CF2246A farmpiggie and Subset Sum

For a permutation $ ^{\text{∗}} $ $ p $ of even length, you can do the following process: - Initialize a counter $ c = 0. $ - For each $ i $ from $ 1 $ to $ n, $ either add $ i \cdot p_i $ to $ c $ , subtract $ i \cdot p_i $ from $ c $ , or do nothing. Let the final value of the counter be $ c_{\mathrm{final}}. $ Formally, for each $ i \in \{1,\ldots,n\}, $ consider the set $ S_i = \{-i \cdot p_i, 0, i \cdot p_i\} $ and choose some $ x_i \in S_i. $ Set $ c_{\mathrm{final}} = \sum_{i = 1}^{n}x_i. $ You are given a single even integer $ n $ . Find any permutation of length $ n $ so that regardless of the operations chosen, the final value $ c_{\mathrm{final}} $ will not be $ 1. $ $ ^{\text{∗}} $ A permutation of length $ n $ is an array consisting of $ n $ distinct integers from $ 1 $ to $ n $ in arbitrary order. For example, $ [2,3,1,5,4] $ is a permutation, but $ [1,2,2] $ is not a permutation ( $ 2 $ appears twice in the array), and $ [1,3,4] $ is also not a permutation ( $ n=3 $ but there is $ 4 $ in the array).

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 25 $ ). The description of the test cases follows. The first and only line of each test case contains a single even integer $ n \, (2 \le n \le 50) $ — the length of the desired permutation.

For each test case, output $ n $ integers $ p_1, \ldots, p_n \, (1 \le p_i \le n) $ — a permutation satisfying the conditions. If there are multiple solutions, print any of them.

In the first test case, the permutation given in the output is $ [2,1]. $ The counter may be incremented in the following $ 9 $ ways: 1. $ 0 \xrightarrow{+2 \cdot 1} 2 \xrightarrow{+0} 2 $ 2. $ 0 \xrightarrow{+0} 2 \xrightarrow{+1 \cdot 2} 2 $ 3. $ 0 \xrightarrow{-2 \cdot 1} -2 \xrightarrow{+0} -2 $ 4. $ 0 \xrightarrow{+0} 2 \xrightarrow{-1 \cdot 2} -2 $ 5. $ 0 \xrightarrow{-2 \cdot 1} -2 \xrightarrow{+1 \cdot 2} 0 $ 6. $ 0 \xrightarrow{+2 \cdot 1} 2 \xrightarrow{-1 \cdot 2} 0 $ 7. $ 0 \xrightarrow{-2 \cdot 1} -2 \xrightarrow{-1 \cdot 2} -4 $ 8. $ 0 \xrightarrow{+2 \cdot 1} 2 \xrightarrow{+1 \cdot 2} 4. $ 9. $ 0 \xrightarrow{+0} 0 \xrightarrow{+0} 0. $ None of these are $ 1, $ so the permutation satisfies the given condition.We can show that the permutation given in the second test case satisfies the condition. However, the permutation $ [1,2,3,4] $ would not satisfy the condition, since the sequence: $$$ 0 \xrightarrow{+1 \cdot 1} 1 \xrightarrow{+0} 1 \xrightarrow{+0} 1 \xrightarrow{+0} 1 $$$ Results in $ c = 1 $ at the end.