P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 5 · n = 58

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 = 5 · n = 58

ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[0.102435670826, 0.102435670828]	Eckard Specht	Packomania ↗
Site record	0.10243567	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
0.102435668→0.10243567
2026-11-01
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.051136364→0.102435668
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": "1.269890050",
      "y": "1.830636163"
    },
    {
      "turn": "0.000000000",
      "x": "1.051036487",
      "y": "1.759526330"
    },
    {
      "turn": "0.000000000",
      "x": "0.856192263",
      "y": "1.696217604"
    },
    {
      "turn": "0.000000000",
      "x": "0.488262901",
      "y": "1.576670107"
    },
    {
      "turn": "0.000000000",
      "x": "0.293418676",
      "y": "1.513361381"
    },
    {
      "turn": "0.000000000",
      "x": "0.693086485",
      "y": "1.572246752"
    },
    {
      "turn": "0.000000000",
      "x": "1.212844971",
      "y": "1.633866944"
    },
    {
      "turn": "0.000000000",
      "x": "0.293418676",
      "y": "1.308490039"
    },
    {
      "turn": "0.000000000",
      "x": "0.565030361",
      "y": "1.386695534"
    },
    {
      "turn": "0.000000000",
      "x": "0.880231176",
      "y": "1.488885304"
    },
    {
      "turn": "0.000000000",
      "x": "1.417611518",
      "y": "1.627315006"
    },
    {
      "turn": "0.000000000",
      "x": "1.084570038",
      "y": "1.474123999"
    },
    {
      "turn": "0.000000000",
      "x": "0.759799073",
      "y": "1.323149430"
    },
    {
      "turn": "0.000000000",
      "x": "0.470154163",
      "y": "1.204871342"
    },
    {
      "turn": "0.000000000",
      "x": "1.316905879",
      "y": "1.448903568"
    },
    {
      "turn": "0.000000000",
      "x": "0.293418676",
      "y": "1.101252644"
    },
    {
      "turn": "0.000000000",
      "x": "0.969616943",
      "y": "1.304542006"
    },
    {
      "turn": "0.000000000",
      "x": "1.538031871",
      "y": "1.461570609"
    },
    {
      "turn": "0.000000000",
      "x": "1.182577542",
      "y": "1.294216256"
    },
    {
      "turn": "0.000000000",
      "x": "0.647577949",
      "y": "1.102435671"
    },
    {
      "turn": "0.000000000",
      "x": "0.849196589",
      "y": "1.138797609"
    },
    {
      "turn": "0.000000000",
      "x": "0.470842463",
      "y": "0.998816973"
    },
    {
      "turn": "0.000000000",
      "x": "1.437326232",
      "y": "1.283159171"
    },
    {
      "turn": "0.000000000",
      "x": "0.293418676",
      "y": "0.896381302"
    },
    {
      "turn": "0.000000000",
      "x": "1.067624446",
      "y": "1.124634263"
    },
    {
      "turn": "0.000000000",
      "x": "1.658452224",
      "y": "1.295826211"
    },
    {
      "turn": "0.000000000",
      "x": "1.271503657",
