P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 5 · n = 75

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 = 75

ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[0.090526425532, 0.090526425534]	Paolo Amore	Packomania ↗
Site record	0.090526425	condor	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 holdercondor

Solution methodAI · OpenAI: GPT-6.1 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 noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (2 changes)

condorAI · OpenAI: GPT-6.1 Sol
0.090526423→0.090526425
2026-10-27
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.045454545→0.090526423
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",
      "x": "1.410391131",
      "y": "0.342485767"
    },
    {
      "turn": "0",
      "x": "1.552809151",
      "y": "0.538507355"
    },
    {
      "turn": "0",
      "x": "1.693025532",
      "y": "0.731498647"
    },
    {
      "turn": "0",
      "x": "1.799445728",
      "y": "0.877973480"
    },
    {
      "turn": "0",
      "x": "1.274395995",
      "y": "0.155388675"
    },
    {
      "turn": "0",
      "x": "1.850571495",
      "y": "1.051657938"
    },
    {
      "turn": "0",
      "x": "1.372748127",
      "y": "0.519582179"
    },
    {
      "turn": "0",
      "x": "1.512719941",
      "y": "0.715066099"
    },
    {
      "turn": "0",
      "x": "1.229481869",
      "y": "0.335276491"
    },
    {
      "turn": "0",
      "x": "1.588641733",
      "y": "0.879431595"
    },
    {
      "turn": "0",
      "x": "1.095496979",
      "y": "0.213505459"
    },
    {
      "turn": "0",
      "x": "1.669817067",
      "y": "1.041267006"
    },
    {
      "turn": "0",
      "x": "1.735555621",
      "y": "1.209963708"
    },
    {
      "turn": "0",
      "x": "1.332019072",
      "y": "0.695994420"
    },
    {
      "turn": "0",
      "x": "1.191838864",
      "y": "0.512372903"
    },
    {
      "turn": "0",
      "x": "1.405931976",
      "y": "0.861273025"
    },
    {
      "turn": "0",
      "x": "1.057290375",
      "y": "0.391224899"
    },
    {
      "turn": "0",
      "x": "1.481853768",
      "y": "1.025638522"
    },
    {
      "turn": "0",
      "x": "0.923305485",
      "y": "0.269453867"
    },
    {
      "turn": "0",
      "x": "1.555494596",
      "y": "1.191038532"
    },
    {
      "turn": "0",
      "x": "1.151109809",
      "y": "0.688785144"
    },
    {
      "turn": "0",
      "x": "1.629135425",
      "y": "1.356438541"
    },
    {
      "turn": "0",
      "x": "1.225022714",
      "y": "0.854063750"
    },
    {
      "turn": "0",
      "x": "1.003136418",
      "y": "0.584458684"
    },
    {
      "turn": "0",
      "x": "1.300944505",
      "y": "1.018429246"
    },
    {
      "turn": "0",
      "x": "0.885098881",
      "y": "0.447173307"
    },
    {
      "turn": "0",
      "x": "0.751113991",
      "y": "0.325402275"
    },
    {
      "turn": "0",
      "x": "1.374585334",
      "y": "1.183829256"
    },
    {
      "turn": "0",
      "x": "1.448226163",
      "y": "1.349229266"
    },
    {
      "turn": "0",
      "x": "1.044689613",
      "y": "0.835259978"
    },
    {
      "turn": "0",
      "x": "1.510986628",
      "y": "1.519056410"
    },
    {
      "turn": "0",
      "x": "1.118602518",
      "y": "1.000538583"
    },
    {
      "turn": "0",
      "x": "0.896716222",
      "y": "0.730933517"
    },
    {
      "turn": "0",
      "x": "1.194524309",
