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Claim. The second question of Problem 668 is answered no. Four points in the plane cannot have all six distances equal to $1$, so at most five unit distances occur among four points. Five unit distances force two unit equilateral triangles on a common edge, a rhombus with angles $60^\circ$ and $120^\circ$ that is unique up to congruence. So $n=4$ has exactly one incongruent maximizer.

Covers. The second question only; the first, whether the count tends to infinity, is untouched.

Depends on. No page of this wiki.

Claimant and date. The remark is the site's unsigned commentary, whose recommended citation names the curator, T. F. Bloom, as the page's author; the page is filed under that name and dated by the earliest dated revision in the site's history, of 20 October 2025, which already carries it. Anay Aggarwal first posted the observation on the discussion thread on 13 August 2025, and the site's page thanks Aggarwal.

Acceptance. None documented: the site labels the problem OPEN.