P02 · Packing and covering · Classic · Weak baseline · Hard

Equal-circle packing in a unit circle · n = 78

Also known ascircles in a circlepacking equal circles in a circlecci

Place n non-overlapping circles of one common radius inside a circle of radius 1, making that radius as large as possible.

Instancen = 78

ObjectiveMaximize common radius

Record comparison

common radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[0.1014434397193697258715657384075, 0.1014434397193697258715657384085]	See source attribution	packomania.com ↗
Site record	No player record yet	—	—

Formal definition

ContainerA circle of radius 1 centred at (1, 1), so both coordinates run from 0 to 2
SubmissionExactly n circles: one shared radius and n centres
ConstraintsEvery circle lies wholly inside the container; no two overlap in their interiors, tangency allowed
ObjectiveMake the common radius as large as possible

Challenge this record ↗

Layout shown: starter layout

2y0

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778

0x2

VERIFIED CONSTRUCTIONr = 0.101443439

Getting a feel for it

Where the room for improvement is

Optimal packings are jammed contact structures: circles brace against each other and the boundary, with tilted rows, offsets, and the odd rattler touching nothing. Neat grids are almost never optimal.

Where the frontier is

The optima for n = 1..14 and 19 are proved; optimality remains open for the other sizes. This site imports Packomania's public constructions through n = 300 and presents the algorithm-scale cases in one table.

Source

LAYOUT & HISTORY

Starter layout

Inspect the layout, solver notes and record history.

Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.

ANSWER FORMAT

How to write your answer

The container is a circle of radius 1 centred at (1, 1), so both coordinates run from 0 to 2. Coordinates and radii share one unit and are written as plain decimals such as "0.5", to at most nine decimal places.

Submit radius and centers. Write every number as a decimal string, for example "0.2".

