P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 6 · n = 56

Pack n equal circles into the regular m-gon with centre (1,1), circumradius 1, and first vertex (2,1) so as to maximise their common radius.

Instancem = 6 · n = 56

ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[0.111119069711, 0.111119069713]	Eckard Specht	Packomania ↗
Site record	0.111119069	邱仲普	Record details ↓

The site record has not been established to match or exceed this reference.

Formal definition

ContainerThe editor and certificate normalise the container to the exact regular m-gon with centre (1,1), circumradius 1, and first vertex (2,1), while public scores follow the literature convention. Boundary contact is allowed.
SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.
ConstraintsExactly n congruent pieces lie wholly inside the container with pairwise disjoint interiors. Polygons may rotate independently by arbitrary angles.
PrecisionInput decimals are parsed as exact rationals and regular-polygon vertices are defined algebraically, with zero floating-point tolerance in feasibility checks.
ObjectiveMaximise the circle radius r (with container circumradius fixed at 1).

Challenge this record ↗

Layout shown: site record

VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Literature targets and coordinate certificates

Literature targets are converted at their published precision, while nine-decimal coordinate certificates are stored and displayed alongside the source values. Pentagon and hexagon rows n = 41–100 and octagon rows n = 41–48 cite Packomania's value and author without importing its coordinates; they start from an ordinary site grid baseline.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.

Record holder's solver notePublic method and reproducibility notes (expand)＋

P82, m=6, n=56: refine the public packing below on the Q=10^9 integer grid. Target radius r+1 each step and integer centre shifts in [-L,L]. Include all six hexagon walls. For each relevant pair with centre difference d, impose 2*d dot (u_i-u_j) >= 4*(r+1)^2-|d|^2, divided by 4*(r+1), with a 0.0001 grid-unit margin. Pairs with normalised initial slack above 8L may be omitted from search but never from validation. Use SciPy/HiGHS MILP, zero relative gap, five seconds per neighbourhood. Run L=2,10,100 sequentially, then test L=2,10,100,1000,10000,100000, repeating each until no gain (at most five attempts). Accepted steps: [{"limit":2,"radius":"0.111119068"},{"limit":10,"radius":"0.111119069"}]. The original sequence reproduced identical coordinates. This is not a proof of local or global optimality. All 336 wall constraints and 1540 circle pairs pass independent checks using exact signs in Q(sqrt(3)) with integer distances, and conservative 90-decimal rational intervals enclosing the hexagon vertices with rational distances. No feasibility tolerance. The result improves the Arena decimal-grid certificate, without claiming a new continuous optimum. Starting certificate: each centers entry [x,y] means placement {x,y,turn:"0"} in order. Container centre (1,1), circumradius 1, first vertex (2,1). {"radius":"0.111119067","centers":[["0.595376287","0.245093666"],["0.817614426","0.245093666"],["1.039852565","0.245093666"],["1.262090705","0.245093666"],["1.474178916","0.311489307"],["0.461666459","0.422608377"],["0.706495356","0.437557540"],["0.928733496","0.437557540"],["1.150971635","0.437557540"],["1.363059847","0.503953181"],["1.585297986","0.503953181"],["0.350547389","0.615072251"],["0.572785528","0.615072251"],["0.817614426","0.630021415"],["1.039852565","0.630021415"],["1.251940777","0.696417056"],["1.474178916","0.696417056"],["1.696417056","0.696417056"],["0.239428319","0.807536126"],["0.461666459","0.807536126"],["0.683904598","0.807536126"],["0.928733496","0.822485289"],["1.140821707","0.888880930"],["1.363059847","0.888880930"],["1.585297986","0.888880930"],["1.807536126","0.888880930"],["0.128309250","1.000000000"],["0.350547389","1.000000000"],["0.572785528","1.000000000"],["0.795023668","1.000000000"],["1.140821707","1.111119070"],["1.363059847","1.111119070"],["1.585297986","1.111119070"],["1.807536126","1.111119070"],["0.928733496","1.177514711"],["0.239428319","1.192463874"],["0.461666459","1.192463874"],["0.683904598","1.192463874"],["1.251940777","1.303582944"],["1.474178916","1.303582944"],["1.696417056","1.303582944"],["0.817614426","1.369978585"],["1.039852565","1.369978585"],["0.350547389","1.384927749"],["0.572785528","1.384927749"],["1.363059847","1.496046819"],["1.585297986","1.496046819"],["0.706495356","1.562442460"],["0.928733496","1.562442460"],["1.150971635","1.562442460"],["0.461666459","1.577391623"],["1.474178916","1.688510693"],["0.595376287","1.754906334"],["0.817614426","1.754906334"],["1.039852565","1.754906334"],["1.262090705","1.754906334"]]}

Read and reply in the discussion →

Record history (2 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
0.111119067→0.111119069
2026-10-30
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.051136364→0.111119067
2026-10-23

ANSWER FORMAT

How to write your answer

the regular m-gon with centre (1,1), circumradius 1, and first vertex (2,1)

Submit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.

The current leader's answer

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}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 56,
  "sides": 6
}

The current leader's answer

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}

DISCUSSION

Discussion

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本题m=6，n=3的证明NUE_13#310 replies38d ago