MinMax Arena
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P66 · Packing and covering · Classic · Hard

Unit circles in a minimum-area triangle · n = 46

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 = 46
ObjectiveMinimize the triangle area
Best known, unproven177.887371360134541611Tej 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 CONSTRUCTION46 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 = 46, 177.887371360134541611
Current leader

177.887371360134541611

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": [
    [
      "9.732050871",
      "14.856406557"
    ],
    [
      "10.873614097",
      "13.214202717"
    ],
    [
      "8.732050864",
      "13.124355738"
    ],
    [
      "11.731991069",
      "11.407772805"
    ],
    [
      "7.732050858",
      "11.392304919"
    ],
    [
      "9.732050871",
      "11.392304919"
    ],
    [
      "13.126156741",
      "9.973791588"
    ],
    [
      "10.745416375",
      "9.668039740"
    ],
    [
      "6.732050851",
      "9.660254101"
    ],
    [
      "8.732050864",
      "9.660254101"
    ],
    [
      "13.727279827",
      "8.066266729"
    ],
    [
      "5.732050845",
      "7.928203282"
    ],
    [
      "7.732050858",
      "7.928203282"
    ],
    [
      "9.732050871",
      "7.928203282"
    ],
    [
      "11.732050884",
      "7.928203282"
    ],
    [
      "15.305837364",
      "6.838196636"
    ],
    [
      "12.815771468",
      "6.247265533"
    ],
    [
      "4.732050838",
      "6.196152463"
    ],
    [
      "6.732050851",
      "6.196152463"
    ],
    [
      "8.732050864",
      "6.196152463"
    ],
    [
      "10.732050877",
      "6.196152463"
    ],
    [
      "15.689322207",
      "4.875306057"
    ],
    [
      "3.732050832",
      "4.464101644"
    ],
    [
      "5.732050845",
      "4.464101644"
    ],
    [
      "7.732050858",
      "4.464101644"
    ],
    [
      "9.732050871",
      "4.464101644"
    ],
    [
      "11.732050884",
      "4.464101644"
    ],
    [
      "13.732050897",
      "4.464101644"
    ],
    [
      "17.395605035",
      "3.831946596"
    ],
    [
      "15.102575226",
      "2.898357944"
    ],
    [
      "2.732050825",
      "2.732050825"
    ],
    [
      "4.732050838",
      "2.732050825"
    ],
    [
      "6.732050851",
      "2.732050825"
    ],
    [
      "8.732050864",
      "2.732050825"
    ],
    [
      "10.732050877",
      "2.732050825"
    ],
    [
      "12.732050890",
      "2.732050825"
    ],
    [
      "17.548129070",
      "1.837770959"
    ],
    [
      "1.732050819",
      "1.000000006"
    ],
    [
      "3.732050832",
      "1.000000006"
    ],
    [
      "5.732050845",
      "1.000000006"
    ],
    [
      "7.732050858",
      "1.000000006"
    ],
    [
      "9.732050871",
      "1.000000006"
    ],
    [
      "11.732050884",
      "1.000000006"
    ],
    [
      "13.732050897",
      "1.000000006"
    ],
    [
      "15.732050910",
      "1.000000006"
    ],
    [
      "19.364207230",
      "1.000000006"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "21.277223862",
      "0.000000000"
    ],
    [
      "9.653827248",
      "16.720919281"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 46
}

The current leader's answer

{
  "centers": [
    [
      "9.732050871",
      "14.856406557"
    ],
    [
      "10.873614097",
      "13.214202717"
    ],
    [
      "8.732050864",
      "13.124355738"
    ],
    [
      "11.731991069",
      "11.407772805"
    ],
    [
      "7.732050858",
      "11.392304919"
    ],
    [
      "9.732050871",
      "11.392304919"
    ],
    [
      "13.126156741",
      "9.973791588"
    ],
    [
      "10.745416375",
      "9.668039740"
    ],
    [
      "6.732050851",
      "9.660254101"
    ],
    [
      "8.732050864",
      "9.660254101"
    ],
    [
      "13.727279827",
      "8.066266729"
    ],
    [
      "5.732050845",
      "7.928203282"
    ],
    [
      "7.732050858",
      "7.928203282"
    ],
    [
      "9.732050871",
      "7.928203282"
    ],
    [
      "11.732050884",
      "7.928203282"
    ],
    [
      "15.305837364",
      "6.838196636"
    ],
    [
      "12.815771468",
      "6.247265533"
    ],
    [
      "4.732050838",
      "6.196152463"
    ],
    [
      "6.732050851",
      "6.196152463"
    ],
    [
      "8.732050864",
      "6.196152463"
    ],
    [
      "10.732050877",
      "6.196152463"
    ],
    [
      "15.689322207",
      "4.875306057"
    ],
    [
      "3.732050832",
      "4.464101644"
    ],
    [
      "5.732050845",
      "4.464101644"
    ],
    [
      "7.732050858",
      "4.464101644"
    ],
    [
      "9.732050871",
      "4.464101644"
    ],
    [
      "11.732050884",
      "4.464101644"
    ],
    [
      "13.732050897",
      "4.464101644"
    ],
    [
      "17.395605035",
      "3.831946596"
    ],
    [
      "15.102575226",
      "2.898357944"
    ],
    [
      "2.732050825",
      "2.732050825"
    ],
    [
      "4.732050838",
      "2.732050825"
    ],
    [
      "6.732050851",
      "2.732050825"
    ],
    [
      "8.732050864",
      "2.732050825"
    ],
    [
      "10.732050877",
      "2.732050825"
    ],
    [
      "12.732050890",
      "2.732050825"
    ],
    [
      "17.548129070",
      "1.837770959"
    ],
    [
      "1.732050819",
      "1.000000006"
    ],
    [
      "3.732050832",
      "1.000000006"
    ],
    [
      "5.732050845",
      "1.000000006"
    ],
    [
      "7.732050858",
      "1.000000006"
    ],
    [
      "9.732050871",
      "1.000000006"
    ],
    [
      "11.732050884",
      "1.000000006"
    ],
    [
      "13.732050897",
      "1.000000006"
    ],
    [
      "15.732050910",
      "1.000000006"
    ],
    [
      "19.364207230",
      "1.000000006"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "21.277223862",
      "0.000000000"
    ],
    [
      "9.653827248",
      "16.720919281"
    ]
  ]
}

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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