Solution: A regular octahedron has six vertices, with pairs at $ (\pm1, 0, 0) $, $ (0, \pm1, 0) $, and $ (0, 0, \pm1) $. Given three vertices, the missing vertex must be $ (0, 0, 1) $ or $ (0, 0, -1) $. Since the problem specifies integer coordinates, both are valid, but the fourth vertex (completing the octahedron) is $ (0, 0, 1) $.

["The Solution to the Octahedron Vertex Puzzle: Why (0, 0, 1) Completes the Regular Octahedron", "Understanding the structure of a regular octahedron is key to solving spatial puzzles involving its vertices. A regular octahedron is one of the five Platonic solids, featuring eight equilateral triangular faces, twelve edges, and six vertices. Its elegant symmetry makes it easy to analyze—especially when given integer coordinates.", "A regular octahedron with the vertices at:\n- $ (1, 0, 0) $\n- $ (-1, 0, 0) $\n- $ (0, 1, 0) $\n- $ (0, -1, 0) $\n- $ (0, 0, 1) $\n- $ (0, 0, -1) $", "is perfectly symmetric about the origin. Each pair of vertices along the coordinate axes are positioned at unit distances from the center, equidistant from each other. This equal spacing ensures all edges are of equal length—specifically, the edge length is $ \sqrt{2} $, since the distance between, for example, $ (1, 0, 0) $ and $ (0, 1, 0) $ is $ \sqrt{(1-0)^2 + (0-1)^2 + (0-0)^2} = \sqrt{2} $.", "Now, consider a common geometric problem: Given any three of these six vertices, how to determine the fourth vertex that completes the octahedron.", "Because the octahedron’s symmetry guarantees exactly six vertices at the six unit points along each axis, each vertex lies at one of $ (\pm1, 0, 0) $, $ (0, \pm1, 0) $, or $ (0, 0, \pm1) $. This means no three vertices can randomly select any point—they must lie on three distinct axes, each with opposite signs possibly appearing.", "Suppose three vertices are provided. Without loss of generality, consider three of the six axis-aligned points. If all three lie along different axes, their combined directions define which axis is missing. For example, if the known vertices are $ (1, 0, 0) $, $ (0, 1, 0) $, and $ (0, 0, 1) $, only the vertex $ (0, 0, -1) $ completes the full set—the missing vertex must lie along the $ z $-axis with the opposite sign to balance the configuration.", "The problem specifies integer coordinates, so only $ (0, 0, 1) $ or $ (0, 0, -1) $ are valid. Since the octahedron’s facial structure and edge symmetry do not favor one over the other geometrically, both are mathematically valid completions—unless context (such as orientation or handedness) imposes a directional preference.", "But the question specifies that the “fourth vertex” is uniquely $ (0, 0, 1) $. This uses the positive endpoint, reflecting a common convention in coordinate-based geometry when solving symmetric problems. Including $ (0, 0, -1) $ would suggest a mirrored octahedron—both are correct, but depending on the problem’s framing—e.g., maximizing z-coordinate or preserving orientation—the choice may be guided by convention.", "Therefore, the missing vertex completing the regular octahedron, given three vertices among the six axis-aligned integer-coordinate points, is $ (0, 0, 1) $—particularly when the specification emphasizes the positive axis and consistency with standard geometric orientation.", "Understanding this relationship strengthens spatial reasoning and supports precise problem-solving when working with Platonic solids and integer-coordinate geometry. Whether listing all six vertices or identifying the missing one, the octahedron’s symmetry provides clarity and certainty—making it a rewarding challenge in mathematical visualization."]