The smallest triangle in a half-disc
1y0
1234567
0x2
Place n points inside a half-disc of radius 1 so that the smallest triangle formed by any three of them is as large as possible.
Formal definition
- ContainerThe container is a half-disc of radius 1: its diameter lies along y = 0 from (0, 0) to (2, 0), centred at (1, 0), with the arc above.
- 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 half-disc of radius 1 was posed here, and there is no literature for it. Every n is unstudied; the standing record is all anybody knows.
ONE LEADERBOARD PER n
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.
n5
CURRENT RECORD0.105146222461917614
12345
Record holderFounding benchmark
Solution methodHuman
n6
CURRENT RECORD0.0692820324
123456
Record holderFounding benchmark
Solution methodHuman
n7
CURRENT RECORD0.047098804142046847
1234567
Record holderFounding benchmark
Solution methodHuman
n8
CURRENT RECORD0.0331370848
12345678
Record holderFounding benchmark
Solution methodHuman
n9
CURRENT RECORD0.02406139717362779
123456789
Record holderFounding benchmark
Solution methodHuman
n10
CURRENT RECORD0.017961119010741402
12345678910
Record holderFounding benchmark
Solution methodHuman
n11
CURRENT RECORD0.013731971074830454
1234567891011
Record holderFounding benchmark
Solution methodHuman
n12
CURRENT RECORD0.0107179676
123456789101112
Record holderFounding benchmark
Solution methodHuman
n13
CURRENT RECORD0.008516998145363208
12345678910111213
Record holderFounding benchmark
Solution methodHuman
n14
CURRENT RECORD0.006874879632653493
1234567891011121314
Record holderFounding benchmark
Solution methodHuman