But it’s only possible if the 5 positions allow a 3-element non-adjacent subset — which requires that the positions span at least 5 positions (e.g., 1,3,5), and in any 5 consecutive positions, there is exactly **one** such set? Not quite — for positions {1,2,3,4,5}, non-adjacent triples: 1,3,5 only? But 1,4,5 has adjacent, 2,4,5 has adjacent. Yes, only {1,3,5} if starting at 1, but depending on layout.

Sun, 03 May 2026 02:29:42 +0000

Understanding the Unique Subset Condition in Position Arrangement When designing flexible systems based on sequential position sets—whether for scheduling, game mechanics, or data indexing—the concept of selecting non-adjacent subsets with strict placement rules plays a crucial role. A key condition often arises: a 3-element non-adjacent subset must exist only if the five positions permit at least one such valid group, and that group must span all five positions non-adjacently. But this leads to a nuanced paradox: only certain arrangements allow exactly one such triple under non-adjacency constraints. ### What Does It Mean for a 3-Element Subset to Be Non-Adjacent? A non-adjacent subset means no two selected elements are immediately next to each other in the sequence. For five positions—say labeled 1 through 5—any valid 3-element subset must avoid adjacent indices. Consider the example positions {1,2,3,4,5}. Let’s analyze all combinations of three positions: