P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 6 · n = 49

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 = 49

ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[0.11644161311, 0.116441613112]	Paolo Amore	Packomania ↗
Site record	0.116441612	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
0.11644161→0.116441612
2026-11-01
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.058441558→0.11644161
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": "0.666383727",
      "y": "0.250416209"
    },
    {
      "turn": "0.000000000",
      "x": "0.899266953",
      "y": "0.250416209"
    },
    {
      "turn": "0.000000000",
      "x": "1.132150180",
      "y": "0.250416209"
    },
    {
      "turn": "0.000000000",
      "x": "1.365033406",
      "y": "0.250416209"
    },
    {
      "turn": "0.000000000",
      "x": "0.483780033",
      "y": "0.394951630"
    },
    {
      "turn": "0.000000000",
      "x": "1.528524353",
      "y": "0.416263450"
    },
    {
      "turn": "0.000000000",
      "x": "0.782825340",
      "y": "0.452098999"
    },
    {
      "turn": "0.000000000",
      "x": "1.015708567",
      "y": "0.452098999"
    },
    {
      "turn": "0.000000000",
      "x": "1.248591793",
      "y": "0.452098999"
    },
    {
      "turn": "0.000000000",
      "x": "0.367338420",
      "y": "0.596634420"
    },
    {
      "turn": "0.000000000",
      "x": "0.600221646",
      "y": "0.596634420"
    },
    {
      "turn": "0.000000000",
      "x": "1.412082740",
      "y": "0.617946240"
    },
    {
      "turn": "0.000000000",
      "x": "1.644965966",
      "y": "0.617946240"
    },
    {
      "turn": "0.000000000",
      "x": "0.899266953",
      "y": "0.653781789"
    },
    {
      "turn": "0.000000000",
      "x": "1.157460483",
      "y": "0.667900635"
    },
    {
      "turn": "0.000000000",
      "x": "0.250896806",
      "y": "0.798317210"
    },
    {
      "turn": "0.000000000",
      "x": "0.483780033",
      "y": "0.798317210"
    },
    {
      "turn": "0.000000000",
      "x": "0.716663259",
      "y": "0.798317210"
    },
    {
      "turn": "0.000000000",
      "x": "1.528524353",
      "y": "0.819629030"
    },
    {
      "turn": "0.000000000",
      "x": "1.761407579",
      "y": "0.819629030"
    },
    {
      "turn": "0.000000000",
      "x": "1.015708567",
      "y": "0.855464579"
    },
    {
      "turn": "0.000000000",
      "x": "1.296840451",
      "y": "0.856285352"
    },
    {
      "turn": "0.000000000",
      "x": "0.134455193",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.367338420",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.600221646",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.833104872",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "1.611074050",
      "y": "1.037390700"
    },
    {
      "turn": "0.000000000",
      "x": "1.843957276",
      "y": "1.037390700"
    },
    {
      "turn": "0.000000000",
      "x": "1.154939936",
      "y": "1.042144045"
    },
    {
      "turn": "0.000000000",
      "x": "1.383497345",
      "y": "1.086821867"
    },
    {
      "turn": "0.000000000",
      "x": "0.972336241",
      "y": "1.186679466"
    },
    {
      "turn": "0.000000000",
      "x": "0.250896806",
      "y": "1.201682790"
    },
    {
      "turn": "0.000000000",
      "x": "0.483780033",
      "y": "1.201682790"
    },
    {
      "turn": "0.000000000",
      "x": "0.716663259",
      "y": "1.201682790"
    },
    {
      "turn": "0.000000000",
      "x": "1.727515663",
      "y": "1.239073490"
    },
    {
      "turn": "0.000000000",
      "x": "1.179408963",
      "y": "1.298294291"
    },
    {
      "turn": "0.000000000",
      "x": "1.413301703",
      "y": "1.317790041"
    },
    {
      "turn": "0.000000000",
      "x": "0.855894628",
      "y": "1.388362256"
    },
    {
      "turn": "0.000000000",
      "x": "0.367338420",
      "y": "1.403365580"
    },
    {
      "turn": "0.000000000",
      "x": "0.622446667",
      "y": "1.416021047"
    },
    {
      "turn": "0.000000000",
      "x": "1.611074050",
      "y": "1.440756280"
    },
    {
      "turn": "0.000000000",
      "x": "1.059927778",
      "y": "1.500634561"
    },
    {
      "turn": "0.000000000",
      "x": "1.292253299",
      "y": "1.516741997"
    },
    {
      "turn": "0.000000000",
      "x": "0.483780033",
      "y": "1.605048370"
    },
    {
      "turn": "0.000000000",
      "x": "0.860680534",
      "y": "1.621196300"
    },
    {
      "turn": "0.000000000",
      "x": "1.494632437",
      "y": "1.642439070"
    },
    {
      "turn": "0.000000000",
      "x": "0.666383727",
      "y": "1.749583791"
    },
    {
      "turn": "0.000000000",
      "x": "1.054977342",
      "y": "1.749583791"
    },
    {
      "turn": "0.000000000",
      "x": "1.287860568",
      "y": "1.749583791"
    }
  ],
  "radius": "0.116441612"
}

