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

Equilateral triangles in a disc · n = 37

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

Instancen = 37
ObjectiveMinimize container circle radius
Best known, unproven2.46349Source-precision interval: 2.46349–2.4635 (values inside match)exhibited site certificate 2.463494364Compiled by Erich Friedman; Found by Thomas Schadt in August 2026. Source expression: r = 2.46349+.

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 = 37, 2.463494364
Current leader

2.463494364

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": "0.010107077",
      "x": "0.995772435",
      "y": "1.791541459"
    },
    {
      "turn": "-0.563952348",
      "x": "0.761457984",
      "y": "1.786804507"
    },
    {
      "turn": "-0.563952348",
      "x": "1.307907703",
      "y": "1.701769857"
    },
    {
      "turn": "0.010107077",
      "x": "1.543905186",
      "y": "1.623255242"
    },
    {
      "turn": "0.010107077",
      "x": "0.648403079",
      "y": "1.581513764"
    },
    {
      "turn": "-0.563952348",
      "x": "0.414088629",
      "y": "1.576776811"
    },
    {
      "turn": "-0.563952348",
      "x": "0.960445962",
      "y": "1.496312072"
    },
    {
      "turn": "0.010107077",
      "x": "1.194943633",
      "y": "1.491985982"
    },
    {
      "turn": "-0.563952348",
      "x": "1.665164733",
      "y": "1.422701451"
    },
    {
      "turn": "0.010107077",
      "x": "0.301033724",
      "y": "1.371486068"
    },
    {
      "turn": "0.010107077",
      "x": "0.847391057",
      "y": "1.291021329"
    },
    {
      "turn": "-0.563952348",
      "x": "1.316252496",
      "y": "1.288992740"
    },
    {
      "turn": "-0.563952348",
      "x": "0.613076606",
      "y": "1.286284377"
    },
    {
      "turn": "0.010107077",
      "x": "1.552109828",
      "y": "1.217410708"
    },
    {
      "turn": "-0.563952348",
      "x": "1.834506162",
      "y": "1.114284015"
    },
    {
      "turn": "-0.563952348",
      "x": "0.968650604",
      "y": "1.090467538"
    },
    {
      "turn": "0.010107077",
      "x": "0.500021702",
      "y": "1.080993633"
    },
    {
      "turn": "0.010107077",
      "x": "1.203268150",
      "y": "1.080211836"
    },
    {
      "turn": "-0.563952348",
      "x": "0.265707251",
      "y": "1.076256681"
    },
    {
      "turn": "0.010107077",
      "x": "1.721451257",
      "y": "0.908993271"
    },
    {
      "turn": "-0.563952348",
      "x": "1.487136806",
      "y": "0.904256319"
    },
    {
      "turn": "0.010107077",
      "x": "0.855595699",
      "y": "0.885176795"
    },
    {
      "turn": "-0.563952348",
      "x": "0.621281249",
      "y": "0.880439843"
    },
    {
      "turn": "0.010107077",
      "x": "0.175067330",
      "y": "0.858621865"
    },
    {
      "turn": "-0.558209257",
      "x": "1.135938678",
      "y": "0.759974809"
    },
    {
      "turn": "0.010107077",
      "x": "1.374081902",
      "y": "0.698965576"
    },
    {
      "turn": "0.010107077",
      "x": "0.530641328",
      "y": "0.662805027"
    },
    {
      "turn": "-0.563952348",
      "x": "0.296326877",
      "y": "0.658068075"
    },
    {
      "turn": "-0.563952348",
      "x": "1.690305411",
      "y": "0.616291590"
    },
    {
      "turn": "0.010107077",
      "x": "1.025452713",
      "y": "0.551255825"
    },
    {
      "turn": "-0.563952348",
      "x": "0.791139889",
      "y": "0.546438425"
    },
    {
      "turn": "0.010107077",
      "x": "1.577250507",
      "y": "0.411000847"
    },
    {
      "turn": "-0.563952348",
      "x": "1.342936056",
      "y": "0.406263894"
    },
    {
      "turn": "0.463609357",
      "x": "0.442989161",
      "y": "0.342130937"
    },
    {
      "turn": "0.010107077",
      "x": "0.715689062",
      "y": "0.320438879"
    },
