MinMax Arena
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P66 · Packing and covering · Classic · Hard

Unit circles in a minimum-area triangle · n = 50

Also known ascircles in arbitrary trianglesunit circles in a minimum-area trianglecircle packing in a triangle of variable shape

Place n radius-1 circles with disjoint interiors inside an arbitrary triangle. The triangle's three vertices are part of the answer. Minimize its area.

Instancen = 50
ObjectiveMinimize the triangle area
Best known, unproven194.498962963929252736Tej Stead configuration listed by Erich Friedman's table; coordinates reconstructed here from the published figure and verified exactly on the nine-decimal grid

Formal definition

  • ContainerThe container is a non-degenerate triangle whose shape is not fixed in advance; its vertices and all centres lie in the 0-to-200 coordinate frame
  • SubmissionThree triangle vertices in counter-clockwise order and exactly n circle centres
  • CirclesEvery circle has fixed radius 1; circle interiors are pairwise disjoint and tangency is allowed
  • ContainmentEvery circle lies wholly inside the triangle and may be tangent to an edge
  • ObjectiveMake the triangle area as small as possible
Challenge this record ↗
VERIFIED CONSTRUCTION50 circles with radius 1

Getting a feel for it

The container is a variable

Ordinary packing moves the circles; here all three edge directions move too. A tiny edge rotation can release several contacts at once—or destabilize the whole packing.

The hard part is the contact structure

A candidate optimum is usually jammed by circle-circle and circle-edge tangencies. The real search is not uniform spacing but finding the right contact graph among many possibilities.

Starting beyond the trivial cases

Friedman's public table gives the current best-known constructions through n=50. The n=1 and n=2 rows are proved and n=3 is still the elementary equilateral arrangement, so the arena starts at n=4; matching a published construction reaches the frontier, while beating it sets a new record.

Source
The record-holding arrangement for Unit circles in a minimum-area triangle n = 50, 194.498962963929252736
Current leader

194.498962963929252736

the triangle area

Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
Challenge this record ↗
ANSWER FORMAT

How to write your answer

The coordinate unit is the circle radius: every circle has radius 1. Put all vertices and centres in the box 0≤x,y≤200 and list the vertices counter-clockwise.

Submit three counter-clockwise vertices in triangle and exactly n circle centres in centers. Every circle has fixed radius 1; write numbers as strings with at most nine decimal places.

The current leader's answer

{
  "centers": [
    [
      "10.767331992",
      "16.607428260"
    ],
    [
      "9.732050817",
      "14.856406475"
    ],
    [
      "11.732049601",
      "14.854198609"
    ],
    [
      "8.732050816",
      "13.124355666"
    ],
    [
      "10.785596341",
      "13.092317162"
    ],
    [
      "12.775925251",
      "12.895871650"
    ],
    [
      "7.732050815",
      "11.392304857"
    ],
    [
      "9.732050817",
      "11.392304857"
    ],
    [
      "11.716757486",
      "11.145445691"
    ],
    [
      "13.716704987",
      "11.130954231"
    ],
    [
      "6.732050814",
      "9.660254048"
    ],
    [
      "8.732050816",
      "9.660254048"
    ],
    [
      "12.704181264",
      "9.406194619"
    ],
    [
      "10.705444786",
      "9.335113694"
    ],
    [
      "14.700523739",
      "9.285294946"
    ],
    [
      "5.732050813",
      "7.928203238"
    ],
    [
      "7.732050815",
      "7.928203238"
    ],
    [
      "10.354067073",
      "7.366222201"
    ],
    [
      "12.357153993",
      "7.435050104"
    ],
    [
      "14.387520009",
      "7.309939618"
    ],
    [
      "16.201947731",
      "6.468600181"
    ],
    [
      "6.732050814",
      "6.196152429"
    ],
    [
      "8.732050816",
      "6.196152429"
    ],
    [
      "4.732050812",
      "6.196152429"
    ],
    [
      "11.354067074",
      "5.634171392"
    ],
    [
      "13.353737016",
      "5.597837772"
    ],
    [
      "15.142727464",
      "4.703682762"
    ],
    [
      "17.142727466",
      "4.703682762"
    ],
    [
      "3.732050811",
      "4.464101620"
    ],
    [
      "5.732050813",
      "4.464101620"
    ],
    [
      "7.732050815",
      "4.464101620"
    ],
    [
      "9.732050817",
      "4.464101620"
    ],
    [
      "12.395594889",
      "3.842285531"
    ],
    [
      "14.184585337",
      "2.948130520"
    ],
    [
      "16.184585339",
      "2.948130520"
    ],
    [
      "18.176175901",
      "2.764917420"
    ],
    [
      "2.732050810",
      "2.732050810"
    ],
    [
      "4.732050812",
      "2.732050810"
    ],
    [
      "6.732050814",
      "2.732050810"
    ],
    [
      "8.732050816",
      "2.732050810"
    ],
    [
      "10.732050818",
      "2.732050810"
    ],
    [
      "7.732050815",
      "1.000000001"
    ],
    [
      "9.732050817",
      "1.000000001"
    ],
    [
      "11.732050819",
      "1.000000001"
    ],
    [
      "19.116955637",
      "1.000000001"
    ],
    [
      "1.732050809",
      "1.000000001"
    ],
    [
      "3.732050811",
      "1.000000001"
    ],
    [
      "5.732050813",
      "1.000000001"
    ],
    [
      "13.732050821",
      "1.000000001"
    ],
    [
      "15.732050823",
      "1.000000001"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "20.783197756",
      "0.000000000"
    ],
    [
      "10.806232029",
      "18.716942912"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 50
}

