P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 48

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 = 8 · n = 48

ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[0.1220283207, 0.122028320702]	Paolo Amore	Packomania ↗
Site record	0.12202832	NUE_13	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 holderNUE_13

Solution methodAI · DeepSeek: DeepSeek V4.1 Flash

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 noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (2 changes)

NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.122028318→0.12202832
2026-11-01
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.058441558→0.122028318
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

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.095291402",
      "y": "0.171553494"
    },
    {
      "turn": "0.000000000",
      "x": "1.529719085",
      "y": "0.351499332"
    },
    {
      "turn": "0.000000000",
      "x": "1.287449953",
      "y": "0.322015235"
    },
    {
      "turn": "0.000000000",
      "x": "0.842162845",
      "y": "0.197460793"
    },
    {
      "turn": "0.000000000",
      "x": "1.679208215",
      "y": "0.544415491"
    },
    {
      "turn": "0.000000000",
      "x": "1.383050542",
      "y": "0.546568506"
    },
    {
      "turn": "0.000000000",
      "x": "0.989932675",
      "y": "0.391697044"
    },
    {
      "turn": "0.000000000",
      "x": "1.774184729",
      "y": "0.773709079"
    },
    {
      "turn": "0.000000000",
      "x": "1.532539672",
      "y": "0.739484665"
    },
    {
      "turn": "0.000000000",
      "x": "1.141701952",
      "y": "0.582824570"
    },
    {
      "turn": "0.000000000",
      "x": "0.747834232",
      "y": "0.422551346"
    },
    {
      "turn": "0.000000000",
      "x": "0.516966327",
      "y": "0.333189277"
    },
    {
      "turn": "0.000000000",
      "x": "1.867581162",
      "y": "0.999188015"
    },
    {
      "turn": "0.000000000",
      "x": "1.291191082",
      "y": "0.775740728"
    },
    {
      "turn": "0.000000000",
      "x": "0.897606946",
      "y": "0.617616508"
    },
    {
      "turn": "0.000000000",
      "x": "1.625612459",
      "y": "0.967332231"
    },
    {
      "turn": "0.000000000",
      "x": "0.560867607",
      "y": "0.579417930"
    },
    {
      "turn": "0.000000000",
      "x": "1.049376222",
      "y": "0.808744033"
    },
    {
      "turn": "0.000000000",
      "x": "0.318637241",
      "y": "0.549617019"
    },
    {
      "turn": "0.000000000",
      "x": "1.383956557",
      "y": "1.001479983"
    },
    {
      "turn": "0.000000000",
      "x": "0.709439877",
      "y": "0.773041082"
    },
    {
      "turn": "0.000000000",
      "x": "1.774759192",
      "y": "1.224904045"
    },
    {
      "turn": "0.000000000",
      "x": "1.532528826",
      "y": "1.195103134"
    },
    {
      "turn": "0.000000000",
      "x": "1.138790847",
      "y": "1.035831393"
    },
    {
      "turn": "0.000000000",
      "x": "0.861209153",
      "y": "0.964168607"
    },
    {
      "turn": "0.000000000",
      "x": "0.467471174",
      "y": "0.804896866"
    },
    {
      "turn": "0.000000000",
      "x": "0.225240808",
      "y": "0.775095955"
    },
    {
      "turn": "0.000000000",
      "x": "1.290560123",
      "y": "1.226958918"
    },
    {
      "turn": "0.000000000",
      "x": "0.616043443",
      "y": "0.998520017"
    },
    {
      "turn": "0.000000000",
      "x": "1.681362759",
      "y": "1.450382981"
    },
    {
      "turn": "0.000000000",
      "x": "0.950623778",
      "y": "1.191255967"
    },
    {
      "turn": "0.000000000",
      "x": "1.439132393",
      "y": "1.420582070"
    },
    {
      "turn": "0.000000000",
      "x": "0.374387541",
      "y": "1.032667769"
    },
    {
      "turn": "0.000000000",
      "x": "1.102393054",
      "y": "1.382383492"
    },
    {
      "turn": "0.000000000",
      "x": "0.708808918",
      "y": "1.224259272"
    },
    {
      "turn": "0.000000000",
      "x": "0.132418838",
      "y": "1.000811985"
    },
    {
      "turn": "0.000000000",
      "x": "1.483033673",
      "y": "1.666810723"
    },
    {
      "turn": "0.000000000",
      "x": "1.252165768",
      "y": "1.577448654"
    },
    {
      "turn": "0.000000000",
      "x": "0.858298048",
      "y": "1.417175430"
    },
    {
      "turn": "0.000000000",
      "x": "0.467460328",
      "y": "1.260515335"
    },
    {
      "turn": "0.000000000",
      "x": "0.225815271",
      "y": "1.226290921"
    },
    {
      "turn": "0.000000000",
      "x": "1.010067325",
      "y": "1.608302956"
    },
    {
      "turn": "0.000000000",
      "x": "0.616949458",
      "y": "1.453431494"
    },
    {
      "turn": "0.000000000",
      "x": "0.320791785",
      "y": "1.455584509"
    },
    {
      "turn": "0.000000000",
      "x": "1.157837155",
      "y": "1.802539207"
    },
    {
      "turn": "0.000000000",
      "x": "0.712550047",
      "y": "1.677984765"
    },
    {
      "turn": "0.000000000",
      "x": "0.470280915",
      "y": "1.648500668"
    },
    {
      "turn": "0.000000000",
      "x": "0.904708598",
      "y": "1.828446506"
    }
  ],
  "radius": "0.122028320"
}

