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

Equilateral triangles in a disc · n = 12

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

Instancen = 12
ObjectiveMinimize container circle radius
Best known, unproven1.507Source-precision interval: 1.507–1.508 (values inside match)exhibited site certificate 1.507145206Compiled by Erich Friedman; Found by David W. Cantrell in October 2005. Source expression: r = 1.507+.

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 = 12, 1.507145206
Current leader

1.507145206

container circle radius

Matches the best known
Answer sourceDavid W. Cantrell
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": "0.340824764",
      "x": "0.900035708",
      "y": "1.745153226"
    },
    {
      "turn": "-0.197635689",
      "x": "1.254310327",
      "y": "1.599426544"
    },
    {
      "turn": "0.515252062",
      "x": "0.473953341",
      "y": "1.383745798"
    },
    {
      "turn": "0.340824764",
      "x": "1.221247753",
      "y": "1.208483978"
    },
    {
      "turn": "-0.269252420",
      "x": "1.668572563",
      "y": "1.175550296"
    },
    {
      "turn": "-0.047860594",
      "x": "0.251607896",
      "y": "1.071801453"
    },
    {
      "turn": "-0.195294841",
      "x": "0.854446853",
      "y": "1.054375067"
    },
    {
      "turn": "0.266646813",
      "x": "1.667641283",
      "y": "0.792476016"
    },
    {
      "turn": "0.266646813",
      "x": "1.028237789",
      "y": "0.637864590"
    },
    {
      "turn": "0.298779996",
      "x": "0.371401454",
      "y": "0.587555994"
    },
    {
      "turn": "-0.269252420",
      "x": "1.392503290",
      "y": "0.502323031"
    },
    {
      "turn": "-0.269252420",
      "x": "0.725471513",
      "y": "0.395834996"
    }
  ],
  "radius": "0.383075411"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 12,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.340824764",
      "x": "0.900035708",
      "y": "1.745153226"
    },
    {
      "turn": "-0.197635689",
      "x": "1.254310327",
      "y": "1.599426544"
    },
    {
      "turn": "0.515252062",
      "x": "0.473953341",
      "y": "1.383745798"
    },
    {
      "turn": "0.340824764",
      "x": "1.221247753",
      "y": "1.208483978"
    },
    {
      "turn": "-0.269252420",
      "x": "1.668572563",
      "y": "1.175550296"
    },
    {
      "turn": "-0.047860594",
      "x": "0.251607896",
      "y": "1.071801453"
    },
    {
      "turn": "-0.195294841",
      "x": "0.854446853",
      "y": "1.054375067"
    },
    {
      "turn": "0.266646813",
      "x": "1.667641283",
      "y": "0.792476016"
    },
    {
      "turn": "0.266646813",
      "x": "1.028237789",
      "y": "0.637864590"
    },
    {
      "turn": "0.298779996",
      "x": "0.371401454",
      "y": "0.587555994"
    },
    {
      "turn": "-0.269252420",
      "x": "1.392503290",
      "y": "0.502323031"
    },
    {
      "turn": "-0.269252420",
      "x": "0.725471513",
      "y": "0.395834996"
    }
  ],
  "radius": "0.383075411"
}

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?