      "y": "1.164904080"
    },
    {
      "turn": "0",
      "x": "0.778678685",
      "y": "0.593648140"
    },
    {
      "turn": "0",
      "x": "1.268165138",
      "y": "1.330304089"
    },
    {
      "turn": "0",
      "x": "0.581273055",
      "y": "0.388125406"
    },
    {
      "turn": "0",
      "x": "1.330925603",
      "y": "1.500131234"
    },
    {
      "turn": "0",
      "x": "0.925832134",
      "y": "0.971836090"
    },
    {
      "turn": "0",
      "x": "1.404566432",
      "y": "1.665531243"
    },
    {
      "turn": "0",
      "x": "0.599413683",
      "y": "0.568267163"
    },
    {
      "turn": "0",
      "x": "0.999745038",
      "y": "1.137114695"
    },
    {
      "turn": "0",
      "x": "0.752620230",
      "y": "0.840553223"
    },
    {
      "turn": "0",
      "x": "1.088104114",
      "y": "1.311378913"
    },
    {
      "turn": "0",
      "x": "0.407001120",
      "y": "0.437211324"
    },
    {
      "turn": "0",
      "x": "1.150864579",
      "y": "1.481206057"
    },
    {
      "turn": "0",
      "x": "0.595392408",
      "y": "0.749275351"
    },
    {
      "turn": "0",
      "x": "1.224505408",
      "y": "1.646606067"
    },
    {
      "turn": "0",
      "x": "0.751753732",
      "y": "1.021604000"
    },
    {
      "turn": "0",
      "x": "1.298146237",
      "y": "1.812006077"
    },
    {
      "turn": "0",
      "x": "0.907968000",
      "y": "1.293182341"
    },
    {
      "turn": "0",
      "x": "0.440123146",
      "y": "0.655969530"
    },
    {
      "turn": "0",
      "x": "0.595392408",
      "y": "0.930328202"
    },
    {
      "turn": "0",
      "x": "0.970728465",
      "y": "1.463009485"
    },
    {
      "turn": "0",
      "x": "0.751171631",
      "y": "1.202655916"
    },
    {
      "turn": "0",
      "x": "0.281509431",
      "y": "0.567717918"
    },
    {
      "turn": "0",
      "x": "1.044369293",
      "y": "1.628409495"
    },
    {
      "turn": "0",
      "x": "0.438305800",
      "y": "0.839297195"
    },
    {
      "turn": "0",
      "x": "1.118010122",
      "y": "1.793809505"
    },
    {
      "turn": "0",
      "x": "0.594667123",
      "y": "1.111625844"
    },
    {
      "turn": "0",
      "x": "0.751171631",
      "y": "1.383708767"
    },
    {
      "turn": "0",
      "x": "0.281509431",
      "y": "0.748770769"
    },
    {
      "turn": "0",
      "x": "0.438305800",
      "y": "1.020350046"
    },
    {
      "turn": "0",
      "x": "0.813932096",
      "y": "1.553535911"
    },
    {
      "turn": "0",
      "x": "0.594375263",
      "y": "1.293182341"
    },
    {
      "turn": "0",
      "x": "0.887572925",
      "y": "1.718935921"
    },
    {
      "turn": "0",
      "x": "0.281509431",
      "y": "0.929823620"
    },
    {
      "turn": "0",
      "x": "0.437870755",
      "y": "1.202152269"
    },
    {
      "turn": "0",
      "x": "0.594375263",
      "y": "1.474235192"
    },
    {
      "turn": "0",
      "x": "0.281509431",
      "y": "1.110876471"
    },
    {
      "turn": "0",
      "x": "0.657135728",
      "y": "1.644062336"
    },
    {
      "turn": "0",
      "x": "0.432105605",
      "y": "1.393931438"
    },
    {
      "turn": "0",
      "x": "0.281509431",
      "y": "1.293428068"
    },
    {
      "turn": "0",
      "x": "0.443779089",
      "y": "1.574738562"
    },
    {
      "turn": "0",
      "x": "0.281509431",
      "y": "1.494434807"
    }
  ],
  "radius": "0.090526425"
}

