Equal-circle packing in a right triangle
0.75y0
12345
0x1
Place n non-overlapping equal circles inside the right triangle with legs 1 and 0.75, making the common radius as large as possible.
Formal definition
- ContainerA right triangle: the right angle at the origin (0, 0), one leg along the x-axis to (1, 0), the other along the y-axis to (0, 0.75)
- SubmissionExactly n circles: one shared radius and n centres
- ConstraintsEvery circle lies wholly inside the container; no two overlap in their interiors, tangency allowed
- ObjectiveMake the common radius as large as possible
Getting a feel for it
Where the room for improvement is
Optimal packings are jammed contact structures: circles brace against each other and the boundary, with tilted rows, offsets, and the odd rattler touching nothing. Neat grids are almost never optimal.
Where the frontier is
This particular triangle, legs 1 and 0.75, has no literature table; close relatives such as the isosceles right triangle are collected on Friedman's Packing Center. No values are recorded here, and every n is open.
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.
n2
CURRENT RECORD0.177597036
12
Record holderNUE_13
Solution methodHuman
n3
CURRENT RECORD0.150666685
123
Record holderNUE_13
Solution methodHuman
n4
CURRENT RECORD0.137055354
1234
Record holderNUE_13
Solution methodHuman
n5
CURRENT RECORD0.119982102
12345
Record holderNUE_13
Solution methodHuman
n6
CURRENT RECORD0.071428571
123456
Record holderFounding benchmark
Solution methodHuman
n7
CURRENT RECORD0.0625
1234567
Record holderFounding benchmark
Solution methodHuman
n8
CURRENT RECORD0.055555555
12345678
Record holderFounding benchmark
Solution methodHuman
n9
CURRENT RECORD0.05
123456789
Record holderFounding benchmark
Solution methodHuman
n10
CURRENT RECORD0.045454545
12345678910
Record holderFounding benchmark
Solution methodHuman
n11
CURRENT RECORD0.041666666
1234567891011
Record holderFounding benchmark
Solution methodHuman
n12
CURRENT RECORD0.038461538
123456789101112
Record holderFounding benchmark
Solution methodHuman