The smallest triangle in a disc
2y0
1234567
0x2
Place n points inside a circle 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 circle of radius 1 centred at (1, 1), so coordinates run from 0 to 2.
- 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
Heilbronn in a disc: MathWorld collects the known values, some in closed form, none proven; the proof techniques from the square version have not been carried over.
SourceONE 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 RECORD0.657163889381144976best known 1.314327780297834015
12345
Record holderFounding benchmark
Solution methodHuman
n6
CURRENT RECORD0.433012701639535125best known 0.866025403784438646
123456
Record holderFounding benchmark
Solution methodHuman
n7
CURRENT RECORD0.294367525908179724best known 0.58873505275423601
1234567
Record holderFounding benchmark
Solution methodHuman
n8
CURRENT RECORD0.216941869025609061best known 0.43388373911755812
12345678
Record holderFounding benchmark
Solution methodHuman
n9
CURRENT RECORD0.096245588931469668best known 0.347527496674425
123456789
Record holderFounding benchmark
Solution methodHuman
n10
CURRENT RECORD0.071844476057464545best known 0.300767466360870593
12345678910
Record holderFounding benchmark
Solution methodHuman
n11
CURRENT RECORD0.054927884410070292best known 0.219546642560923
1234567891011
Record holderFounding benchmark
Solution methodHuman
n12
CURRENT RECORD0.042871870780912164best known 0.209830765392198
123456789101112
Record holderFounding benchmark
Solution methodHuman
n13
CURRENT RECORD0.034067992868178005best known 0.171316471464200
12345678910111213
Record holderFounding benchmark
Solution methodHuman
n14
CURRENT RECORD0.027499518584008526best known 0.151714502121802
1234567891011121314
Record holderFounding benchmark
Solution methodHuman