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

Instance parameters

{
  "n": 75,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "1.410391131",
      "y": "0.342485767"
    },
    {
      "turn": "0",
      "x": "1.552809151",
      "y": "0.538507355"
    },
    {
      "turn": "0",
      "x": "1.693025532",
      "y": "0.731498647"
    },
    {
      "turn": "0",
      "x": "1.799445728",
      "y": "0.877973480"
    },
    {
      "turn": "0",
      "x": "1.274395995",
      "y": "0.155388675"
    },
    {
      "turn": "0",
      "x": "1.850571495",
      "y": "1.051657938"
    },
    {
      "turn": "0",
      "x": "1.372748127",
      "y": "0.519582179"
    },
    {
      "turn": "0",
      "x": "1.512719941",
      "y": "0.715066099"
    },
    {
      "turn": "0",
      "x": "1.229481869",
      "y": "0.335276491"
    },
    {
      "turn": "0",
      "x": "1.588641733",
      "y": "0.879431595"
    },
    {
      "turn": "0",
      "x": "1.095496979",
      "y": "0.213505459"
    },
    {
      "turn": "0",
      "x": "1.669817067",
      "y": "1.041267006"
    },
    {
      "turn": "0",
      "x": "1.735555621",
      "y": "1.209963708"
    },
    {
      "turn": "0",
      "x": "1.332019072",
      "y": "0.695994420"
    },
    {
      "turn": "0",
      "x": "1.191838864",
      "y": "0.512372903"
    },
    {
      "turn": "0",
      "x": "1.405931976",
      "y": "0.861273025"
    },
    {
      "turn": "0",
      "x": "1.057290375",
      "y": "0.391224899"
    },
    {
      "turn": "0",
      "x": "1.481853768",
      "y": "1.025638522"
    },
    {
      "turn": "0",
      "x": "0.923305485",
      "y": "0.269453867"
    },
    {
      "turn": "0",
      "x": "1.555494596",
      "y": "1.191038532"
    },
    {
      "turn": "0",
      "x": "1.151109809",
      "y": "0.688785144"
    },
    {
      "turn": "0",
      "x": "1.629135425",
      "y": "1.356438541"
    },
    {
      "turn": "0",
      "x": "1.225022714",
      "y": "0.854063750"
    },
    {
      "turn": "0",
      "x": "1.003136418",
      "y": "0.584458684"
    },
    {
      "turn": "0",
      "x": "1.300944505",
      "y": "1.018429246"
    },
    {
      "turn": "0",
      "x": "0.885098881",
      "y": "0.447173307"
    },
    {
      "turn": "0",
      "x": "0.751113991",
      "y": "0.325402275"
    },
    {
      "turn": "0",
      "x": "1.374585334",
      "y": "1.183829256"
    },
    {
      "turn": "0",
      "x": "1.448226163",
      "y": "1.349229266"
    },
    {
      "turn": "0",
      "x": "1.044689613",
      "y": "0.835259978"
    },
    {
      "turn": "0",
      "x": "1.510986628",
      "y": "1.519056410"
    },
    {
      "turn": "0",
      "x": "1.118602518",
      "y": "1.000538583"
    },
    {
      "turn": "0",
      "x": "0.896716222",
      "y": "0.730933517"
    },
    {
      "turn": "0",
      "x": "1.194524309",
      "y": "1.164904080"
    },
    {
      "turn": "0",
      "x": "0.778678685",
      "y": "0.593648140"
    },
    {
      "turn": "0",
      "x": "1.268165138",
      "y": "1.330304089"
    },
    {
      "turn": "0",
      "x": "0.581273055",
      "y": "0.388125406"
    },
    {
      "turn": "0",
      "x": "1.330925603",
      "y": "1.500131234"
    },
    {
      "turn": "0",
      "x": "0.925832134",
      "y": "0.971836090"
    },
    {
      "turn": "0",
      "x": "1.404566432",
      "y": "1.665531243"
    },
    {
      "turn": "0",
      "x": "0.599413683",
      "y": "0.568267163"
    },
    {
      "turn": "0",
      "x": "0.999745038",
      "y": "1.137114695"
    },
    {
      "turn": "0",
      "x": "0.752620230",
      "y": "0.840553223"
    },
    {
      "turn": "0",
      "x": "1.088104114",
      "y": "1.311378913"
    },
    {
      "turn": "0",
      "x": "0.407001120",
      "y": "0.437211324"
    },
    {
      "turn": "0",
      "x": "1.150864579",
      "y": "1.481206057"
    },
    {
      "turn": "0",
      "x": "0.595392408",
      "y": "0.749275351"
    },
    {
      "turn": "0",
      "x": "1.224505408",
      "y": "1.646606067"
    },
    {
      "turn": "0",
      "x": "0.751753732",
      "y": "1.021604000"
    },
    {
      "turn": "0",
      "x": "1.298146237",
      "y": "1.812006077"
    },
    {
      "turn": "0",
      "x": "0.907968000",
      "y": "1.293182341"
    },
    {
      "turn": "0",
      "x": "0.440123146",
      "y": "0.655969530"
    },
    {
      "turn": "0",
      "x": "0.595392408",
      "y": "0.930328202"
    },
    {
      "turn": "0",
      "x": "0.970728465",
      "y": "1.463009485"
    },
    {
      "turn": "0",
      "x": "0.751171631",
      "y": "1.202655916"
    },
    {
      "turn": "0",
      "x": "0.281509431",
      "y": "0.567717918"
    },
    {
      "turn": "0",
      "x": "1.044369293",
      "y": "1.628409495"
    },
    {
      "turn": "0",
      "x": "0.438305800",
      "y": "0.839297195"
    },
    {
      "turn": "0",
      "x": "1.118010122",
      "y": "1.793809505"
    },
    {
      "turn": "0",
      "x": "0.594667123",
      "y": "1.111625844"
    },
    {
      "turn": "0",
      "x": "0.751171631",
      "y": "1.383708767"
    },
    {
      "turn": "0",
      "x": "0.281509431",
      "y": "0.748770769"
    },
    {
      "turn": "0",
      "x": "0.438305800",
      "y": "1.020350046"
    },
    {
      "turn": "0",
      "x": "0.813932096",
      "y": "1.553535911"
    },
    {
      "turn": "0",
      "x": "0.594375263",
      "y": "1.293182341"
    },
    {
      "turn": "0",
      "x": "0.887572925",
      "y": "1.718935921"
    },
    {
      "turn": "0",
      "x": "0.281509431",
      "y": "0.929823620"
    },
    {
      "turn": "0",
      "x": "0.437870755",
      "y": "1.202152269"
    },
    {
      "turn": "0",
      "x": "0.594375263",
      "y": "1.474235192"
    },
    {
      "turn": "0",
      "x": "0.281509431",
      "y": "1.110876471"
    },
    {
      "turn": "0",
      "x": "0.657135728",
      "y": "1.644062336"
    },
    {
      "turn": "0",
      "x": "0.432105605",
      "y": "1.393931438"
    },
    {
      "turn": "0",
      "x": "0.281509431",
      "y": "1.293428068"
    },
    {
      "turn": "0",
      "x": "0.443779089",
      "y": "1.574738562"
    },
    {
      "turn": "0",
      "x": "0.281509431",
      "y": "1.494434807"
    }
  ],
  "radius": "0.090526425"
}

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