P87 · Packing and covering · Classic · Medium

Equal discs covering a disc · n = 98

Cover the largest possible disc with n unit-radius discs. Maximise the covered disc radius.

Instancen = 98

ObjectiveMaximize covered disc radius

Record comparison

covered disc radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	8.578781052	int3	Record details ↓

Formal definition

ContainerThe editor and certificate normalise the target disc to the closed unit disc centred at (1,1), while public scores are reported in the literature's unit-covering-disc scale.
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.
ConstraintsThe n distinct centres lie inside the target disc. Covering discs may overlap and protrude, and their union must cover every point of the target disc.
PrecisionInput decimals are parsed as exact rationals, with zero floating-point tolerance in feasibility checks.
ObjectiveCovered disc radius R = 1/r, where larger is better.

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.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderint3

Solution methodAI · Google: Gemini 3.8 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 (6 changes)

int3AI · Google: Gemini 3.8 Flash
8.578739692→8.578781052
2026-10-13
int3AI · Google: Gemini 3.8 Flash
8.577613913→8.578739692
2026-10-13
matrixfangAI · OpenAI: GPT-6.1 Sol
8.569254331→8.577613913
2026-10-12
pikachu345AI · OpenAI: GPT-6 Astra
8.567836224→8.569254331
2026-10-05
pikachu345AI · OpenAI: GPT-6 Astra
8.54878991→8.567836224
2026-10-05
pikachu345AI · OpenAI: GPT-6 Astra
8.520357378→8.54878991
2026-10-05

ANSWER FORMAT

How to write your answer

the closed unit disc centred at (1,1)

