| Problem: Distinct Pairwise XOR Set | |
| Time Limit: 1 second | |
| Memory Limit: 512 MB | |
| Description | |
| Given an integer n, find a subset S ⊆ {1, 2, ..., n} such that: | |
| 1) For all pairs (a, b) with a, b ∈ S and a < b, the values (a XOR b) are all distinct (i.e., no two different unordered pairs produce the same XOR). | |
| 2) |S| ≥ floor(sqrt(n / 2)). | |
| Input | |
| A single integer n (1 ≤ n ≤ 10^7). | |
| Output | |
| - First line: an integer m — the size of the set S. | |
| - Second line: m distinct integers in the range [1, n] — the elements of S, in any order. | |
| Notes | |
| - Any valid S is accepted. You do NOT need to maximize m; you only need m ≥ floor(sqrt(n/2)). | |
| - The pairwise XOR distinctness means the set {a_i XOR a_j | 1 ≤ i < j ≤ m} has size m*(m-1)/2. | |
| - Multiple correct outputs may exist for the same n. | |
| - Print out the sequence with the longest length. | |
| Sample | |
| Input | |
| 49 | |
| Output | |
| 4 | |
| 1 2 3 4 | |