = 4$ . Then \[ $4, 3, 1, 2$ \] is a funny permutation, since:

- to the right of $p_1=4$ is $p_2=p_1-1=4-1=3$ ;
- to the left of $p_2=3$ is $p_1=p_2+1=3+1=4$ ;
- to the right of $p_3=1$ is $p_4=p_3+1=1+1=2$ ;
- to the left of $p_4=2$ is $p_3=p_4-1=2-1=1$ .
- for all $i$ is $p_i \ne i$ .

For a given positive integer $n$ , output any funny permutation of length $n$ , or output -1 if funny permutation of length $n$ does not exist.

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

The description of the test cases follows.

Each test case consists of f single line containing one integer $n$ ( $2 \le n \le 2 \cdot 10^5$ ).

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$ .

For each test case, print on a separate line:

- any funny permutation $p$ of length $n$ ;
- or the number -1 if the permutation you are looking for does not exist.