Cyclic antimonotonic permutations

A permutation is a sequence of integers which contains each integer from 1 to n exactly once. In this problem we are looking for permutations with special properties:

1. Antimonotonic: for each consecutive 3 values p_i-1, p_i, p_i+1 (1 < i < n), p_i should either be the smallest or the biggest of the three values.
2. Cyclic: The permutation should consist of only one cycle, that is, when we use p_i as a pointer from i to p_i, it should be possible to start at position 1 and follow the pointers and reach all n positions before returning to position 1.

The input file contains several test cases. Each test case consists of a line containing an integer n, (3 ≤ n ≤ 10⁶), the number of integers in the permutation. Input is terminated by n=0.

For each test case print a permutation of the integers 1 to n which is both antimonotonic and cyclic. In case there are multiple solutions, you may print any one. Separate all integers by whitespace characters.

3
5
10
0

3 1 2
4 5 2 3 1
6 10 2 9 3 5 4 7 1 8

2009/2010 Ulm Local Contest