      "y": "1.104496269"
    },
    {
      "turn": "0.000000000",
      "x": "0.648743905",
      "y": "0.897204237"
    },
    {
      "turn": "0.000000000",
      "x": "0.947204093",
      "y": "0.958889866"
    },
    {
      "turn": "0.000000000",
      "x": "0.470842463",
      "y": "0.793945631"
    },
    {
      "turn": "0.000000000",
      "x": "1.475122516",
      "y": "1.081804490"
    },
    {
      "turn": "0.000000000",
      "x": "0.293418676",
      "y": "0.691509961"
    },
    {
      "turn": "0.000000000",
      "x": "1.150953128",
      "y": "0.937474980"
    },
    {
      "turn": "0.000000000",
      "x": "1.679736833",
      "y": "1.092063526"
    },
    {
      "turn": "0.000000000",
      "x": "0.825690036",
      "y": "0.793945631"
    },
    {
      "turn": "0.000000000",
      "x": "0.648266249",
      "y": "0.691509961"
    },
    {
      "turn": "0.000000000",
      "x": "1.354702163",
      "y": "0.916060093"
    },
    {
      "turn": "0.000000000",
      "x": "1.030532774",
      "y": "0.771730583"
    },
    {
      "turn": "0.000000000",
      "x": "0.470842463",
      "y": "0.589074290"
    },
    {
      "turn": "0.000000000",
      "x": "1.866949357",
      "y": "1.008854527"
    },
    {
      "turn": "0.000000000",
      "x": "0.293418676",
      "y": "0.486638619"
    },
    {
      "turn": "0.000000000",
      "x": "1.558451198",
      "y": "0.894645207"
    },
    {
      "turn": "0.000000000",
      "x": "1.234281809",
      "y": "0.750315696"
    },
    {
      "turn": "0.000000000",
      "x": "0.909018717",
      "y": "0.606786348"
    },
    {
      "turn": "0.000000000",
      "x": "1.755105690",
      "y": "0.837205872"
    },
    {
      "turn": "0.000000000",
      "x": "0.731594931",
      "y": "0.504350677"
    },
    {
      "turn": "0.000000000",
      "x": "1.438030844",
      "y": "0.728900810"
    },
    {
      "turn": "0.000000000",
      "x": "1.113861456",
      "y": "0.584571299"
    },
    {
      "turn": "0.000000000",
      "x": "0.554171145",
      "y": "0.401915006"
    },
    {
      "turn": "0.000000000",
      "x": "1.634685337",
      "y": "0.671461475"
    },
    {
      "turn": "0.000000000",
      "x": "1.317610491",
      "y": "0.563156413"
    },
    {
      "turn": "0.000000000",
      "x": "0.992347399",
      "y": "0.419627064"
    },
    {
      "turn": "0.000000000",
      "x": "0.814923613",
      "y": "0.317191393"
    },
    {
      "turn": "0.000000000",
      "x": "1.514264984",
      "y": "0.505717078"
    },
    {
      "turn": "0.000000000",
      "x": "1.197190138",
      "y": "0.397412016"
    },
    {
      "turn": "0.000000000",
      "x": "1.393844630",
      "y": "0.339972681"
    },
    {
      "turn": "0.000000000",
      "x": "1.075676081",
      "y": "0.232467781"
    },
    {
      "turn": "0.000000000",
      "x": "1.273220090",
      "y": "0.174020419"
    }
  ],
  "radius": "0.102435670"
}