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": "1.728531810",
      "y": "1.372122035"
    },
    {
      "turn": "0",
      "x": "1.733540467",
      "y": "1.170584147"
    },
    {
      "turn": "0",
      "x": "1.735875520",
      "y": "0.993037261"
    },
    {
      "turn": "0",
      "x": "1.715040869",
      "y": "0.825994874"
    },
    {
      "turn": "0",
      "x": "1.669166185",
      "y": "0.654461025"
    },
    {
      "turn": "0",
      "x": "1.565784342",
      "y": "1.505254636"
    },
    {
      "turn": "0",
      "x": "1.559306091",
      "y": "1.297685543"
    },
    {
      "turn": "0",
      "x": "1.549234897",
      "y": "1.105998473"
    },
    {
      "turn": "0",
      "x": "1.537882882",
      "y": "0.932911726"
    },
    {
      "turn": "0",
      "x": "1.506371463",
      "y": "0.762339349"
    },
    {
      "turn": "0",
      "x": "1.469049615",
      "y": "0.574051176"
    },
    {
      "turn": "0",
      "x": "1.424343838",
      "y": "0.371250117"
    },
    {
      "turn": "0",
      "x": "1.414631954",
      "y": "1.625288782"
    },
    {
      "turn": "0",
      "x": "1.394406707",
      "y": "1.420971881"
    },
    {
      "turn": "0",
      "x": "1.373014897",
      "y": "1.223162301"
    },
    {
      "turn": "0",
      "x": "1.352376976",
      "y": "1.040670497"
    },
    {
      "turn": "0",
      "x": "1.331591210",
      "y": "0.874020505"
    },
    {
      "turn": "0",
      "x": "1.306771094",
      "y": "0.692050407"
    },
    {
      "turn": "0",
      "x": "1.278919145",
      "y": "0.495046593"
    },
    {
      "turn": "0",
      "x": "1.248338070",
      "y": "0.292021397"
    },
    {
      "turn": "0",
      "x": "1.254666881",
      "y": "1.751255939"
    },
    {
      "turn": "0",
      "x": "1.234459525",
      "y": "1.546946496"
    },
    {
      "turn": "0",
      "x": "1.208198454",
      "y": "1.346975923"
    },
    {
      "turn": "0",
      "x": "1.178572907",
      "y": "1.156133570"
    },
    {
      "turn": "0",
      "x": "1.146085298",
      "y": "0.981779237"
    },
    {
      "turn": "0",
      "x": "1.134750583",
      "y": "0.804786433"
    },
    {
      "turn": "0",
      "x": "1.116591469",
      "y": "0.612513911"
    },
    {
      "turn": "0",
      "x": "1.092928799",
      "y": "0.412219259"
    },
    {
      "turn": "0",
      "x": "1.062332256",
      "y": "0.209205654"
    },
    {
      "turn": "0",
      "x": "1.077069663",
      "y": "1.684933232"
    },
    {
      "turn": "0",
      "x": "1.050826484",
      "y": "1.484970117"
    },
    {
      "turn": "0",
      "x": "1.016331675",
      "y": "1.291966771"
    },
    {
      "turn": "0",
      "x": "0.984413932",
      "y": "1.103312423"
    },
    {
      "turn": "0",
      "x": "0.959514564",
      "y": "0.903681058"
    },
    {
      "turn": "0",
      "x": "0.944114909",
      "y": "0.712966497"
    },
    {
      "turn": "0",
      "x": "0.930145073",
      "y": "0.517403137"
    },
    {
      "turn": "0",
      "x": "0.906466935",
      "y": "0.317120076"
    },
    {
      "turn": "0",
      "x": "0.919328366",
      "y": "1.783621358"
    },
    {
      "turn": "0",
      "x": "0.889494826",
      "y": "1.587954182"
    },
    {
      "turn": "0",
      "x": "0.854954229",
      "y": "1.397038619"
    },
    {
      "turn": "0",
      "x": "0.819140106",
      "y": "1.223987308"
    },
    {
      "turn": "0",
      "x": "0.799253649",
      "y": "1.025038350"
    },
    {
      "turn": "0",
      "x": "0.769662572",
      "y": "0.827299816"
    },
    {
      "turn": "0",
      "x": "0.761872525",
      "y": "0.650753158"
    },
    {
      "turn": "0",
      "x": "0.748459955",
      "y": "0.457202376"
    },
    {
      "turn": "0",
      "x": "0.729317418",
      "y": "0.260201744"
    },
    {
      "turn": "0",
      "x": "0.727414948",
      "y": "1.705425325"
    },
    {
      "turn": "0",
      "x": "0.692518733",
      "y": "1.514356433"
    },
    {
      "turn": "0",
      "x": "0.646925261",
      "y": "1.340348415"
    },
    {
