P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 6 · n = 69

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 = 69

ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[0.099881706144, 0.099881706146]	Eckard Specht	Packomania ↗
Site record	0.099881705	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
0.099881703→0.099881705
2026-11-01
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.045454545→0.099881703
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": "0.700354882",
      "y": "0.233856302"
    },
    {
      "turn": "0.000000000",
      "x": "0.900118294",
      "y": "0.233856302"
    },
    {
      "turn": "0.000000000",
      "x": "1.099881706",
      "y": "0.233856302"
    },
    {
      "turn": "0.000000000",
      "x": "1.299645118",
      "y": "0.233856302"
    },
    {
      "turn": "0.000000000",
      "x": "0.514860284",
      "y": "0.307999241"
    },
    {
      "turn": "0.000000000",
      "x": "1.485139716",
      "y": "0.307999241"
    },
    {
      "turn": "0.000000000",
      "x": "0.800236588",
      "y": "0.406856492"
    },
    {
      "turn": "0.000000000",
      "x": "1.000000000",
      "y": "0.406856492"
    },
    {
      "turn": "0.000000000",
      "x": "1.199763412",
      "y": "0.406856492"
    },
    {
      "turn": "0.000000000",
      "x": "0.414978578",
      "y": "0.480999431"
    },
    {
      "turn": "0.000000000",
      "x": "0.614741991",
      "y": "0.480999431"
    },
    {
      "turn": "0.000000000",
      "x": "1.385258009",
      "y": "0.480999431"
    },
    {
      "turn": "0.000000000",
      "x": "1.585021422",
      "y": "0.480999431"
    },
    {
      "turn": "0.000000000",
      "x": "0.900118294",
      "y": "0.579856682"
    },
    {
      "turn": "0.000000000",
      "x": "1.099881706",
      "y": "0.579856682"
    },
    {
      "turn": "0.000000000",
      "x": "0.315096872",
      "y": "0.653999620"
    },
    {
      "turn": "0.000000000",
      "x": "0.514860284",
      "y": "0.653999620"
    },
    {
      "turn": "0.000000000",
      "x": "0.714623697",
      "y": "0.653999620"
    },
    {
      "turn": "0.000000000",
      "x": "1.285376303",
      "y": "0.653999620"
    },
    {
      "turn": "0.000000000",
      "x": "1.485139716",
      "y": "0.653999620"
    },
    {
      "turn": "0.000000000",
      "x": "1.684903128",
      "y": "0.653999620"
    },
    {
      "turn": "0.000000000",
      "x": "1.000000000",
      "y": "0.752856872"
    },
    {
      "turn": "0.000000000",
      "x": "0.215215166",
      "y": "0.826999810"
    },
    {
      "turn": "0.000000000",
      "x": "0.414978578",
      "y": "0.826999810"
    },
    {
      "turn": "0.000000000",
      "x": "0.614741991",
      "y": "0.826999810"
    },
    {
      "turn": "0.000000000",
      "x": "0.814505403",
      "y": "0.826999810"
    },
    {
      "turn": "0.000000000",
      "x": "1.185494597",
      "y": "0.826999810"
    },
    {
      "turn": "0.000000000",
      "x": "1.385258009",
      "y": "0.826999810"
    },
    {
      "turn": "0.000000000",
      "x": "1.585021422",
      "y": "0.826999810"
    },
    {
      "turn": "0.000000000",
      "x": "1.784784834",
      "y": "0.826999810"
    },
    {
      "turn": "0.000000000",