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

Instance parameters

{
  "n": 58,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.269890050",
      "y": "1.830636163"
    },
    {
      "turn": "0.000000000",
      "x": "1.051036487",
      "y": "1.759526330"
    },
    {
      "turn": "0.000000000",
      "x": "0.856192263",
      "y": "1.696217604"
    },
    {
      "turn": "0.000000000",
      "x": "0.488262901",
      "y": "1.576670107"
    },
    {
      "turn": "0.000000000",
      "x": "0.293418676",
      "y": "1.513361381"
    },
    {
      "turn": "0.000000000",
      "x": "0.693086485",
      "y": "1.572246752"
    },
    {
      "turn": "0.000000000",
      "x": "1.212844971",
      "y": "1.633866944"
    },
    {
      "turn": "0.000000000",
      "x": "0.293418676",
      "y": "1.308490039"
    },
    {
      "turn": "0.000000000",
      "x": "0.565030361",
      "y": "1.386695534"
    },
    {
      "turn": "0.000000000",
      "x": "0.880231176",
      "y": "1.488885304"
    },
    {
      "turn": "0.000000000",
      "x": "1.417611518",
      "y": "1.627315006"
    },
    {
      "turn": "0.000000000",
      "x": "1.084570038",
      "y": "1.474123999"
    },
    {
      "turn": "0.000000000",
      "x": "0.759799073",
      "y": "1.323149430"
    },
    {
      "turn": "0.000000000",
      "x": "0.470154163",
      "y": "1.204871342"
    },
    {
      "turn": "0.000000000",
      "x": "1.316905879",
      "y": "1.448903568"
    },
    {
      "turn": "0.000000000",
      "x": "0.293418676",
      "y": "1.101252644"
    },
    {
      "turn": "0.000000000",
      "x": "0.969616943",
      "y": "1.304542006"
    },
    {
      "turn": "0.000000000",
      "x": "1.538031871",
      "y": "1.461570609"
    },
    {
      "turn": "0.000000000",
      "x": "1.182577542",
      "y": "1.294216256"
    },
    {
      "turn": "0.000000000",
      "x": "0.647577949",
      "y": "1.102435671"
    },
    {
      "turn": "0.000000000",
      "x": "0.849196589",
      "y": "1.138797609"
    },
    {
      "turn": "0.000000000",
      "x": "0.470842463",
      "y": "0.998816973"
    },
    {
      "turn": "0.000000000",
      "x": "1.437326232",
      "y": "1.283159171"
    },
    {
      "turn": "0.000000000",
      "x": "0.293418676",
      "y": "0.896381302"
    },
    {
      "turn": "0.000000000",
      "x": "1.067624446",
      "y": "1.124634263"
    },
    {
      "turn": "0.000000000",
      "x": "1.658452224",
      "y": "1.295826211"
    },
    {
      "turn": "0.000000000",
      "x": "1.271503657",
      "y": "1.104496269"
    },
    {
      "turn": "0.000000000",
      "x": "0.648743905",
      "y": "0.897204237"
    },
    {
      "turn": "0.000000000",
      "x": "0.947204093",
      "y": "0.958889866"
    },
    {
      "turn": "0.000000000",
      "x": "0.470842463",
      "y": "0.793945631"
    },
    {
      "turn": "0.000000000",
      "x": "1.475122516",
      "y": "1.081804490"
    },
    {
      "turn": "0.000000000",
      "x": "0.293418676",
      "y": "0.691509961"
    },
    {
      "turn": "0.000000000",
      "x": "1.150953128",
      "y": "0.937474980"
    },
    {
      "turn": "0.000000000",
      "x": "1.679736833",
      "y": "1.092063526"
    },
    {
      "turn": "0.000000000",
      "x": "0.825690036",
      "y": "0.793945631"
    },
    {
      "turn": "0.000000000",
      "x": "0.648266249",
      "y": "0.691509961"
    },
    {
      "turn": "0.000000000",
      "x": "1.354702163",
      "y": "0.916060093"
    },
    {
      "turn": "0.000000000",
      "x": "1.030532774",
      "y": "0.771730583"
    },
    {
      "turn": "0.000000000",
      "x": "0.470842463",
      "y": "0.589074290"
    },
    {
      "turn": "0.000000000",
      "x": "1.866949357",
      "y": "1.008854527"
    },
    {
      "turn": "0.000000000",
      "x": "0.293418676",
      "y": "0.486638619"
    },
    {
      "turn": "0.000000000",
      "x": "1.558451198",
      "y": "0.894645207"
    },
    {
      "turn": "0.000000000",
      "x": "1.234281809",
      "y": "0.750315696"
    },
    {
      "turn": "0.000000000",
      "x": "0.909018717",
      "y": "0.606786348"
    },
    {
      "turn": "0.000000000",
      "x": "1.755105690",
      "y": "0.837205872"
    },
    {
      "turn": "0.000000000",
      "x": "0.731594931",
      "y": "0.504350677"
    },
    {
      "turn": "0.000000000",
      "x": "1.438030844",
      "y": "0.728900810"
    },
    {
      "turn": "0.000000000",
      "x": "1.113861456",
      "y": "0.584571299"
    },
    {
      "turn": "0.000000000",
      "x": "0.554171145",
      "y": "0.401915006"
    },
    {
      "turn": "0.000000000",
      "x": "1.634685337",
      "y": "0.671461475"
    },
    {
      "turn": "0.000000000",
      "x": "1.317610491",
      "y": "0.563156413"
    },
    {
      "turn": "0.000000000",
      "x": "0.992347399",
      "y": "0.419627064"
    },
    {
      "turn": "0.000000000",
      "x": "0.814923613",
      "y": "0.317191393"
    },
    {
      "turn": "0.000000000",
      "x": "1.514264984",
      "y": "0.505717078"
    },
    {
      "turn": "0.000000000",
      "x": "1.197190138",
      "y": "0.397412016"
    },
    {
      "turn": "0.000000000",
      "x": "1.393844630",
      "y": "0.339972681"
    },
    {
      "turn": "0.000000000",
      "x": "1.075676081",
      "y": "0.232467781"
    },
    {
      "turn": "0.000000000",
      "x": "1.273220090",
      "y": "0.174020419"
    }
  ],
  "radius": "0.102435670"
}

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