P25 · Extremal configurations · Formed here

The smallest triangle in a plus sign

Place n points inside a plus sign (a 3 × 3 square with all four corner squares removed) so that the smallest triangle formed by any three of them is as large as possible.

Formal definition

ContainerThe container is a square of side 3 with all four of its 1 × 1 corners removed, leaving a plus sign: the origin (0, 0) is its lower-left corner and (3, 3) its upper-right.
SubmissionExactly n points, no three collinear
ConstraintsEvery point lies inside the container or on its boundary
ObjectiveMake the smallest triangle over all triples as large as possible; compared internally by twice the area, exactly

Getting a feel for it

Where the room for improvement is

Even spreading is not enough: no three points may come close to collinear, and near-collinearity is exactly what tidy arrangements love to do. Optima are often asymmetric and hard even to guess.

Where the frontier is

Our own variant: Heilbronn's problem in a plus sign (a 3 × 3 square with all four corner squares removed) was posed here, and there is no literature for it. Every n is unstudied; the standing record is all anybody knows.