      "x": "0.115333460",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.315096872",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.514860284",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.714623697",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.914387109",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "1.285376303",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "1.485139716",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "1.684903128",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "1.884666540",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "1.099881706",
      "y": "1.074142938"
    },
    {
      "turn": "0.000000000",
      "x": "0.215215166",
      "y": "1.173000190"
    },
    {
      "turn": "0.000000000",
      "x": "0.414978578",
      "y": "1.173000190"
    },
    {
      "turn": "0.000000000",
      "x": "0.614741991",
      "y": "1.173000190"
    },
    {
      "turn": "0.000000000",
      "x": "0.814505403",
      "y": "1.173000190"
    },
    {
      "turn": "0.000000000",
      "x": "1.385258009",
      "y": "1.173000190"
    },
    {
      "turn": "0.000000000",
      "x": "1.585021422",
      "y": "1.173000190"
    },
    {
      "turn": "0.000000000",
      "x": "1.784784834",
      "y": "1.173000190"
    },
    {
      "turn": "0.000000000",
      "x": "1.000000000",
      "y": "1.247143128"
    },
    {
      "turn": "0.000000000",
      "x": "1.199763412",
      "y": "1.247143128"
    },
    {
      "turn": "0.000000000",
      "x": "0.315096872",
      "y": "1.346000380"
    },
    {
      "turn": "0.000000000",
      "x": "0.514860284",
      "y": "1.346000380"
    },
    {
      "turn": "0.000000000",
      "x": "0.714623697",
      "y": "1.346000380"
    },
    {
      "turn": "0.000000000",
      "x": "1.485139716",
      "y": "1.346000380"
    },
    {
      "turn": "0.000000000",
      "x": "1.684903128",
      "y": "1.346000380"
    },
    {
      "turn": "0.000000000",
      "x": "0.900118294",
      "y": "1.420143318"
    },
    {
      "turn": "0.000000000",
      "x": "1.099881706",
      "y": "1.420143318"
    },
    {
      "turn": "0.000000000",
      "x": "1.299645118",
      "y": "1.420143318"
    },
    {
      "turn": "0.000000000",
      "x": "0.414978578",
      "y": "1.519000569"
    },
    {
      "turn": "0.000000000",
      "x": "0.614741991",
      "y": "1.519000569"
    },
    {
      "turn": "0.000000000",
      "x": "1.585021422",
      "y": "1.519000569"
    },
    {
      "turn": "0.000000000",
      "x": "0.800236588",
      "y": "1.593143508"
    },
    {
      "turn": "0.000000000",
      "x": "1.000000000",
      "y": "1.593143508"
    },
    {
      "turn": "0.000000000",
      "x": "1.199763412",
      "y": "1.593143508"
    },
    {
      "turn": "0.000000000",
      "x": "1.399526825",
      "y": "1.593143508"
    },
    {
      "turn": "0.000000000",
      "x": "0.514860284",
      "y": "1.692000759"
    },
    {
      "turn": "0.000000000",
      "x": "0.700354882",
      "y": "1.766143698"
    },
    {
      "turn": "0.000000000",
      "x": "0.900118294",
      "y": "1.766143698"
    },
    {
      "turn": "0.000000000",
      "x": "1.099881706",
      "y": "1.766143698"
    },
    {
      "turn": "0.000000000",
      "x": "1.299645118",
      "y": "1.766143698"
    }
  ],
  "radius": "0.099881705"
}