      "turn": "0",
      "x": "0.633875825",
      "y": "1.144902581"
    },
    {
      "turn": "0",
      "x": "0.609499980",
      "y": "0.949468560"
    },
    {
      "turn": "0",
      "x": "0.574141845",
      "y": "0.756805200"
    },
    {
      "turn": "0",
      "x": "0.575596549",
      "y": "0.576929028"
    },
    {
      "turn": "0",
      "x": "0.562491062",
      "y": "0.383142247"
    },
    {
      "turn": "0",
      "x": "0.530874402",
      "y": "1.635901288"
    },
    {
      "turn": "0",
      "x": "0.482321779",
      "y": "1.460574235"
    },
    {
      "turn": "0",
      "x": "0.461660979",
      "y": "1.261263682"
    },
    {
      "turn": "0",
      "x": "0.444122155",
      "y": "1.069332785"
    },
    {
      "turn": "0",
      "x": "0.413979252",
      "y": "0.878973939"
    },
    {
      "turn": "0",
      "x": "0.385049997",
      "y": "0.680694667"
    },
    {
      "turn": "0",
      "x": "0.389049219",
      "y": "0.498812990"
    },
    {
      "turn": "0",
      "x": "1.614806882",
      "y": "0.460327944"
    },
    {
      "turn": "0",
      "x": "0.302140687",
      "y": "1.383592099"
    },
    {
      "turn": "0",
      "x": "0.271893789",
      "y": "1.185585072"
    },
    {
      "turn": "0",
      "x": "0.248615044",
      "y": "0.998946923"
    },
    {
      "turn": "0",
      "x": "0.229298373",
      "y": "0.799576601"
    },
    {
      "turn": "0",
      "x": "1.595294205",
      "y": "1.713610004"
    },
    {
      "turn": "0",
      "x": "1.447464479",
      "y": "1.819783169"
    },
    {
      "turn": "0",
      "x": "1.274501645",
      "y": "1.939356261"
    },
    {
      "turn": "0",
      "x": "1.100511780",
      "y": "1.890534905"
    },
    {
      "turn": "0",
      "x": "0.942275890",
      "y": "1.974422162"
    },
    {
      "turn": "0",
      "x": "0.749176613",
      "y": "1.897255266"
    },
    {
      "turn": "0",
      "x": "0.578864035",
      "y": "1.831685730"
    },
    {
      "turn": "0",
      "x": "0.382029336",
      "y": "1.759566987"
    },
    {
      "turn": "0",
      "x": "0.321440009",
      "y": "1.587287143"
    },
    {
      "turn": "0",
      "x": "0.162602559",
      "y": "1.501339864"
    },
    {
      "turn": "0",
      "x": "0.111542638",
      "y": "1.297072491"
    },
    {
      "turn": "0",
      "x": "0.076237840",
      "y": "1.115217752"
    },
    {
      "turn": "0",
      "x": "0.065791647",
      "y": "0.930262474"
    },
    {
      "turn": "0",
      "x": "0.065112634",
      "y": "0.719711244"
    },
    {
      "turn": "0",
      "x": "0.197972422",
      "y": "0.597381577"
    },
    {
      "turn": "0",
      "x": "0.214387991",
      "y": "0.415497160"
    },
    {
      "turn": "0",
      "x": "0.387477641",
      "y": "0.297239017"
    },
    {
      "turn": "0",
      "x": "0.536466385",
      "y": "0.191844094"
    },
    {
      "turn": "0",
      "x": "0.704695027",
      "y": "0.069609898"
    },
    {
      "turn": "0",
      "x": "0.878688647",
      "y": "0.112059264"
    },
    {
      "turn": "0",
      "x": "1.035356172",
      "y": "0.021995983"
    },
    {
      "turn": "0",
      "x": "1.232390812",
      "y": "0.095421027"
    },
    {
      "turn": "0",
      "x": "1.401770909",
      "y": "0.162029552"
    },
    {
      "turn": "0",
      "x": "1.554300108",
      "y": "0.254386889"
    },
    {
      "turn": "0",
      "x": "1.740312065",
      "y": "0.351932867"
    },
    {
      "turn": "0",
      "x": "1.814664917",
      "y": "0.549417266"
    },
    {
      "turn": "0",
      "x": "1.881452952",
      "y": "0.714080134"
    },
    {
      "turn": "0",
      "x": "1.921152719",
      "y": "0.885107778"
    },
    {
      "turn": "0",
      "x": "1.924679099",
      "y": "1.060647227"
    },
    {
      "turn": "0",
      "x": "1.900383617",
      "y": "1.236670586"
    },
    {
      "turn": "0",
      "x": "1.876817267",
      "y": "1.446368117"
    },
    {
      "turn": "0",
      "x": "1.720464525",
      "y": "1.586616109"
    }
  ],
  "radius": "0.116566677"
}