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

Instance parameters

{
  "n": 49,
  "sides": 6
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.666383727",
      "y": "0.250416209"
    },
    {
      "turn": "0.000000000",
      "x": "0.899266953",
      "y": "0.250416209"
    },
    {
      "turn": "0.000000000",
      "x": "1.132150180",
      "y": "0.250416209"
    },
    {
      "turn": "0.000000000",
      "x": "1.365033406",
      "y": "0.250416209"
    },
    {
      "turn": "0.000000000",
      "x": "0.483780033",
      "y": "0.394951630"
    },
    {
      "turn": "0.000000000",
      "x": "1.528524353",
      "y": "0.416263450"
    },
    {
      "turn": "0.000000000",
      "x": "0.782825340",
      "y": "0.452098999"
    },
    {
      "turn": "0.000000000",
      "x": "1.015708567",
      "y": "0.452098999"
    },
    {
      "turn": "0.000000000",
      "x": "1.248591793",
      "y": "0.452098999"
    },
    {
      "turn": "0.000000000",
      "x": "0.367338420",
      "y": "0.596634420"
    },
    {
      "turn": "0.000000000",
      "x": "0.600221646",
      "y": "0.596634420"
    },
    {
      "turn": "0.000000000",
      "x": "1.412082740",
      "y": "0.617946240"
    },
    {
      "turn": "0.000000000",
      "x": "1.644965966",
      "y": "0.617946240"
    },
    {
      "turn": "0.000000000",
      "x": "0.899266953",
      "y": "0.653781789"
    },
    {
      "turn": "0.000000000",
      "x": "1.157460483",
      "y": "0.667900635"
    },
    {
      "turn": "0.000000000",
      "x": "0.250896806",
      "y": "0.798317210"
    },
    {
      "turn": "0.000000000",
      "x": "0.483780033",
      "y": "0.798317210"
    },
    {
      "turn": "0.000000000",
      "x": "0.716663259",
      "y": "0.798317210"
    },
    {
      "turn": "0.000000000",
      "x": "1.528524353",
      "y": "0.819629030"
    },
    {
      "turn": "0.000000000",
      "x": "1.761407579",
      "y": "0.819629030"
    },
    {
      "turn": "0.000000000",
      "x": "1.015708567",
      "y": "0.855464579"
    },
    {
      "turn": "0.000000000",
      "x": "1.296840451",
      "y": "0.856285352"
    },
    {
      "turn": "0.000000000",
      "x": "0.134455193",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.367338420",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.600221646",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.833104872",
      "y": "1.000000000"
    },
    {
      "turn": "0.000000000",
      "x": "1.611074050",
      "y": "1.037390700"
    },
    {
      "turn": "0.000000000",
      "x": "1.843957276",
      "y": "1.037390700"
    },
    {
      "turn": "0.000000000",
      "x": "1.154939936",
      "y": "1.042144045"
    },
    {
      "turn": "0.000000000",
      "x": "1.383497345",
      "y": "1.086821867"
    },
    {
      "turn": "0.000000000",
      "x": "0.972336241",
      "y": "1.186679466"
    },
    {
      "turn": "0.000000000",
      "x": "0.250896806",
      "y": "1.201682790"
    },
    {
      "turn": "0.000000000",
      "x": "0.483780033",
      "y": "1.201682790"
    },
    {
      "turn": "0.000000000",
      "x": "0.716663259",
      "y": "1.201682790"
    },
    {
      "turn": "0.000000000",
      "x": "1.727515663",
      "y": "1.239073490"
    },
    {
      "turn": "0.000000000",
      "x": "1.179408963",
      "y": "1.298294291"
    },
    {
      "turn": "0.000000000",
      "x": "1.413301703",
      "y": "1.317790041"
    },
    {
      "turn": "0.000000000",
      "x": "0.855894628",
      "y": "1.388362256"
    },
    {
      "turn": "0.000000000",
      "x": "0.367338420",
      "y": "1.403365580"
    },
    {
      "turn": "0.000000000",
      "x": "0.622446667",
      "y": "1.416021047"
    },
    {
      "turn": "0.000000000",
      "x": "1.611074050",
      "y": "1.440756280"
    },
    {
      "turn": "0.000000000",
      "x": "1.059927778",
      "y": "1.500634561"
    },
    {
      "turn": "0.000000000",
      "x": "1.292253299",
      "y": "1.516741997"
    },
    {
      "turn": "0.000000000",
      "x": "0.483780033",
      "y": "1.605048370"
    },
    {
      "turn": "0.000000000",
      "x": "0.860680534",
      "y": "1.621196300"
    },
    {
      "turn": "0.000000000",
      "x": "1.494632437",
      "y": "1.642439070"
    },
    {
      "turn": "0.000000000",
      "x": "0.666383727",
      "y": "1.749583791"
    },
    {
      "turn": "0.000000000",
      "x": "1.054977342",
      "y": "1.749583791"
    },
    {
      "turn": "0.000000000",
      "x": "1.287860568",
      "y": "1.749583791"
    }
  ],
  "radius": "0.116441612"
}

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