    {
      "turn": "0.010107077",
      "x": "1.212956872",
      "y": "0.210293458"
    },
    {
      "turn": "-0.563952348",
      "x": "0.978642421",
      "y": "0.205556506"
    }
  ],
  "radius": "0.234362326"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 37,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.010107077",
      "x": "0.995772435",
      "y": "1.791541459"
    },
    {
      "turn": "-0.563952348",
      "x": "0.761457984",
      "y": "1.786804507"
    },
    {
      "turn": "-0.563952348",
      "x": "1.307907703",
      "y": "1.701769857"
    },
    {
      "turn": "0.010107077",
      "x": "1.543905186",
      "y": "1.623255242"
    },
    {
      "turn": "0.010107077",
      "x": "0.648403079",
      "y": "1.581513764"
    },
    {
      "turn": "-0.563952348",
      "x": "0.414088629",
      "y": "1.576776811"
    },
    {
      "turn": "-0.563952348",
      "x": "0.960445962",
      "y": "1.496312072"
    },
    {
      "turn": "0.010107077",
      "x": "1.194943633",
      "y": "1.491985982"
    },
    {
      "turn": "-0.563952348",
      "x": "1.665164733",
      "y": "1.422701451"
    },
    {
      "turn": "0.010107077",
      "x": "0.301033724",
      "y": "1.371486068"
    },
    {
      "turn": "0.010107077",
      "x": "0.847391057",
      "y": "1.291021329"
    },
    {
      "turn": "-0.563952348",
      "x": "1.316252496",
      "y": "1.288992740"
    },
    {
      "turn": "-0.563952348",
      "x": "0.613076606",
      "y": "1.286284377"
    },
    {
      "turn": "0.010107077",
      "x": "1.552109828",
      "y": "1.217410708"
    },
    {
      "turn": "-0.563952348",
      "x": "1.834506162",
      "y": "1.114284015"
    },
    {
      "turn": "-0.563952348",
      "x": "0.968650604",
      "y": "1.090467538"
    },
    {
      "turn": "0.010107077",
      "x": "0.500021702",
      "y": "1.080993633"
    },
    {
      "turn": "0.010107077",
      "x": "1.203268150",
      "y": "1.080211836"
    },
    {
      "turn": "-0.563952348",
      "x": "0.265707251",
      "y": "1.076256681"
    },
    {
      "turn": "0.010107077",
      "x": "1.721451257",
      "y": "0.908993271"
    },
    {
      "turn": "-0.563952348",
      "x": "1.487136806",
      "y": "0.904256319"
    },
    {
      "turn": "0.010107077",
      "x": "0.855595699",
      "y": "0.885176795"
    },
    {
      "turn": "-0.563952348",
      "x": "0.621281249",
      "y": "0.880439843"
    },
    {
      "turn": "0.010107077",
      "x": "0.175067330",
      "y": "0.858621865"
    },
    {
      "turn": "-0.558209257",
      "x": "1.135938678",
      "y": "0.759974809"
    },
    {
      "turn": "0.010107077",
      "x": "1.374081902",
      "y": "0.698965576"
    },
    {
      "turn": "0.010107077",
      "x": "0.530641328",
      "y": "0.662805027"
    },
    {
      "turn": "-0.563952348",
      "x": "0.296326877",
      "y": "0.658068075"
    },
    {
      "turn": "-0.563952348",
      "x": "1.690305411",
      "y": "0.616291590"
    },
    {
      "turn": "0.010107077",
      "x": "1.025452713",
      "y": "0.551255825"
    },
    {
      "turn": "-0.563952348",
      "x": "0.791139889",
      "y": "0.546438425"
    },
    {
      "turn": "0.010107077",
      "x": "1.577250507",
      "y": "0.411000847"
    },
    {
      "turn": "-0.563952348",
      "x": "1.342936056",
      "y": "0.406263894"
    },
    {
      "turn": "0.463609357",
      "x": "0.442989161",
      "y": "0.342130937"
    },
    {
      "turn": "0.010107077",
      "x": "0.715689062",
      "y": "0.320438879"
    },
    {
      "turn": "0.010107077",
      "x": "1.212956872",
      "y": "0.210293458"
    },
    {
      "turn": "-0.563952348",
      "x": "0.978642421",
      "y": "0.205556506"
    }
  ],
  "radius": "0.234362326"
}

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?