P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 6 · n = 50

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 = 50

ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[0.115694712595, 0.115694712597]	Eckard Specht	Packomania ↗
Site record	0.115694712	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
0.11569471→0.115694712
2026-11-01
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.051136364→0.11569471
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": "0.657570545",
      "y": "0.249669309"
    },
    {
      "turn": "0.000000000",
      "x": "0.888959970",
      "y": "0.249669309"
    },
    {
      "turn": "0.000000000",
      "x": "1.120349395",
      "y": "0.249669309"
    },
    {
      "turn": "0.000000000",
      "x": "1.351738821",
      "y": "0.249669309"
    },
    {
      "turn": "0.000000000",
      "x": "0.480676885",
      "y": "0.398832639"
    },
    {
      "turn": "0.000000000",
      "x": "1.523023268",
      "y": "0.405241492"
    },
    {
      "turn": "0.000000000",
      "x": "0.773265258",
      "y": "0.450058429"
    },
    {
      "turn": "0.000000000",
      "x": "1.004654683",
      "y": "0.450058429"
    },
    {
      "turn": "0.000000000",
      "x": "1.236044108",
      "y": "0.450058429"
    },
    {
      "turn": "0.000000000",
      "x": "0.364982172",
      "y": "0.599221759"
    },
    {
      "turn": "0.000000000",
      "x": "0.596371597",
      "y": "0.599221759"
    },
    {
      "turn": "0.000000000",
      "x": "1.407328556",
      "y": "0.605630612"
    },
    {
      "turn": "0.000000000",
      "x": "1.638717981",
      "y": "0.605630612"
    },
    {
      "turn": "0.000000000",
      "x": "0.888959970",
      "y": "0.650447550"
    },
    {
      "turn": "0.000000000",
      "x": "1.120349395",
      "y": "0.650447550"
    },
    {
      "turn": "0.000000000",
      "x": "0.249287460",
      "y": "0.799610880"
    },
    {
      "turn": "0.000000000",
      "x": "0.480676885",
      "y": "0.799610880"
    },
    {
      "turn": "0.000000000",
      "x": "0.712066310",
      "y": "0.799610880"
    },
    {
      "turn": "0.000000000",
      "x": "1.291633843",
      "y": "0.806019732"
    },
    {
      "turn": "0.000000000",
      "x": "1.523023268",
      "y": "0.806019732"
    },
    {
      "turn": "0.000000000",
      "x": "1.754412693",
      "y": "0.806019732"
    },
    {
      "turn": "0.000000000",
      "x": "1.004654683",
      "y": "0.850836670"
    },
    {
      "turn": "0.000000000",
      "x": "0.133592747",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.364982172",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.596371597",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.827761022",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "1.396552864",
      "y": "1.012255193"
    },
    {
      "turn": "0.000000000",
      "x": "1.627942289",
      "y": "1.012255193"
    },
    {
      "turn": "0.000000000",
      "x": "1.859331714",
      "y": "1.012255193"
    },
    {
      "turn": "0.000000000",
      "x": "1.165221711",
      "y": "1.017447880"
    },
    {
      "turn": "0.000000000",
      "x": "0.988328051",
      "y": "1.166611211"
    },
    {
      "turn": "0.000000000",
      "x": "0.249287460",
      "y": "1.200389120"
    },
    {
      "turn": "0.000000000",
      "x": "0.480676885",
      "y": "1.200389120"
    },
    {
      "turn": "0.000000000",
      "x": "1.512247576",
      "y": "1.212644313"
    },
    {
      "turn": "0.000000000",
      "x": "1.743637001",
      "y": "1.212644313"
    },
    {
      "turn": "0.000000000",
      "x": "0.750096969",
      "y": "1.225032094"
    },
    {
      "turn": "0.000000000",
      "x": "1.251783926",
      "y": "1.232177290"
    },
    {
      "turn": "0.000000000",
      "x": "0.943942735",
      "y": "1.393703724"
    },
    {
      "turn": "0.000000000",
      "x": "0.364982172",
      "y": "1.400778241"
    },
    {
      "turn": "0.000000000",
      "x": "0.596371597",
      "y": "1.400778241"
    },
    {
      "turn": "0.000000000",
      "x": "1.396552864",
      "y": "1.413033434"
    },
    {
      "turn": "0.000000000",
      "x": "1.627942289",
      "y": "1.413033434"
    },
    {
      "turn": "0.000000000",
      "x": "1.168231220",
      "y": "1.450587305"
    },
    {
      "turn": "0.000000000",
      "x": "0.773265258",
      "y": "1.549941571"
    },
    {
      "turn": "0.000000000",
      "x": "0.480676885",
      "y": "1.601167361"
    },
    {
      "turn": "0.000000000",
      "x": "1.512247576",
      "y": "1.613422554"
    },
    {
      "turn": "0.000000000",
      "x": "1.107333698",
      "y": "1.673819376"
    },
    {
      "turn": "0.000000000",
      "x": "0.657570545",
      "y": "1.750330691"
    },
    {
      "turn": "0.000000000",
      "x": "0.888959970",
      "y": "1.750330691"
    },
    {
      "turn": "0.000000000",
      "x": "1.325707425",
      "y": "1.750330691"
    }
  ],
  "radius": "0.115694712"
}

