MinMax Arena
中/EN
P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 38

Pack n unit-side equilateral triangles into a disc. Contact is allowed; interiors cannot overlap. Minimise the container circle radius.

Instancen = 38
ObjectiveMinimize container circle radius
Best known, unproven2.49003Source-precision interval: 2.49003–2.49004 (values inside match)exhibited site certificate 2.490030016Compiled by Erich Friedman; Found by Thomas Schadt in August 2026. Source expression: r = 2.49003+.

Formal definition

  • ContainerThe editor and certificate normalize to the closed unit disc centred at (1,1). This is a coordinate convention only; scores use the literature's unit-piece scale. Boundary contact is allowed.
  • SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.
  • ConstraintsExactly n congruent shapes lie wholly in the container with pairwise disjoint interiors. Polygon rotations are independent, not limited to quarter turns.
  • PrecisionInput decimals are exact rationals; regular-polygon vertices are algebraic, not rounded templates. Feasibility has no floating-point tolerance. A finite-decimal certificate is not a continuous optimality proof.
  • Objectivecontainer circle radius = 1/(2 sin(pi/3) r), smaller is better. Pages, leaderboards and literature share these units; conversion is automatic.
Open the full editor ↗
VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Source and certificate

External targets are converted with their printed precision; decimal coordinate certificates are separate. Reconstruction loss never lowers the literature target, and a source construction is not automatically an optimality proof.

Source
The record-holding arrangement for Equilateral triangles in a disc n = 38, 2.490030016
Current leader

2.490030016

container circle radius

Matches the best known
Answer sourceThomas Schadt
Solution methodPublished reference construction
Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

ANSWER FORMAT

How to write your answer

the closed unit disc centred at (1,1)