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

Instance parameters

{
  "n": 98,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "1.728531810",
      "y": "1.372122035"
    },
    {
      "turn": "0",
      "x": "1.733540467",
      "y": "1.170584147"
    },
    {
      "turn": "0",
      "x": "1.735875520",
      "y": "0.993037261"
    },
    {
      "turn": "0",
      "x": "1.715040869",
      "y": "0.825994874"
    },
    {
      "turn": "0",
      "x": "1.669166185",
      "y": "0.654461025"
    },
    {
      "turn": "0",
      "x": "1.565784342",
      "y": "1.505254636"
    },
    {
      "turn": "0",
      "x": "1.559306091",
      "y": "1.297685543"
    },
    {
      "turn": "0",
      "x": "1.549234897",
      "y": "1.105998473"
    },
    {
      "turn": "0",
      "x": "1.537882882",
      "y": "0.932911726"
    },
    {
      "turn": "0",
      "x": "1.506371463",
      "y": "0.762339349"
    },
    {
      "turn": "0",
      "x": "1.469049615",
      "y": "0.574051176"
    },
    {
      "turn": "0",
      "x": "1.424343838",
      "y": "0.371250117"
    },
    {
      "turn": "0",
      "x": "1.414631954",
      "y": "1.625288782"
    },
    {
      "turn": "0",
      "x": "1.394406707",
      "y": "1.420971881"
    },
    {
      "turn": "0",
      "x": "1.373014897",
      "y": "1.223162301"
    },
    {
      "turn": "0",
      "x": "1.352376976",
      "y": "1.040670497"
    },
    {
      "turn": "0",
      "x": "1.331591210",
      "y": "0.874020505"
    },
    {
      "turn": "0",
      "x": "1.306771094",
      "y": "0.692050407"
    },
    {
      "turn": "0",
      "x": "1.278919145",
      "y": "0.495046593"
    },
    {
      "turn": "0",
      "x": "1.248338070",
      "y": "0.292021397"
    },
    {
      "turn": "0",
      "x": "1.254666881",
      "y": "1.751255939"
    },
    {
      "turn": "0",
      "x": "1.234459525",
      "y": "1.546946496"
    },
    {
      "turn": "0",
      "x": "1.208198454",
      "y": "1.346975923"
    },
    {
      "turn": "0",
      "x": "1.178572907",
      "y": "1.156133570"
    },
    {
      "turn": "0",
      "x": "1.146085298",
      "y": "0.981779237"
    },
    {
      "turn": "0",
      "x": "1.134750583",
      "y": "0.804786433"
    },
    {
      "turn": "0",
      "x": "1.116591469",
      "y": "0.612513911"
    },
    {
      "turn": "0",
      "x": "1.092928799",
      "y": "0.412219259"
    },
    {
      "turn": "0",
      "x": "1.062332256",
      "y": "0.209205654"
    },
    {
      "turn": "0",
      "x": "1.077069663",
      "y": "1.684933232"
    },
    {
      "turn": "0",
      "x": "1.050826484",
      "y": "1.484970117"
    },
    {
      "turn": "0",
      "x": "1.016331675",
      "y": "1.291966771"
    },
    {
      "turn": "0",
      "x": "0.984413932",
      "y": "1.103312423"
    },
    {
      "turn": "0",
      "x": "0.959514564",
      "y": "0.903681058"
    },
    {
      "turn": "0",
      "x": "0.944114909",
      "y": "0.712966497"
    },
    {
      "turn": "0",
      "x": "0.930145073",
      "y": "0.517403137"
    },
    {
      "turn": "0",
      "x": "0.906466935",
      "y": "0.317120076"
    },
    {
      "turn": "0",
      "x": "0.919328366",
      "y": "1.783621358"
    },
    {
      "turn": "0",
      "x": "0.889494826",
      "y": "1.587954182"
    },
    {
      "turn": "0",
      "x": "0.854954229",
      "y": "1.397038619"
    },
    {
      "turn": "0",
      "x": "0.819140106",
      "y": "1.223987308"
    },
    {
      "turn": "0",
      "x": "0.799253649",
      "y": "1.025038350"
    },
    {
      "turn": "0",
      "x": "0.769662572",
      "y": "0.827299816"
    },
    {
      "turn": "0",
      "x": "0.761872525",
      "y": "0.650753158"
    },
    {
      "turn": "0",
      "x": "0.748459955",
      "y": "0.457202376"
    },
    {
      "turn": "0",
      "x": "0.729317418",
      "y": "0.260201744"
    },
    {
      "turn": "0",
      "x": "0.727414948",
      "y": "1.705425325"
    },
    {
      "turn": "0",
      "x": "0.692518733",
      "y": "1.514356433"
    },
    {
      "turn": "0",
      "x": "0.646925261",
      "y": "1.340348415"
    },
    {
