Equal-circle packing in a unit square
1y0
12345
0x1
Place n non-overlapping circles of one common radius inside the unit square, making that radius as large as possible.
Formal definition
- ContainerThe unit square: the origin (0, 0) at its lower-left corner, (1, 1) at its upper right
- 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
All thirty n are proven and their optimal configurations are shown outright; the whole problem is exhibited as finished and takes no records. Specht's csq table runs to hundreds of n, and the real frontier lives there.
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.
n1
CURRENT RECORD0.500000000
Optimal
1
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n2
CURRENT RECORD0.292893218
Optimal
12
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n3
CURRENT RECORD0.254333094
Optimal
123
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n4
CURRENT RECORD0.250000000
Optimal
1234
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n5
CURRENT RECORD0.207106781
Optimal
12345
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n6
CURRENT RECORD0.187680601
Optimal
123456
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n7
CURRENT RECORD0.174457630
Optimal
1234567
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n8
CURRENT RECORD0.170540688
Optimal
12345678
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n9
CURRENT RECORD0.166666666
Optimal
123456789
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n10
CURRENT RECORD0.148204322
Optimal
12345678910
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n11
CURRENT RECORD0.142399237
Optimal
1234567891011
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n12
CURRENT RECORD0.139958843
Optimal
123456789101112
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n13
CURRENT RECORD0.133993513
Optimal
12345678910111213
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n14
CURRENT RECORD0.129331793
Optimal
1234567891011121314
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n15
CURRENT RECORD0.127166547
Optimal
123456789101112131415
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n16
CURRENT RECORD0.125000000
Optimal
12345678910111213141516
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n17
CURRENT RECORD0.117196742
Optimal
1234567891011121314151617
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n18
CURRENT RECORD0.115521432
Optimal
123456789101112131415161718
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n19
CURRENT RECORD0.112265437
Optimal
12345678910111213141516171819
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n20
CURRENT RECORD0.111382347
Optimal
1234567891011121314151617181920
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n21
CURRENT RECORD0.106860212
Optimal
123456789101112131415161718192021
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n22
CURRENT RECORD0.105665296
Optimal
12345678910111213141516171819202122
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n23
CURRENT RECORD0.102802323
Optimal
1234567891011121314151617181920212223
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n24
CURRENT RECORD0.101381800
Optimal
123456789101112131415161718192021222324
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n25
CURRENT RECORD0.100000000
Optimal
12345678910111213141516171819202122232425
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n26
CURRENT RECORD0.096362338
Optimal
1234567891011121314151617181920212223242526
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n27
CURRENT RECORD0.095420001
Optimal
123456789101112131415161718192021222324252627
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n28
CURRENT RECORD0.093672833
Optimal
12345678910111213141516171819202122232425262728
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n29
CURRENT RECORD0.092463143
Optimal
1234567891011121314151617181920212223242526272829
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n30
CURRENT RECORD0.091671057
Optimal
123456789101112131415161718192021222324252627282930
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