My first instinct was to sort the points and check side lengths, which works for axis-aligned cases but falls apart once the square is rotated.
First clarify the definition of a square (four equal sides, four right angles, non-degenerate). Then propose an efficient algorithm: compute all six pairwise squared distances, sort them, and check that the four smallest are equal and positive, and the two largest are equal and exactly twice the smallest. This avoids floating-point issues and handles any point order.
Pro tip: Mention that using squared distances avoids floating-point precision problems, and explicitly handle duplicate points (zero distances) to reject degenerate cases. Also, note that this method works for any input order and runs in O(1) time.
Confirm that the four points must be distinct and form a non-degenerate square. Ask whether the points are given as integers or floats, and whether the square can be rotated.
Decide to compute all pairwise squared distances (6 total) and use the property that a square has four equal sides and two equal diagonals that are twice the side length squared.
Compute the six squared distances, sort them, and verify that the first four are equal and positive, and the last two are equal and exactly twice the first value.
State that the algorithm runs in O(1) time and space. Walk through test cases: a valid square, a rectangle, a rhombus, and points with duplicates.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.