The current leader's answer

{
  "centers": [
    [
      "10.767331992",
      "16.607428260"
    ],
    [
      "9.732050817",
      "14.856406475"
    ],
    [
      "11.732049601",
      "14.854198609"
    ],
    [
      "8.732050816",
      "13.124355666"
    ],
    [
      "10.785596341",
      "13.092317162"
    ],
    [
      "12.775925251",
      "12.895871650"
    ],
    [
      "7.732050815",
      "11.392304857"
    ],
    [
      "9.732050817",
      "11.392304857"
    ],
    [
      "11.716757486",
      "11.145445691"
    ],
    [
      "13.716704987",
      "11.130954231"
    ],
    [
      "6.732050814",
      "9.660254048"
    ],
    [
      "8.732050816",
      "9.660254048"
    ],
    [
      "12.704181264",
      "9.406194619"
    ],
    [
      "10.705444786",
      "9.335113694"
    ],
    [
      "14.700523739",
      "9.285294946"
    ],
    [
      "5.732050813",
      "7.928203238"
    ],
    [
      "7.732050815",
      "7.928203238"
    ],
    [
      "10.354067073",
      "7.366222201"
    ],
    [
      "12.357153993",
      "7.435050104"
    ],
    [
      "14.387520009",
      "7.309939618"
    ],
    [
      "16.201947731",
      "6.468600181"
    ],
    [
      "6.732050814",
      "6.196152429"
    ],
    [
      "8.732050816",
      "6.196152429"
    ],
    [
      "4.732050812",
      "6.196152429"
    ],
    [
      "11.354067074",
      "5.634171392"
    ],
    [
      "13.353737016",
      "5.597837772"
    ],
    [
      "15.142727464",
      "4.703682762"
    ],
    [
      "17.142727466",
      "4.703682762"
    ],
    [
      "3.732050811",
      "4.464101620"
    ],
    [
      "5.732050813",
      "4.464101620"
    ],
    [
      "7.732050815",
      "4.464101620"
    ],
    [
      "9.732050817",
      "4.464101620"
    ],
    [
      "12.395594889",
      "3.842285531"
    ],
    [
      "14.184585337",
      "2.948130520"
    ],
    [
      "16.184585339",
      "2.948130520"
    ],
    [
      "18.176175901",
      "2.764917420"
    ],
    [
      "2.732050810",
      "2.732050810"
    ],
    [
      "4.732050812",
      "2.732050810"
    ],
    [
      "6.732050814",
      "2.732050810"
    ],
    [
      "8.732050816",
      "2.732050810"
    ],
    [
      "10.732050818",
      "2.732050810"
    ],
    [
      "7.732050815",
      "1.000000001"
    ],
    [
      "9.732050817",
      "1.000000001"
    ],
    [
      "11.732050819",
      "1.000000001"
    ],
    [
      "19.116955637",
      "1.000000001"
    ],
    [
      "1.732050809",
      "1.000000001"
    ],
    [
      "3.732050811",
      "1.000000001"
    ],
    [
      "5.732050813",
      "1.000000001"
    ],
    [
      "13.732050821",
      "1.000000001"
    ],
    [
      "15.732050823",
      "1.000000001"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "20.783197756",
      "0.000000000"
    ],
    [
      "10.806232029",
      "18.716942912"
    ]
  ]
}

Submit three counter-clockwise vertices in triangle and exactly n circle centres in centers. Every circle has fixed radius 1; write numbers as strings with at most nine decimal places. · Verifier v1.0.0

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