      "turn": "0",
      "x": "0.633875825",
      "y": "1.144902581"
    },
    {
      "turn": "0",
      "x": "0.609499980",
      "y": "0.949468560"
    },
    {
      "turn": "0",
      "x": "0.574141845",
      "y": "0.756805200"
    },
    {
      "turn": "0",
      "x": "0.575596549",
      "y": "0.576929028"
    },
    {
      "turn": "0",
      "x": "0.562491062",
      "y": "0.383142247"
    },
    {
      "turn": "0",
      "x": "0.530874402",
      "y": "1.635901288"
    },
    {
      "turn": "0",
      "x": "0.482321779",
      "y": "1.460574235"
    },
    {
      "turn": "0",
      "x": "0.461660979",
      "y": "1.261263682"
    },
    {
      "turn": "0",
      "x": "0.444122155",
      "y": "1.069332785"
    },
    {
      "turn": "0",
      "x": "0.413979252",
      "y": "0.878973939"
    },
    {
      "turn": "0",
      "x": "0.385049997",
      "y": "0.680694667"
    },
    {
      "turn": "0",
      "x": "0.389049219",
      "y": "0.498812990"
    },
    {
      "turn": "0",
      "x": "1.614806882",
      "y": "0.460327944"
    },
    {
      "turn": "0",
      "x": "0.302140687",
      "y": "1.383592099"
    },
    {
      "turn": "0",
      "x": "0.271893789",
      "y": "1.185585072"
    },
    {
      "turn": "0",
      "x": "0.248615044",
      "y": "0.998946923"
    },
    {
      "turn": "0",
      "x": "0.229298373",
      "y": "0.799576601"
    },
    {
      "turn": "0",
      "x": "1.595294205",
      "y": "1.713610004"
    },
    {
      "turn": "0",
      "x": "1.447464479",
      "y": "1.819783169"
    },
    {
      "turn": "0",
      "x": "1.274501645",
      "y": "1.939356261"
    },
    {
      "turn": "0",
      "x": "1.100511780",
      "y": "1.890534905"
    },
    {
      "turn": "0",
      "x": "0.942275890",
      "y": "1.974422162"
    },
    {
      "turn": "0",
      "x": "0.749176613",
      "y": "1.897255266"
    },
    {
      "turn": "0",
      "x": "0.578864035",
      "y": "1.831685730"
    },
    {
      "turn": "0",
      "x": "0.382029336",
      "y": "1.759566987"
    },
    {
      "turn": "0",
      "x": "0.321440009",
      "y": "1.587287143"
    },
    {
      "turn": "0",
      "x": "0.162602559",
      "y": "1.501339864"
    },
    {
      "turn": "0",
      "x": "0.111542638",
      "y": "1.297072491"
    },
    {
      "turn": "0",
      "x": "0.076237840",
      "y": "1.115217752"
    },
    {
      "turn": "0",
      "x": "0.065791647",
      "y": "0.930262474"
    },
    {
      "turn": "0",
      "x": "0.065112634",
      "y": "0.719711244"
    },
    {
      "turn": "0",
      "x": "0.197972422",
      "y": "0.597381577"
    },
    {
      "turn": "0",
      "x": "0.214387991",
      "y": "0.415497160"
    },
    {
      "turn": "0",
      "x": "0.387477641",
      "y": "0.297239017"
    },
    {
      "turn": "0",
      "x": "0.536466385",
      "y": "0.191844094"
    },
    {
      "turn": "0",
      "x": "0.704695027",
      "y": "0.069609898"
    },
    {
      "turn": "0",
      "x": "0.878688647",
      "y": "0.112059264"
    },
    {
      "turn": "0",
      "x": "1.035356172",
      "y": "0.021995983"
    },
    {
      "turn": "0",
      "x": "1.232390812",
      "y": "0.095421027"
    },
    {
      "turn": "0",
      "x": "1.401770909",
      "y": "0.162029552"
    },
    {
      "turn": "0",
      "x": "1.554300108",
      "y": "0.254386889"
    },
    {
      "turn": "0",
      "x": "1.740312065",
      "y": "0.351932867"
    },
    {
      "turn": "0",
      "x": "1.814664917",
      "y": "0.549417266"
    },
    {
      "turn": "0",
      "x": "1.881452952",
      "y": "0.714080134"
    },
    {
      "turn": "0",
      "x": "1.921152719",
      "y": "0.885107778"
    },
    {
      "turn": "0",
      "x": "1.924679099",
      "y": "1.060647227"
    },
    {
      "turn": "0",
      "x": "1.900383617",
      "y": "1.236670586"
    },
    {
      "turn": "0",
      "x": "1.876817267",
      "y": "1.446368117"
    },
    {
      "turn": "0",
      "x": "1.720464525",
      "y": "1.586616109"
    }
  ],
  "radius": "0.116566677"
}

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.

No threads yet. Start the first discussion.