Starter answer

{
  "centers": [
    [
      "1.028559715",
      "0.101897425"
    ],
    [
      "1.229319633",
      "0.131198298"
    ],
    [
      "0.679449819",
      "0.160564789"
    ],
    [
      "1.418388371",
      "0.204792441"
    ],
    [
      "0.870438429",
      "0.229022344"
    ],
    [
      "0.499294967",
      "0.253877784"
    ],
    [
      "1.059592481",
      "0.302396935"
    ],
    [
      "1.586126824",
      "0.318927876"
    ],
    [
      "1.26031201",
      "0.331973217"
    ],
    [
      "0.676077591",
      "0.363423641"
    ],
    [
      "0.344667071",
      "0.3852296"
    ],
    [
      "0.891017558",
      "0.43094961"
    ],
    [
      "1.428025112",
      "0.446145898"
    ],
    [
      "0.518025059",
      "0.49063403"
    ],
    [
      "1.080213856",
      "0.504233118"
    ],
    [
      "1.617218807",
      "0.519418211"
    ],
    [
      "0.223449376",
      "0.547923657"
    ],
    [
      "0.73296763",
      "0.558171194"
    ],
    [
      "1.270654923",
      "0.574199438"
    ],
    [
      "1.803970546",
      "0.59870802"
    ],
    [
      "0.922161324",
      "0.631443507"
    ],
    [
      "1.459848617",
      "0.647471751"
    ],
    [
      "0.574915098",
      "0.685381583"
    ],
    [
      "0.37202942",
      "0.68607957"
    ],
    [
      "1.111355019",
      "0.70471582"
    ],
    [
      "0.141821804",
      "0.733665481"
    ],
    [
      "1.668540411",
      "0.74977676"
    ],
    [
      "0.764108793",
      "0.758653896"
    ],
    [
      "1.301796085",
      "0.77468214"
    ],
    [
      "1.873505762",
      "0.789306917"
    ],
    [
      "0.953302487",
      "0.831926209"
    ],
    [
      "1.49098978",
      "0.847954453"
    ],
    [
      "0.493287527",
      "0.871123407"
    ],
    [
      "0.290401848",
      "0.871821395"
    ],
    [
      "1.142496182",
      "0.905198522"
    ],
    [
      "0.682481221",
      "0.94439572"
    ],
    [
      "0.102876913",
      "0.949264823"
    ],
    [
      "1.699681573",
      "0.950259462"
    ],
    [
      "1.332937248",
      "0.975164842"
    ],
    [
      "1.898507886",
      "0.990647374"
    ],
    [
      "0.885930561",
      "1.023300474"
    ],
    [
      "1.522130943",
      "1.048437155"
    ],
    [
      "0.504773744",
      "1.073684886"
    ],
    [
      "0.301888065",
      "1.074382874"
    ],
    [
      "1.075124255",
      "1.096572787"
    ],
    [
      "0.722354871",
      "1.143325803"
    ],
    [
      "0.11436313",
      "1.151826302"
    ],
    [
      "1.267384581",
      "1.167169878"
    ],
    [
      "1.675100257",
      "1.181716842"
    ],
    [
      "1.877702257",
      "1.192464647"
    ],
    [
      "0.916901406",
      "1.225429861"
    ],
    [
      "1.456578275",
      "1.24044219"
    ],
    [
      "0.561411522",
      "1.268505952"
    ],
    [
      "0.358525843",
      "1.26920394"
    ],
    [
      "1.110883014",
      "1.29628357"
    ],
    [
      "0.749577703",
      "1.344378043"
    ],
    [
      "0.171000908",
      "1.346647368"
    ],
    [
      "1.609547589",
      "1.373721877"
    ],
    [
      "1.313065858",
      "1.384026186"
    ],
    [
      "1.812149589",
      "1.384469682"
    ],
    [
      "0.95414844",
      "1.425114307"
    ],
    [
      "0.457427675",
      "1.446352221"
    ],
    [
      "1.144086418",
      "1.496435062"
    ],
    [
      "1.469457804",
      "1.52047997"
    ],
    [
      "0.26990274",
      "1.523795649"
    ],
    [
      "0.789270442",
      "1.543344303"
    ],
    [
      "1.705191891",
      "1.556873674"
    ],
    [
      "0.593551343",
      "1.596796339"
    ],
    [
      "1.297881714",
      "1.628760769"
    ],
    [
      "0.406026408",
      "1.674239768"
    ],
    [
      "1.106386643",
      "1.695788549"
    ],
    [
      "1.562282081",
      "1.700887119"
    ],
    [
      "0.904044532",
      "1.710646407"
    ],
    [
      "0.590176204",
      "1.799655143"
    ],
    [
      "1.390705992",
      "1.809167918"
    ],
    [
      "0.780024432",
      "1.871214463"
    ],
    [
      "1.19921092",
      "1.876195698"
    ],
    [
      "0.981231848",
      "1.898355032"
    ]
  ],
  "radius": "0.101443439"
}

