P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 6 · n = 53

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 = 53

ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[0.11225352737, 0.112253527372]	Eckard Specht	Packomania ↗
Site record	0.112253526	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=53. Refine the public packing below on the Q=10^9 grid. Target radius r+1 per step and integer centre shifts in [-L,L]. Include all six hexagon walls. For each pair with centre difference d, impose 2*d dot (u_i-u_j) >= 4*(r+1)^2-|d|^2, divided by 4*(r+1), plus a 0.0001 grid-unit margin. Pairs with normalised slack above 8L may be omitted from search, never from final verification. Use SciPy/HiGHS MILP with zero relative gap and a five-second limit. Run L=2,10,100 sequentially; then test L=2,10,100,1000,10000,100000, repeating at each L while it improves (up to five attempts). Accepted steps: [{"limit":2,"radius":"0.112253525"},{"limit":10,"radius":"0.112253526"}]. The original sequence replayed exactly; larger neighbourhoods gave no further gain. This is not an optimality proof. All 318 wall constraints and 1378 circle pairs pass two independent checks: exact signs in Q(sqrt(3)) and integer squared distances; conservative 90-decimal rational intervals enclosing the exact hexagon vertices and exact rational pair distances. No feasibility tolerance. This 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 the given order. Container centre (1,1), circumradius 1, first vertex (2,1). {"radius":"0.112253524","centers":[["0.652687175","0.246228124"],["0.877194230","0.246228124"],["1.101701284","0.246228124"],["1.326208339","0.246228124"],["1.509797227","0.375450946"],["0.481171744","0.391093146"],["0.764940702","0.440656936"],["0.989447757","0.440656936"],["1.213954812","0.440656936"],["1.397543700","0.569879759"],["1.622050754","0.569879759"],["0.368918217","0.585521959"],["0.593425272","0.585521959"],["0.877194230","0.635085749"],["1.101701284","0.635085749"],["1.285290172","0.764308572"],["1.509797227","0.764308572"],["1.734304282","0.764308572"],["0.256664690","0.779950772"],["0.481171744","0.779950772"],["0.705678799","0.779950772"],["0.986759785","0.831119483"],["1.173036645","0.958737384"],["1.397543700","0.958737384"],["1.622050754","0.958737384"],["1.846557809","0.958737384"],["0.144411162","0.974379584"],["0.368918217","0.974379584"],["0.815173405","0.975946560"],["0.591958599","1.000000000"],["1.000605116","1.102510766"],["1.285290172","1.153166197"],["1.509797227","1.153166197"],["1.734304282","1.153166197"],["0.233743422","1.180348429"],["0.458250477","1.180348429"],["0.733629862","1.208227043"],["0.989445535","1.326740294"],["1.210683341","1.364914251"],["1.435190396","1.364914251"],["1.659697450","1.364914251"],["0.345996950","1.374777242"],["0.570504005","1.374777242"],["0.788760754","1.427383408"],["1.083029897","1.550538448"],["1.322936868","1.559343064"],["1.547443923","1.559343064"],["0.458250477","1.569206054"],["0.786125186","1.651874992"],["0.586074086","1.753771876"],["0.986176286","1.753771876"],["1.210683341","1.753771876"],["1.435190396","1.753771876"]]}

Read and reply in the discussion →

Record history (2 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
0.112253524→0.112253526
2026-10-30
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.051136364→0.112253524
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": 53,
  "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