Equal circles in an L
2y0
1234567
0x2
Place n non-overlapping circles of one common radius inside the L-shaped region, making that radius as large as possible.
Formal definition
- ContainerA 2 × 2 square with its top-right 1 × 1 removed: the origin (0, 0) at the lower-left; the notch is where x and y both exceed 1
- SubmissionExactly n circles: one shared radius and n centres
- ConstraintsEvery circle lies wholly inside the L and clear of the notch; 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
Our own variant: equal-circle packing in an L 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.
n4
CURRENT RECORD0.365737043
1234
Record holderEraser
Solution methodHuman
n5
CURRENT RECORD0.305592154
12345
Record holderAnonymous
Solution methodHuman
n6
CURRENT RECORD0.2949
123456
Record holderAnonymous
Solution methodHuman
n7
CURRENT RECORD0.291013627
1234567
Record holderNUE_13
Solution methodHuman
n8
CURRENT RECORD0.272869229
12345678
Record holderNUE_13
Solution methodHuman
n9
CURRENT RECORD0.258792284
123456789
Record holderNUE_13
Solution methodHuman
n10
CURRENT RECORD0.1
12345678910
Record holderFounding benchmark
Solution methodHuman
n11
CURRENT RECORD0.09090909
1234567891011
Record holderFounding benchmark
Solution methodHuman
n12
CURRENT RECORD0.25
123456789101112
Record holderNUE_13
Solution methodHuman
n13
CURRENT RECORD0.076923076
12345678910111213
Record holderFounding benchmark
Solution methodHuman
n14
CURRENT RECORD0.071428571
1234567891011121314
Record holderFounding benchmark
Solution methodHuman
n15
CURRENT RECORD0.066666666
123456789101112131415
Record holderFounding benchmark
Solution methodHuman
n16
CURRENT RECORD0.0625
12345678910111213141516
Record holderFounding benchmark
Solution methodHuman
n17
CURRENT RECORD0.058823529
1234567891011121314151617
Record holderFounding benchmark
Solution methodHuman
n18
CURRENT RECORD0.055555555
123456789101112131415161718
Record holderFounding benchmark
Solution methodHuman