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

Instance parameters

{
  "n": 78
}

Starter answer

{
  "centers": [
    [
      "1.028559715",
      "0.101897425"
    ],
    [
      "1.229319633",
      "0.131198298"
    ],
    [
      "0.679449819",
      "0.160564789"
    ],
    [
      "1.418388371",
      "0.204792441"
    ],
    [
      "0.870438429",
      "0.229022344"
    ],
    [
      "0.499294967",
      "0.253877784"
    ],
    [
      "1.059592481",
      "0.302396935"
    ],
    [
      "1.586126824",
      "0.318927876"
    ],
    [
      "1.26031201",
      "0.331973217"
    ],
    [
      "0.676077591",
      "0.363423641"
    ],
    [
      "0.344667071",
      "0.3852296"
    ],
    [
      "0.891017558",
      "0.43094961"
    ],
    [
      "1.428025112",
      "0.446145898"
    ],
    [
      "0.518025059",
      "0.49063403"
    ],
    [
      "1.080213856",
      "0.504233118"
    ],
    [
      "1.617218807",
      "0.519418211"
    ],
    [
      "0.223449376",
      "0.547923657"
    ],
    [
      "0.73296763",
      "0.558171194"
    ],
    [
      "1.270654923",
      "0.574199438"
    ],
    [
      "1.803970546",
      "0.59870802"
    ],
    [
      "0.922161324",
      "0.631443507"
    ],
    [
      "1.459848617",
      "0.647471751"
    ],
    [
      "0.574915098",
      "0.685381583"
    ],
    [
      "0.37202942",
      "0.68607957"
    ],
    [
      "1.111355019",
      "0.70471582"
    ],
    [
      "0.141821804",
      "0.733665481"
    ],
    [
      "1.668540411",
      "0.74977676"
    ],
    [
      "0.764108793",
      "0.758653896"
    ],
    [
      "1.301796085",
      "0.77468214"
    ],
    [
      "1.873505762",
      "0.789306917"
    ],
    [
      "0.953302487",
      "0.831926209"
    ],
    [
      "1.49098978",
      "0.847954453"
    ],
    [
      "0.493287527",
      "0.871123407"
    ],
    [
      "0.290401848",
      "0.871821395"
    ],
    [
      "1.142496182",
      "0.905198522"
    ],
    [
      "0.682481221",
      "0.94439572"
    ],
    [
      "0.102876913",
      "0.949264823"
    ],
    [
      "1.699681573",
      "0.950259462"
    ],
    [
      "1.332937248",
      "0.975164842"
    ],
    [
      "1.898507886",
      "0.990647374"
    ],
    [
      "0.885930561",
      "1.023300474"
    ],
    [
      "1.522130943",
      "1.048437155"
    ],
    [
      "0.504773744",
      "1.073684886"
    ],
    [
      "0.301888065",
      "1.074382874"
    ],
    [
      "1.075124255",
      "1.096572787"
    ],
    [
      "0.722354871",
      "1.143325803"
    ],
    [
      "0.11436313",
      "1.151826302"
    ],
    [
      "1.267384581",
      "1.167169878"
    ],
    [
      "1.675100257",
      "1.181716842"
    ],
    [
      "1.877702257",
      "1.192464647"
    ],
    [
      "0.916901406",
      "1.225429861"
    ],
    [
      "1.456578275",
      "1.24044219"
    ],
    [
      "0.561411522",
      "1.268505952"
    ],
    [
      "0.358525843",
      "1.26920394"
    ],
    [
      "1.110883014",
      "1.29628357"
    ],
    [
      "0.749577703",
      "1.344378043"
    ],
    [
      "0.171000908",
      "1.346647368"
    ],
    [
      "1.609547589",
      "1.373721877"
    ],
    [
      "1.313065858",
      "1.384026186"
    ],
    [
      "1.812149589",
      "1.384469682"
    ],
    [
      "0.95414844",
      "1.425114307"
    ],
    [
      "0.457427675",
      "1.446352221"
    ],
    [
      "1.144086418",
      "1.496435062"
    ],
    [
      "1.469457804",
      "1.52047997"
    ],
    [
      "0.26990274",
      "1.523795649"
    ],
    [
      "0.789270442",
      "1.543344303"
    ],
    [
      "1.705191891",
      "1.556873674"
    ],
    [
      "0.593551343",
      "1.596796339"
    ],
    [
      "1.297881714",
      "1.628760769"
    ],
    [
      "0.406026408",
      "1.674239768"
    ],
    [
      "1.106386643",
      "1.695788549"
    ],
    [
      "1.562282081",
      "1.700887119"
    ],
    [
      "0.904044532",
      "1.710646407"
    ],
    [
      "0.590176204",
      "1.799655143"
    ],
    [
      "1.390705992",
      "1.809167918"
    ],
    [
      "0.780024432",
      "1.871214463"
    ],
    [
      "1.19921092",
      "1.876195698"
    ],
    [
      "0.981231848",
      "1.898355032"
    ]
  ],
  "radius": "0.101443439"
}

Submit radius and centers. Write every number as a decimal string, for example "0.2". · Verifier v1.0.0

DISCUSSION

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