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Claim. The second question of Problem 1082 has a negative answer: there is a set of eight points in the plane, no three on a line, in which no point sees $\lfloor 8/2\rfloor=4$ distinct distances. The set, Harborth's configuration $H_8$, consists of the four vertices of a square and the four apexes of the equilateral triangles erected on its sides, all facing outward (or all inward, which on the same square gives a similar set, smaller by the factor $(\sqrt3-1)/\sqrt2$). With the square's vertices at $(\pm2,0)$ and $(0,\pm2)$ the outward apexes are $(\pm(1+\sqrt3),\pm(1+\sqrt3))$; each vertex of the square is at distances $2\sqrt2$, $4$ and $2+2\sqrt3$ from the other seven points, each apex at distances $2\sqrt2$, $2+2\sqrt3$ and $2\sqrt2(1+\sqrt3)$, so every point sees exactly three distinct distances, and no three of the eight points are collinear. The whole set determines four distinct distances, so it is not a counterexample to the first question.

Covers. The second question, for the single value $n=8$: a set with no three points on a line need not contain a point seeing $\lfloor n/2\rfloor$ distinct distances. Nothing is claimed about the first question, which asks for the number of distinct distances the whole set determines.

Depends on. No page of this wiki.

Source and credit. The configuration first appeared in the literature in P. Erdős and P. Fishburn, Distinct distances in finite planar sets, Discrete Mathematics 175 (1997), 97–132, who credit it to Heiko Harborth; it is the subject of P. C. Fishburn, A remarkable eight-point planar configuration, Discrete Mathematics 252 (2002), 103–122, which studies its distance structure in detail. The site's remarks give the same account and credit Harborth. The claimant slug names the paper that published the construction; the construction itself is Harborth's.

Acceptance. Refereed: both papers are journal publications in Discrete Mathematics, cited above with their volumes and pages. Not reviewed: the site's label for the problem is FALSIFIABLE, which settles neither question, so the curator's remark crediting the configuration is not an acceptance of the problem or of its part. Later independent findings of the same answer are disclosed here: Xichuan's forty-two-point example on the site's thread (19 December 2025), which gets no page because it is a forum post, and the eight-point Lean proof found by a DeepMind prover agent (25 February 2026), which constructs this same configuration and has [[problems/distance_problems/E1082/claims/2026_02_25_deepmind|its own claim page]].