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

Instance parameters

{
  "n": 48,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.095291402",
      "y": "0.171553494"
    },
    {
      "turn": "0.000000000",
      "x": "1.529719085",
      "y": "0.351499332"
    },
    {
      "turn": "0.000000000",
      "x": "1.287449953",
      "y": "0.322015235"
    },
    {
      "turn": "0.000000000",
      "x": "0.842162845",
      "y": "0.197460793"
    },
    {
      "turn": "0.000000000",
      "x": "1.679208215",
      "y": "0.544415491"
    },
    {
      "turn": "0.000000000",
      "x": "1.383050542",
      "y": "0.546568506"
    },
    {
      "turn": "0.000000000",
      "x": "0.989932675",
      "y": "0.391697044"
    },
    {
      "turn": "0.000000000",
      "x": "1.774184729",
      "y": "0.773709079"
    },
    {
      "turn": "0.000000000",
      "x": "1.532539672",
      "y": "0.739484665"
    },
    {
      "turn": "0.000000000",
      "x": "1.141701952",
      "y": "0.582824570"
    },
    {
      "turn": "0.000000000",
      "x": "0.747834232",
      "y": "0.422551346"
    },
    {
      "turn": "0.000000000",
      "x": "0.516966327",
      "y": "0.333189277"
    },
    {
      "turn": "0.000000000",
      "x": "1.867581162",
      "y": "0.999188015"
    },
    {
      "turn": "0.000000000",
      "x": "1.291191082",
      "y": "0.775740728"
    },
    {
      "turn": "0.000000000",
      "x": "0.897606946",
      "y": "0.617616508"
    },
    {
      "turn": "0.000000000",
      "x": "1.625612459",
      "y": "0.967332231"
    },
    {
      "turn": "0.000000000",
      "x": "0.560867607",
      "y": "0.579417930"
    },
    {
      "turn": "0.000000000",
      "x": "1.049376222",
      "y": "0.808744033"
    },
    {
      "turn": "0.000000000",
      "x": "0.318637241",
      "y": "0.549617019"
    },
    {
      "turn": "0.000000000",
      "x": "1.383956557",
      "y": "1.001479983"
    },
    {
      "turn": "0.000000000",
      "x": "0.709439877",
      "y": "0.773041082"
    },
    {
      "turn": "0.000000000",
      "x": "1.774759192",
      "y": "1.224904045"
    },
    {
      "turn": "0.000000000",
      "x": "1.532528826",
      "y": "1.195103134"
    },
    {
      "turn": "0.000000000",
      "x": "1.138790847",
      "y": "1.035831393"
    },
    {
      "turn": "0.000000000",
      "x": "0.861209153",
      "y": "0.964168607"
    },
    {
      "turn": "0.000000000",
      "x": "0.467471174",
      "y": "0.804896866"
    },
    {
      "turn": "0.000000000",
      "x": "0.225240808",
      "y": "0.775095955"
    },
    {
      "turn": "0.000000000",
      "x": "1.290560123",
      "y": "1.226958918"
    },
    {
      "turn": "0.000000000",
      "x": "0.616043443",
      "y": "0.998520017"
    },
    {
      "turn": "0.000000000",
      "x": "1.681362759",
      "y": "1.450382981"
    },
    {
      "turn": "0.000000000",
      "x": "0.950623778",
      "y": "1.191255967"
    },
    {
      "turn": "0.000000000",
      "x": "1.439132393",
      "y": "1.420582070"
    },
    {
      "turn": "0.000000000",
      "x": "0.374387541",
      "y": "1.032667769"
    },
    {
      "turn": "0.000000000",
      "x": "1.102393054",
      "y": "1.382383492"
    },
    {
      "turn": "0.000000000",
      "x": "0.708808918",
      "y": "1.224259272"
    },
    {
      "turn": "0.000000000",
      "x": "0.132418838",
      "y": "1.000811985"
    },
    {
      "turn": "0.000000000",
      "x": "1.483033673",
      "y": "1.666810723"
    },
    {
      "turn": "0.000000000",
      "x": "1.252165768",
      "y": "1.577448654"
    },
    {
      "turn": "0.000000000",
      "x": "0.858298048",
      "y": "1.417175430"
    },
    {
      "turn": "0.000000000",
      "x": "0.467460328",
      "y": "1.260515335"
    },
    {
      "turn": "0.000000000",
      "x": "0.225815271",
      "y": "1.226290921"
    },
    {
      "turn": "0.000000000",
      "x": "1.010067325",
      "y": "1.608302956"
    },
    {
      "turn": "0.000000000",
      "x": "0.616949458",
      "y": "1.453431494"
    },
    {
      "turn": "0.000000000",
      "x": "0.320791785",
      "y": "1.455584509"
    },
    {
      "turn": "0.000000000",
      "x": "1.157837155",
      "y": "1.802539207"
    },
    {
      "turn": "0.000000000",
      "x": "0.712550047",
      "y": "1.677984765"
    },
    {
      "turn": "0.000000000",
      "x": "0.470280915",
      "y": "1.648500668"
    },
    {
      "turn": "0.000000000",
      "x": "0.904708598",
      "y": "1.828446506"
    }
  ],
  "radius": "0.122028320"
}

DISCUSSION

Discussion

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

本题m=6，n=3的证明NUE_13#310 replies38d ago