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "turn": "0.473692034",
      "x": "0.905198593",
      "y": "1.819234404"
    },
    {
      "turn": "-0.081397218",
      "x": "1.134011156",
      "y": "1.781736550"
    },
    {
      "turn": "0.473692034",
      "x": "0.529505653",
      "y": "1.677323693"
    },
    {
      "turn": "0.473692034",
      "x": "1.422277223",
      "y": "1.642768745"
    },
    {
      "turn": "-0.081397218",
      "x": "0.758318217",
      "y": "1.639825839"
    },
    {
      "turn": "-0.111670733",
      "x": "1.648113036",
      "y": "1.536594057"
    },
    {
      "turn": "0.473692034",
      "x": "1.046584284",
      "y": "1.500858035"
    },
    {
      "turn": "-0.081397218",
      "x": "0.382625277",
      "y": "1.497915128"
    },
    {
      "turn": "-0.081397218",
      "x": "1.275396847",
      "y": "1.463360180"
    },
    {
      "turn": "0.473692034",
      "x": "0.670891344",
      "y": "1.358947324"
    },
    {
      "turn": "-0.081397218",
      "x": "0.899703907",
      "y": "1.321449469"
    },
    {
      "turn": "0.437474206",
      "x": "1.535373875",
      "y": "1.258780542"
    },
    {
      "turn": "0.473692034",
      "x": "0.295198404",
      "y": "1.217036613"
    },
    {
      "turn": "-0.111670733",
      "x": "1.761527014",
      "y": "1.207633345"
    },
    {
      "turn": "-0.081397218",
      "x": "0.524010968",
      "y": "1.179538758"
    },
    {
      "turn": "0.473692034",
      "x": "1.169726676",
      "y": "1.175590613"
    },
    {
      "turn": "-0.111670733",
      "x": "1.399189820",
      "y": "1.068918753"
    },
    {
      "turn": "0.473692034",
      "x": "0.794033737",
      "y": "1.033679903"
    },
    {
      "turn": "-0.081397218",
      "x": "0.158171216",
      "y": "1.029561320"
    },
    {
      "turn": "-0.081397218",
      "x": "1.022846300",
      "y": "0.996182048"
    },
    {
      "turn": "0.437474206",
      "x": "1.830308812",
      "y": "0.986205383"
    },
    {
      "turn": "0.473692034",
      "x": "0.416281400",
      "y": "0.879202716"
    },
    {
      "turn": "-0.081397218",
      "x": "0.647153360",
      "y": "0.854271337"
    },
    {
      "turn": "0.437474206",
      "x": "1.467971617",
      "y": "0.847490790"
    },
    {
      "turn": "-0.111670733",
      "x": "1.694124757",
      "y": "0.796343593"
    },
    {
      "turn": "0.473692034",
      "x": "1.104778487",
      "y": "0.779275628"
    },
    {
      "turn": "-0.081397218",
      "x": "0.269401024",
      "y": "0.699794151"
    },
    {
      "turn": "0.473692034",
      "x": "0.729085548",
      "y": "0.637364917"
    },
    {
      "turn": "-0.081397218",
      "x": "0.957898111",
      "y": "0.599867063"
    },
    {
      "turn": "-0.081397218",
      "x": "1.320142663",
      "y": "0.581606829"
    },
    {
      "turn": "0.212679502",
      "x": "1.698867145",
      "y": "0.492494444"
    },
    {
      "turn": "0.473692034",
      "x": "0.351333211",
      "y": "0.482887731"
    },
    {
      "turn": "-0.081397218",
      "x": "0.582205171",
      "y": "0.457956352"
    },
    {
      "turn": "0.473692034",
      "x": "1.053475430",
      "y": "0.388114826"
    },
    {
      "turn": "0.473692034",
      "x": "1.402074850",
      "y": "0.364700409"
    },
    {
      "turn": "0.473692034",
      "x": "0.677782490",
      "y": "0.246204116"
    },
    {
      "turn": "-0.081397218",
      "x": "0.906595053",
      "y": "0.208706261"
    },
    {
      "turn": "-0.081397218",
      "x": "1.255194474",
      "y": "0.185291844"
    }
  ],
  "radius": "0.231864783"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 38,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.473692034",
      "x": "0.905198593",
      "y": "1.819234404"
    },
    {
      "turn": "-0.081397218",
      "x": "1.134011156",
      "y": "1.781736550"
    },
    {
      "turn": "0.473692034",
      "x": "0.529505653",
      "y": "1.677323693"
    },
    {
      "turn": "0.473692034",
      "x": "1.422277223",
      "y": "1.642768745"
    },
    {
      "turn": "-0.081397218",
      "x": "0.758318217",
      "y": "1.639825839"
    },
    {
      "turn": "-0.111670733",
      "x": "1.648113036",
      "y": "1.536594057"
    },
    {
      "turn": "0.473692034",
      "x": "1.046584284",
      "y": "1.500858035"
    },
    {
      "turn": "-0.081397218",
      "x": "0.382625277",
      "y": "1.497915128"
    },
    {
      "turn": "-0.081397218",
      "x": "1.275396847",
      "y": "1.463360180"
    },
    {
      "turn": "0.473692034",
      "x": "0.670891344",
      "y": "1.358947324"
    },
    {
      "turn": "-0.081397218",
      "x": "0.899703907",
      "y": "1.321449469"
    },
    {
      "turn": "0.437474206",
      "x": "1.535373875",
      "y": "1.258780542"
    },
    {
      "turn": "0.473692034",
      "x": "0.295198404",
      "y": "1.217036613"
    },
    {
      "turn": "-0.111670733",
      "x": "1.761527014",
      "y": "1.207633345"
    },
    {
      "turn": "-0.081397218",
      "x": "0.524010968",
      "y": "1.179538758"
    },
    {
      "turn": "0.473692034",
      "x": "1.169726676",
      "y": "1.175590613"
    },
    {
      "turn": "-0.111670733",
      "x": "1.399189820",
      "y": "1.068918753"
    },
    {
      "turn": "0.473692034",
      "x": "0.794033737",
      "y": "1.033679903"
    },
    {
      "turn": "-0.081397218",
      "x": "0.158171216",
      "y": "1.029561320"
    },
    {
      "turn": "-0.081397218",
      "x": "1.022846300",
      "y": "0.996182048"
    },
    {
      "turn": "0.437474206",
      "x": "1.830308812",
      "y": "0.986205383"
    },
    {
      "turn": "0.473692034",
      "x": "0.416281400",
      "y": "0.879202716"
    },
    {
      "turn": "-0.081397218",
      "x": "0.647153360",
      "y": "0.854271337"
    },
    {
      "turn": "0.437474206",
      "x": "1.467971617",
      "y": "0.847490790"
    },
    {
      "turn": "-0.111670733",
      "x": "1.694124757",
      "y": "0.796343593"
    },
    {
      "turn": "0.473692034",
      "x": "1.104778487",
      "y": "0.779275628"
    },
    {
      "turn": "-0.081397218",
      "x": "0.269401024",
      "y": "0.699794151"
    },
    {
      "turn": "0.473692034",
      "x": "0.729085548",
      "y": "0.637364917"
    },
    {
      "turn": "-0.081397218",
      "x": "0.957898111",
      "y": "0.599867063"
    },
    {
      "turn": "-0.081397218",
      "x": "1.320142663",
      "y": "0.581606829"
    },
    {
      "turn": "0.212679502",
      "x": "1.698867145",
      "y": "0.492494444"
    },
    {
      "turn": "0.473692034",
      "x": "0.351333211",
      "y": "0.482887731"
    },
    {
      "turn": "-0.081397218",
      "x": "0.582205171",
      "y": "0.457956352"
    },
    {
      "turn": "0.473692034",
      "x": "1.053475430",
      "y": "0.388114826"
    },
    {
      "turn": "0.473692034",
      "x": "1.402074850",
      "y": "0.364700409"
    },
    {
      "turn": "0.473692034",
      "x": "0.677782490",
      "y": "0.246204116"
    },
    {
      "turn": "-0.081397218",
      "x": "0.906595053",
      "y": "0.208706261"
    },
    {
      "turn": "-0.081397218",
      "x": "1.255194474",
      "y": "0.185291844"
    }
  ],
  "radius": "0.231864783"
}

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4. · Verifier v1.0.0

DISCUSSION

Discussion

Talk strategy, share methods, ask why you are stuck. Every signed-in user can post. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. New posts appear after an automated review.

No threads yet. Be the first to talk about this problem?