MinMax Arena
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P66 · Packing and covering · Classic · Hard

Unit circles in a minimum-area triangle · n = 49

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 = 49
ObjectiveMinimize the triangle area
Best known, unproven84 + 61√3the best a certificate can hold 189.655099647113421046Graham–Lubachevsky equilateral-triangle construction, still listed by Friedman's current table as best known among arbitrary triangles; public numerical data was recovered algebraically and quantized to the nine-decimal grid hereSubmissions are decimal, so a record chases the writable value. If you believe you have the exact closed form, send it in. minmaxarena.com@gmail.com

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 CONSTRUCTION49 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 = 49, 189.655099647113421046
Current leader

189.655099647113421046

the triangle area

Matches the best known
Answer sourceRonald L. Graham and Boris D. Lubachevsky
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": [
    [
      "3.732050812",
      "1.000000001"
    ],
    [
      "2.732050811",
      "2.732050810"
    ],
    [
      "1.732050810",
      "1.000000001"
    ],
    [
      "16.196152439",
      "6.196152429"
    ],
    [
      "11.464101627",
      "10.928203241"
    ],
    [
      "7.464101623",
      "8.928203239"
    ],
    [
      "19.196152442",
      "1.000000001"
    ],
    [
      "13.464101629",
      "10.928203241"
    ],
    [
      "13.196152436",
      "1.000000001"
    ],
    [
      "8.732050817",
      "2.732050810"
    ],
    [
      "4.732050813",
      "2.732050810"
    ],
    [
      "7.732050816",
      "4.464101620"
    ],
    [
      "5.732050814",
      "7.928203238"
    ],
    [
      "6.732050815",
      "2.732050810"
    ],
    [
      "9.464101625",
      "5.464101621"
    ],
    [
      "14.196152437",
      "6.196152429"
    ],
    [
      "10.464101626",
      "12.660254051"
    ],
    [
      "13.196152436",
      "4.464101620"
    ],
    [
      "8.464101624",
      "12.660254051"
    ],
    [
      "12.464101628",
      "7.196152430"
    ],
    [
      "7.464101623",
      "10.928203241"
    ],
    [
      "6.732050815",
      "6.196152429"
    ],
    [
      "11.464101627",
      "8.928203239"
    ],
    [
      "12.464101628",
      "12.660254051"
    ],
    [
      "11.464101627",
      "5.464101621"
    ],
    [
      "14.196152437",
      "2.732050810"
    ],
    [
      "5.732050814",
      "1.000000001"
    ],
    [
      "13.464101629",
      "8.928203239"
    ],
    [
      "18.196152441",
      "2.732050810"
    ],
    [
      "10.464101626",
      "16.124355669"
    ],
    [
      "16.196152439",
      "2.732050810"
    ],
    [
      "9.464101625",
      "14.392304860"
    ],
    [
      "12.196152435",
      "2.732050810"
    ],
    [
      "9.464101625",
      "10.928203241"
    ],
    [
      "17.196152440",
      "4.464101620"
    ],
    [
      "15.196152438",
      "4.464101620"
    ],
    [
      "3.732050812",
      "4.464101620"
    ],
    [
      "10.464101626",
      "3.732050811"
    ],
    [
      "15.196152438",
      "1.000000001"
    ],
    [
      "10.464101626",
      "7.196152430"
    ],
    [
      "15.196152438",
      "7.928203238"
    ],
    [
      "7.732050816",
      "1.000000001"
    ],
    [
      "8.464101624",
      "7.196152430"
    ],
    [
      "17.196152440",
      "1.000000001"
    ],
    [
      "11.464101627",
      "14.392304860"
    ],
    [
      "4.732050813",
      "6.196152429"
    ],
    [
      "9.464101625",
      "8.928203239"
    ],
    [
      "5.732050814",
      "4.464101620"
    ],
    [
      "10.464101626",
      "1.267949194"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "20.928203252",
      "0.000000000"
    ],
    [
      "10.464101626",
      "18.124355671"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 49
}

The current leader's answer

{
  "centers": [
    [
      "3.732050812",
      "1.000000001"
    ],
    [
      "2.732050811",
      "2.732050810"
    ],
    [
      "1.732050810",
      "1.000000001"
    ],
    [
      "16.196152439",
      "6.196152429"
    ],
    [
      "11.464101627",
      "10.928203241"
    ],
    [
      "7.464101623",
      "8.928203239"
    ],
    [
      "19.196152442",
      "1.000000001"
    ],
    [
      "13.464101629",
      "10.928203241"
    ],
    [
      "13.196152436",
      "1.000000001"
    ],
    [
      "8.732050817",
      "2.732050810"
    ],
    [
      "4.732050813",
      "2.732050810"
    ],
    [
      "7.732050816",
      "4.464101620"
    ],
    [
      "5.732050814",
      "7.928203238"
    ],
    [
      "6.732050815",
      "2.732050810"
    ],
    [
      "9.464101625",
      "5.464101621"
    ],
    [
      "14.196152437",
      "6.196152429"
    ],
    [
      "10.464101626",
      "12.660254051"
    ],
    [
      "13.196152436",
      "4.464101620"
    ],
    [
      "8.464101624",
      "12.660254051"
    ],
    [
      "12.464101628",
      "7.196152430"
    ],
    [
      "7.464101623",
      "10.928203241"
    ],
    [
      "6.732050815",
      "6.196152429"
    ],
    [
      "11.464101627",
      "8.928203239"
    ],
    [
      "12.464101628",
      "12.660254051"
    ],
    [
      "11.464101627",
      "5.464101621"
    ],
    [
      "14.196152437",
      "2.732050810"
    ],
    [
      "5.732050814",
      "1.000000001"
    ],
    [
      "13.464101629",
      "8.928203239"
    ],
    [
      "18.196152441",
      "2.732050810"
    ],
    [
      "10.464101626",
      "16.124355669"
    ],
    [
      "16.196152439",
      "2.732050810"
    ],
    [
      "9.464101625",
      "14.392304860"
    ],
    [
      "12.196152435",
      "2.732050810"
    ],
    [
      "9.464101625",
      "10.928203241"
    ],
    [
      "17.196152440",
      "4.464101620"
    ],
    [
      "15.196152438",
      "4.464101620"
    ],
    [
      "3.732050812",
      "4.464101620"
    ],
    [
      "10.464101626",
      "3.732050811"
    ],
    [
      "15.196152438",
      "1.000000001"
    ],
    [
      "10.464101626",
      "7.196152430"
    ],
    [
      "15.196152438",
      "7.928203238"
    ],
    [
      "7.732050816",
      "1.000000001"
    ],
    [
      "8.464101624",
      "7.196152430"
    ],
    [
      "17.196152440",
      "1.000000001"
    ],
    [
      "11.464101627",
      "14.392304860"
    ],
    [
      "4.732050813",
      "6.196152429"
    ],
    [
      "9.464101625",
      "8.928203239"
    ],
    [
      "5.732050814",
      "4.464101620"
    ],
    [
      "10.464101626",
      "1.267949194"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "20.928203252",
      "0.000000000"
    ],
    [
      "10.464101626",
      "18.124355671"
    ]
  ]
}

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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