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

Instance parameters

{
  "n": 50,
  "sides": 6
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.657570545",
      "y": "0.249669309"
    },
    {
      "turn": "0.000000000",
      "x": "0.888959970",
      "y": "0.249669309"
    },
    {
      "turn": "0.000000000",
      "x": "1.120349395",
      "y": "0.249669309"
    },
    {
      "turn": "0.000000000",
      "x": "1.351738821",
      "y": "0.249669309"
    },
    {
      "turn": "0.000000000",
      "x": "0.480676885",
      "y": "0.398832639"
    },
    {
      "turn": "0.000000000",
      "x": "1.523023268",
      "y": "0.405241492"
    },
    {
      "turn": "0.000000000",
      "x": "0.773265258",
      "y": "0.450058429"
    },
    {
      "turn": "0.000000000",
      "x": "1.004654683",
      "y": "0.450058429"
    },
    {
      "turn": "0.000000000",
      "x": "1.236044108",
      "y": "0.450058429"
    },
    {
      "turn": "0.000000000",
      "x": "0.364982172",
      "y": "0.599221759"
    },
    {
      "turn": "0.000000000",
      "x": "0.596371597",
      "y": "0.599221759"
    },
    {
      "turn": "0.000000000",
      "x": "1.407328556",
      "y": "0.605630612"
    },
    {
      "turn": "0.000000000",
      "x": "1.638717981",
      "y": "0.605630612"
    },
    {
      "turn": "0.000000000",
      "x": "0.888959970",
      "y": "0.650447550"
    },
    {
      "turn": "0.000000000",
      "x": "1.120349395",
      "y": "0.650447550"
    },
    {
      "turn": "0.000000000",
      "x": "0.249287460",
      "y": "0.799610880"
    },
    {
      "turn": "0.000000000",
      "x": "0.480676885",
      "y": "0.799610880"
    },
    {
      "turn": "0.000000000",
      "x": "0.712066310",
      "y": "0.799610880"
    },
    {
      "turn": "0.000000000",
      "x": "1.291633843",
      "y": "0.806019732"
    },
    {
      "turn": "0.000000000",
      "x": "1.523023268",
      "y": "0.806019732"
    },
    {
      "turn": "0.000000000",
      "x": "1.754412693",
      "y": "0.806019732"
    },
    {
      "turn": "0.000000000",
      "x": "1.004654683",
      "y": "0.850836670"
    },
    {
      "turn": "0.000000000",
      "x": "0.133592747",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.364982172",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.596371597",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.827761022",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "1.396552864",
      "y": "1.012255193"
    },
    {
      "turn": "0.000000000",
      "x": "1.627942289",
      "y": "1.012255193"
    },
    {
      "turn": "0.000000000",
      "x": "1.859331714",
      "y": "1.012255193"
    },
    {
      "turn": "0.000000000",
      "x": "1.165221711",
      "y": "1.017447880"
    },
    {
      "turn": "0.000000000",
      "x": "0.988328051",
      "y": "1.166611211"
    },
    {
      "turn": "0.000000000",
      "x": "0.249287460",
      "y": "1.200389120"
    },
    {
      "turn": "0.000000000",
      "x": "0.480676885",
      "y": "1.200389120"
    },
    {
      "turn": "0.000000000",
      "x": "1.512247576",
      "y": "1.212644313"
    },
    {
      "turn": "0.000000000",
      "x": "1.743637001",
      "y": "1.212644313"
    },
    {
      "turn": "0.000000000",
      "x": "0.750096969",
      "y": "1.225032094"
    },
    {
      "turn": "0.000000000",
      "x": "1.251783926",
      "y": "1.232177290"
    },
    {
      "turn": "0.000000000",
      "x": "0.943942735",
      "y": "1.393703724"
    },
    {
      "turn": "0.000000000",
      "x": "0.364982172",
      "y": "1.400778241"
    },
    {
      "turn": "0.000000000",
      "x": "0.596371597",
      "y": "1.400778241"
    },
    {
      "turn": "0.000000000",
      "x": "1.396552864",
      "y": "1.413033434"
    },
    {
      "turn": "0.000000000",
      "x": "1.627942289",
      "y": "1.413033434"
    },
    {
      "turn": "0.000000000",
      "x": "1.168231220",
      "y": "1.450587305"
    },
    {
      "turn": "0.000000000",
      "x": "0.773265258",
      "y": "1.549941571"
    },
    {
      "turn": "0.000000000",
      "x": "0.480676885",
      "y": "1.601167361"
    },
    {
      "turn": "0.000000000",
      "x": "1.512247576",
      "y": "1.613422554"
    },
    {
      "turn": "0.000000000",
      "x": "1.107333698",
      "y": "1.673819376"
    },
    {
      "turn": "0.000000000",
      "x": "0.657570545",
      "y": "1.750330691"
    },
    {
      "turn": "0.000000000",
      "x": "0.888959970",
      "y": "1.750330691"
    },
    {
      "turn": "0.000000000",
      "x": "1.325707425",
      "y": "1.750330691"
    }
  ],
  "radius": "0.115694712"
}

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