Equal-circle packing in a unit circle
2y0
123456
0x2
Place n non-overlapping circles of one common radius inside a circle of radius 1, making that radius as large as possible.
Formal definition
- ContainerA circle of radius 1 centred at (1, 1), so both coordinates run from 0 to 2
- 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
Proven for n = 2..14 and 19 (the bold marks in Specht's cci survey); 15..18 and everything from 20 up are open, including all n offered here.
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 RECORD0.500000000
Optimal
12
Record holderReference answer
Solution method—
n3
CURRENT RECORD0.464101615
Optimal
123
Record holderReference answer
Solution method—
n4
CURRENT RECORD0.414213562
Optimal
1234
Record holderReference answer
Solution method—
n5
CURRENT RECORD0.370191908
Optimal
12345
Record holderReference answer
Solution method—
n6
CURRENT RECORD0.333333333
Optimal
123456
Record holderReference answer
Solution method—
n7
CURRENT RECORD0.333333333
Optimal
1234567
Record holderReference answer
Solution method—
n8
CURRENT RECORD0.302593388
Optimal
12345678
Record holderReference answer
Solution method—
n9
CURRENT RECORD0.276768653
Optimal
123456789
Record holderReference answer
Solution method—
n10
CURRENT RECORD0.262258923
Optimal
12345678910
Record holderReference answer
Solution method—
n11
CURRENT RECORD0.254854701
Optimal
1234567891011
Record holderReference answer
Solution method—
n12
CURRENT RECORD0.248163470
Optimal
123456789101112
Record holderReference answer
Solution method—
n13
CURRENT RECORD0.236067977
Optimal
12345678910111213
Record holderReference answer
Solution method—
n14
CURRENT RECORD0.231030727
Optimal
1234567891011121314
Record holderReference answer
Solution method—
n15
CURRENT RECORD0.221172538best known 0.221172539
123456789101112131415
Record holderFounding benchmark
Solution methodHuman
n16
CURRENT RECORD0.216664742best known 0.216664742
Matches the best known
12345678910111213141516
Record holderFounding benchmark
Solution methodHuman
n17
CURRENT RECORD0.208679665best known 0.208679665
Matches the best known
1234567891011121314151617
Record holderFounding benchmark
Solution methodHuman
n18
CURRENT RECORD0.205604646best known 0.205604646
Matches the best known
123456789101112131415161718
Record holderFounding benchmark
Solution methodHuman
n19
CURRENT RECORD0.205604646
Optimal
12345678910111213141516171819
Record holderReference answer
Solution method—
n20
CURRENT RECORD0.195224010best known 0.195224011
1234567891011121314151617181920
Record holderFounding benchmark
Solution methodHuman
n21
CURRENT RECORD0.190392146best known 0.190392146
Matches the best known
123456789101112131415161718192021
Record holderFounding benchmark
Solution methodHuman
n22
CURRENT RECORD0.183833026best known 0.183833026
Matches the best known
12345678910111213141516171819202122
Record holderFounding benchmark
Solution methodHuman
n23
CURRENT RECORD0.180336008best known 0.180336009
1234567891011121314151617181920212223
Record holderFounding benchmark
Solution methodHuman
n24
CURRENT RECORD0.176939130best known 0.176939130
Matches the best known
123456789101112131415161718192021222324
Record holderFounding benchmark
Solution methodHuman
n25
CURRENT RECORD0.173827661best known 0.173827661
Matches the best known
12345678910111213141516171819202122232425
Record holderFounding benchmark
Solution methodHuman
n26
CURRENT RECORD0.171580251best known 0.171580252
1234567891011121314151617181920212223242526
Record holderFounding benchmark
Solution methodHuman
n27
CURRENT RECORD0.169307930best known 0.169307931
123456789101112131415161718192021222324252627
Record holderFounding benchmark
Solution methodHuman
n28
CURRENT RECORD0.166252749best known 0.166252750
12345678910111213141516171819202122232425262728
Record holderFounding benchmark
Solution methodHuman
n29
CURRENT RECORD0.162903648best known 0.162903649
1234567891011121314151617181920212223242526272829
Record holderFounding benchmark
Solution methodHuman
n30
CURRENT RECORD0.150000000
123456789101112131415161718192021222324252627282930
Record holderAnonymous
Solution methodHuman