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

Equilateral triangles in a disc · n = 21

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

Instancen = 21
ObjectiveMinimize container circle radius
Best known, unproven1.910Source-precision interval: 1.91–1.911 (values inside match)exhibited site certificate 1.910541299Compiled by Erich Friedman; Found by Maurizio Morandi in May 2008. Source expression: r = 1.910+.

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 = 21, 1.910541299
Current leader

1.910541299

container circle radius

Matches the best known
Answer sourceMaurizio Morandi
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": "0.222940118",
      "x": "1.067078279",
      "y": "1.810764193"
    },
    {
      "turn": "-0.313994495",
      "x": "0.793503235",
      "y": "1.682402623"
    },
    {
      "turn": "0.268652021",
      "x": "1.442551256",
      "y": "1.558702609"
    },
    {
      "turn": "-0.267246610",
      "x": "1.704455096",
      "y": "1.407949911"
    },
    {
      "turn": "0.268652021",
      "x": "0.525980629",
      "y": "1.379833362"
    },
    {
      "turn": "0.268652021",
      "x": "1.035701657",
      "y": "1.356735259"
    },
    {
      "turn": "-0.267246610",
      "x": "0.264473042",
      "y": "1.228394333"
    },
    {
      "turn": "-0.267246610",
      "x": "0.774194071",
      "y": "1.205296229"
    },
    {
      "turn": "-0.267246610",
      "x": "1.297605497",
      "y": "1.205982561"
    },
    {
      "turn": "0.268652021",
      "x": "1.704851350",
      "y": "1.105758183"
    },
    {
      "turn": "0.268652021",
      "x": "0.264869296",
      "y": "0.926202605"
    },
    {
      "turn": "0.268652021",
      "x": "1.298001751",
      "y": "0.903790833"
    },
    {
      "turn": "0.268652021",
      "x": "0.774590325",
      "y": "0.903104501"
    },
    {
      "turn": "-0.267246610",
      "x": "1.036494165",
      "y": "0.752351803"
    },
    {
      "turn": "-0.267246610",
      "x": "1.559905591",
      "y": "0.753038135"
    },
    {
      "turn": "-0.267246610",
      "x": "0.513082738",
      "y": "0.751665471"
    },
    {
      "turn": "0.268652021",
      "x": "1.524132475",
      "y": "0.450798979"
    },
    {
      "turn": "0.268652021",
      "x": "1.000721049",
      "y": "0.450112647"
    },
    {
      "turn": "0.268652021",
      "x": "0.477309623",
      "y": "0.449426315"
    },
    {
      "turn": "-0.267246610",
      "x": "1.262624889",
      "y": "0.299359950"
    },
    {
      "turn": "-0.267246610",
      "x": "0.739213463",
      "y": "0.298673617"
    }
  ],
  "radius": "0.302191986"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 21,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.222940118",
      "x": "1.067078279",
      "y": "1.810764193"
    },
    {
      "turn": "-0.313994495",
      "x": "0.793503235",
      "y": "1.682402623"
    },
    {
      "turn": "0.268652021",
      "x": "1.442551256",
      "y": "1.558702609"
    },
    {
      "turn": "-0.267246610",
      "x": "1.704455096",
      "y": "1.407949911"
    },
    {
      "turn": "0.268652021",
      "x": "0.525980629",
      "y": "1.379833362"
    },
    {
      "turn": "0.268652021",
      "x": "1.035701657",
      "y": "1.356735259"
    },
    {
      "turn": "-0.267246610",
      "x": "0.264473042",
      "y": "1.228394333"
    },
    {
      "turn": "-0.267246610",
      "x": "0.774194071",
      "y": "1.205296229"
    },
    {
      "turn": "-0.267246610",
      "x": "1.297605497",
      "y": "1.205982561"
    },
    {
      "turn": "0.268652021",
      "x": "1.704851350",
      "y": "1.105758183"
    },
    {
      "turn": "0.268652021",
      "x": "0.264869296",
      "y": "0.926202605"
    },
    {
      "turn": "0.268652021",
      "x": "1.298001751",
      "y": "0.903790833"
    },
    {
      "turn": "0.268652021",
      "x": "0.774590325",
      "y": "0.903104501"
    },
    {
      "turn": "-0.267246610",
      "x": "1.036494165",
      "y": "0.752351803"
    },
    {
      "turn": "-0.267246610",
      "x": "1.559905591",
      "y": "0.753038135"
    },
    {
      "turn": "-0.267246610",
      "x": "0.513082738",
      "y": "0.751665471"
    },
    {
      "turn": "0.268652021",
      "x": "1.524132475",
      "y": "0.450798979"
    },
    {
      "turn": "0.268652021",
      "x": "1.000721049",
      "y": "0.450112647"
    },
    {
      "turn": "0.268652021",
      "x": "0.477309623",
      "y": "0.449426315"
    },
    {
      "turn": "-0.267246610",
      "x": "1.262624889",
      "y": "0.299359950"
    },
    {
      "turn": "-0.267246610",
      "x": "0.739213463",
      "y": "0.298673617"
    }
  ],
  "radius": "0.302191986"
}

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?