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Eyal Ackerman, Kevin Buchin, Christian Knauer, Rom Pinchasi and Günter Rote, There are not too many magic configurations, Discrete Comput. Geom. 39 (2008), no. 1–3, 3–16, DOI 10.1007/s00454-007-9023-0, published online 2007-09-19; an earlier version appeared in the Proceedings of the 23rd Annual Symposium on Computational Geometry (SCG '07), 142–149, DOI 10.1145/1247069.1247098, the result's first publication. The symposium opened on 2007-06-06, the date the page name carries. The source card, ackerman_2008_there_are_not_too_many_magic_configurations, cites the authors' manuscript of 2007-02-27.

The result. Call a finite set $P$ of $n$ points in the plane a magic configuration if positive weights can be given to its points so that the weights on every line through at least two points of $P$ have the same sum (the paper normalizes the sum to $1$). Theorem 1 of the paper states that a magic configuration is one of three kinds: at least $n-1$ of its points are collinear; no three of its points are collinear; or $n=7$ and $P$ is, up to a projective transformation, the failed Fano configuration, the seven points formed by the vertices of a triangle, the feet of its three angle bisectors and its incenter. Theorem 1 gives one direction, that a magic configuration is of one of these kinds; the converse, that each kind is magic, is elementary; the paper states it for the failed Fano configuration, whose weights its Figure 1 shows, and not for the other kinds: weight $1/2$ at every point when no three are collinear; weights $1/(n-1)$ on the line and $(n-2)/(n-1)$ at the point off it when $n-1$ points are collinear; any positive weights with sum $1$ when all points are collinear; and weight $1/2$ at the three feet and $1/4$ at the other four points of the failed Fano configuration (each side and each bisector then carries one foot and two quarter-weight points, and each line through two feet is an ordinary line). Together these answer the question of when such weights exist: for exactly these configurations, and for no others. Theorem 1 proves the 1971 conjecture of Murty that the site records, whose four cases (all points on a line; $n-1$ points on a line; no three on a line; the triangle with its angle bisectors and incenter) are the theorem's three kinds with the collinear family split in two. The proof reduces Theorem 1 to Theorem 2, a statement about two point sets whose union's ordinary lines are exactly the lines through two points of the first set, and uses the Sylvester–Gallai theorem, the Kelly–Moser bound on ordinary lines, and a dual argument on great circles of a sphere.

Acceptance. The paper is refereed: it appeared in Discrete and Computational Geometry, volume 39 (2008), after the SCG '07 proceedings version. The site's curator, Thomas Bloom, labels the problem solved and credits Ackerman, Buchin, Knauer, Pinchasi and Rote.

Formalization. Boris Alexeev's repository of Lean proofs of Erdős problems holds a file, linked above at a pinned commit, whose header declares it a Lean formalization of a solution to Problem 735 with Ackerman, Buchin, Knauer, Pinchasi and Rote as the informal authors and the systems Codex and GPT-5.6 Sol as the formal authors; the file was added on 2026-08-20. Its theorem erdos_735 states both directions: a finite set of points of the plane is magic exactly when it is collinear, in general position, a near-pencil (all but one point collinear) or the failed Fano configuration. This corpus has not built the file or audited its statement, so it is a link here and not formalized evidence.