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

Instance parameters

{
  "n": 69,
  "sides": 6
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.700354882",
      "y": "0.233856302"
    },
    {
      "turn": "0.000000000",
      "x": "0.900118294",
      "y": "0.233856302"
    },
    {
      "turn": "0.000000000",
      "x": "1.099881706",
      "y": "0.233856302"
    },
    {
      "turn": "0.000000000",
      "x": "1.299645118",
      "y": "0.233856302"
    },
    {
      "turn": "0.000000000",
      "x": "0.514860284",
      "y": "0.307999241"
    },
    {
      "turn": "0.000000000",
      "x": "1.485139716",
      "y": "0.307999241"
    },
    {
      "turn": "0.000000000",
      "x": "0.800236588",
      "y": "0.406856492"
    },
    {
      "turn": "0.000000000",
      "x": "1.000000000",
      "y": "0.406856492"
    },
    {
      "turn": "0.000000000",
      "x": "1.199763412",
      "y": "0.406856492"
    },
    {
      "turn": "0.000000000",
      "x": "0.414978578",
      "y": "0.480999431"
    },
    {
      "turn": "0.000000000",
      "x": "0.614741991",
      "y": "0.480999431"
    },
    {
      "turn": "0.000000000",
      "x": "1.385258009",
      "y": "0.480999431"
    },
    {
      "turn": "0.000000000",
      "x": "1.585021422",
      "y": "0.480999431"
    },
    {
      "turn": "0.000000000",
      "x": "0.900118294",
      "y": "0.579856682"
    },
    {
      "turn": "0.000000000",
      "x": "1.099881706",
      "y": "0.579856682"
    },
    {
      "turn": "0.000000000",
      "x": "0.315096872",
      "y": "0.653999620"
    },
    {
      "turn": "0.000000000",
      "x": "0.514860284",
      "y": "0.653999620"
    },
    {
      "turn": "0.000000000",
      "x": "0.714623697",
      "y": "0.653999620"
    },
    {
      "turn": "0.000000000",
      "x": "1.285376303",
      "y": "0.653999620"
    },
    {
      "turn": "0.000000000",
      "x": "1.485139716",
      "y": "0.653999620"
    },
    {
      "turn": "0.000000000",
      "x": "1.684903128",
      "y": "0.653999620"
    },
    {
      "turn": "0.000000000",
      "x": "1.000000000",
      "y": "0.752856872"
    },
    {
      "turn": "0.000000000",
      "x": "0.215215166",
      "y": "0.826999810"
    },
    {
      "turn": "0.000000000",
      "x": "0.414978578",
      "y": "0.826999810"
    },
    {
      "turn": "0.000000000",
      "x": "0.614741991",
      "y": "0.826999810"
    },
    {
      "turn": "0.000000000",
      "x": "0.814505403",
      "y": "0.826999810"
    },
    {
      "turn": "0.000000000",
      "x": "1.185494597",
      "y": "0.826999810"
    },
    {
      "turn": "0.000000000",
      "x": "1.385258009",
      "y": "0.826999810"
    },
    {
      "turn": "0.000000000",
      "x": "1.585021422",
      "y": "0.826999810"
    },
    {
      "turn": "0.000000000",
      "x": "1.784784834",
      "y": "0.826999810"
    },
    {
      "turn": "0.000000000",
      "x": "0.115333460",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.315096872",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.514860284",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.714623697",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.914387109",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "1.285376303",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "1.485139716",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "1.684903128",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "1.884666540",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "1.099881706",
      "y": "1.074142938"
    },
    {
      "turn": "0.000000000",
      "x": "0.215215166",
      "y": "1.173000190"
    },
    {
      "turn": "0.000000000",
      "x": "0.414978578",
      "y": "1.173000190"
    },
    {
      "turn": "0.000000000",
      "x": "0.614741991",
      "y": "1.173000190"
    },
    {
      "turn": "0.000000000",
      "x": "0.814505403",
      "y": "1.173000190"
    },
    {
      "turn": "0.000000000",
      "x": "1.385258009",
      "y": "1.173000190"
    },
    {
      "turn": "0.000000000",
      "x": "1.585021422",
      "y": "1.173000190"
    },
    {
      "turn": "0.000000000",
      "x": "1.784784834",
      "y": "1.173000190"
    },
    {
      "turn": "0.000000000",
      "x": "1.000000000",
      "y": "1.247143128"
    },
    {
      "turn": "0.000000000",
      "x": "1.199763412",
      "y": "1.247143128"
    },
    {
      "turn": "0.000000000",
      "x": "0.315096872",
      "y": "1.346000380"
    },
    {
      "turn": "0.000000000",
      "x": "0.514860284",
      "y": "1.346000380"
    },
    {
      "turn": "0.000000000",
      "x": "0.714623697",
      "y": "1.346000380"
    },
    {
      "turn": "0.000000000",
      "x": "1.485139716",
      "y": "1.346000380"
    },
    {
      "turn": "0.000000000",
      "x": "1.684903128",
      "y": "1.346000380"
    },
    {
      "turn": "0.000000000",
      "x": "0.900118294",
      "y": "1.420143318"
    },
    {
      "turn": "0.000000000",
      "x": "1.099881706",
      "y": "1.420143318"
    },
    {
      "turn": "0.000000000",
      "x": "1.299645118",
      "y": "1.420143318"
    },
    {
      "turn": "0.000000000",
      "x": "0.414978578",
      "y": "1.519000569"
    },
    {
      "turn": "0.000000000",
      "x": "0.614741991",
      "y": "1.519000569"
    },
    {
      "turn": "0.000000000",
      "x": "1.585021422",
      "y": "1.519000569"
    },
    {
      "turn": "0.000000000",
      "x": "0.800236588",
      "y": "1.593143508"
    },
    {
      "turn": "0.000000000",
      "x": "1.000000000",
      "y": "1.593143508"
    },
    {
      "turn": "0.000000000",
      "x": "1.199763412",
      "y": "1.593143508"
    },
    {
      "turn": "0.000000000",
      "x": "1.399526825",
      "y": "1.593143508"
    },
    {
      "turn": "0.000000000",
      "x": "0.514860284",
      "y": "1.692000759"
    },
    {
      "turn": "0.000000000",
      "x": "0.700354882",
      "y": "1.766143698"
    },
    {
      "turn": "0.000000000",
      "x": "0.900118294",
      "y": "1.766143698"
    },
    {
      "turn": "0.000000000",
      "x": "1.099881706",
      "y": "1.766143698"
    },
    {
      "turn": "0.000000000",
      "x": "1.299645118",
      "y": "1.766143698"
    }
  ],
  "radius": "0.099881705"
}

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