The most uniform sampling mesh in the unit square
Place n points in the unit square. h is the greatest distance any location has to its nearest point, δ the distance of the closest pair; the score is M = 2h/δ. Make it as small as you can.
Formal definition
- ContainerThe unit square: the origin (0, 0) at its lower-left corner, (1, 1) at its upper right
- SubmissionExactly n points, each coordinate a decimal with at most nine places; no two points may coincide
- Measuresh is the maximum over all locations of the distance to the nearest submitted point; δ is the minimum over all pairs
- ObjectiveMake M = 2h/δ as small as possible. Scored internally as M² = 4h²/δ², exact, rounded up at 10⁻¹⁵
Getting a feel for it
An analogy: siting base stations
Read the n points as n base stations. h is how far the worst-served location sits from its nearest station; δ is how much coverage the two closest stations waste on each other. M charges for both: no big holes, no huddles.
Where the room for improvement is
A square grid sits at M = √2 ≈ 1.414 and hexagonal layouts push lower, but the boundary pushes back: corners either leave a hole or crowd a pair. Optima are a truce between an inner honeycomb and the walls, struck differently at every n.
Where the frontier is
The mesh ratio is a standard uniformity measure in meshless methods, but a per-n table of the smallest M achievable in a square seems absent from the literature; every n here is treated as open. Pointers to sources are welcome.
Current best solutions by n
Each n is an independent record with a page of its own. Open any of them to inspect the current construction, then challenge it.