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

Equilateral triangles in a disc · n = 14

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

Instancen = 14
ObjectiveMinimize container circle radius
Best known, unproven1.60448Source-precision interval: 1.60448–1.60449 (values inside match)exhibited site certificate 1.604489741Compiled by Erich Friedman; Found by Jake Loyd in June 2026. Source expression: r = 1.60448+.

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 = 14, 1.604489741
Current leader

1.604489741

container circle radius

Matches the best known
Answer sourceJake Loyd
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": "0.047245168",
      "x": "1.320535342",
      "y": "1.700428816"
    },
    {
      "turn": "0.568528850",
      "x": "0.927999297",
      "y": "1.684495387"
    },
    {
      "turn": "0.568528850",
      "x": "0.426672860",
      "y": "1.354760795"
    },
    {
      "turn": "-0.006641433",
      "x": "0.786475311",
      "y": "1.349981376"
    },
    {
      "turn": "-0.045936025",
      "x": "1.616453928",
      "y": "1.256188822"
    },
    {
      "turn": "0.567291188",
      "x": "1.232183174",
      "y": "1.186434091"
    },
    {
      "turn": "-0.025441396",
      "x": "0.230708626",
      "y": "1.039169046"
    },
    {
      "turn": "0.517684638",
      "x": "1.767043937",
      "y": "0.929381087"
    },
    {
      "turn": "-0.490378552",
      "x": "0.736170486",
      "y": "0.858833169"
    },
    {
      "turn": "0.067781427",
      "x": "1.107274952",
      "y": "0.800468326"
    },
    {
      "turn": "0.067781427",
      "x": "0.488074604",
      "y": "0.571536995"
    },
    {
      "turn": "0.162191911",
      "x": "1.576246247",
      "y": "0.488840832"
    },
    {
      "turn": "-0.379610998",
      "x": "1.234858549",
      "y": "0.375108314"
    },
    {
      "turn": "-0.490378552",
      "x": "0.708397718",
      "y": "0.287040295"
    }
  ],
  "radius": "0.359834192"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 14,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.047245168",
      "x": "1.320535342",
      "y": "1.700428816"
    },
    {
      "turn": "0.568528850",
      "x": "0.927999297",
      "y": "1.684495387"
    },
    {
      "turn": "0.568528850",
      "x": "0.426672860",
      "y": "1.354760795"
    },
    {
      "turn": "-0.006641433",
      "x": "0.786475311",
      "y": "1.349981376"
    },
    {
      "turn": "-0.045936025",
      "x": "1.616453928",
      "y": "1.256188822"
    },
    {
      "turn": "0.567291188",
      "x": "1.232183174",
      "y": "1.186434091"
    },
    {
      "turn": "-0.025441396",
      "x": "0.230708626",
      "y": "1.039169046"
    },
    {
      "turn": "0.517684638",
      "x": "1.767043937",
      "y": "0.929381087"
    },
    {
      "turn": "-0.490378552",
      "x": "0.736170486",
      "y": "0.858833169"
    },
    {
      "turn": "0.067781427",
      "x": "1.107274952",
      "y": "0.800468326"
    },
    {
      "turn": "0.067781427",
      "x": "0.488074604",
      "y": "0.571536995"
    },
    {
      "turn": "0.162191911",
      "x": "1.576246247",
      "y": "0.488840832"
    },
    {
      "turn": "-0.379610998",
      "x": "1.234858549",
      "y": "0.375108314"
    },
    {
      "turn": "-0.490378552",
      "x": "0.708397718",
      "y": "0.287040295"
    }
  ],
  "radius": "0.359834192"
}

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?