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

Equilateral triangles in a disc · n = 35

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

Instancen = 35
ObjectiveMinimize container circle radius
Best known, unproven2.39906Source-precision interval: 2.39906–2.39907 (values inside match)exhibited site certificate 2.39906468Compiled by Erich Friedman; Found by Thomas Schadt in August 2026. Source expression: r = 2.39906+.

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 = 35, 2.39906468
Current leader

2.39906468

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": "0.160740342",
      "x": "1.120792395",
      "y": "1.838063361"
    },
    {
      "turn": "0.150062122",
      "x": "0.769842796",
      "y": "1.749797091"
    },
    {
      "turn": "-0.393220177",
      "x": "0.539786211",
      "y": "1.679160901"
    },
    {
      "turn": "0.150062122",
      "x": "1.448837095",
      "y": "1.663600400"
    },
    {
      "turn": "-0.393220177",
      "x": "1.220912092",
      "y": "1.586021814"
    },
    {
      "turn": "-0.393220177",
      "x": "1.625038123",
      "y": "1.499683648"
    },
    {
      "turn": "-0.393220177",
      "x": "0.874805319",
      "y": "1.483395764"
    },
    {
      "turn": "0.150062122",
      "x": "0.505789817",
      "y": "1.440048109"
    },
    {
      "turn": "-0.393220177",
      "x": "0.275733231",
      "y": "1.369411919"
    },
    {
      "turn": "0.150062122",
      "x": "1.167056534",
      "y": "1.351468872"
    },
    {
      "turn": "0.150062122",
      "x": "1.571182565",
      "y": "1.265130706"
    },
    {
      "turn": "0.150062122",
      "x": "0.820949761",
      "y": "1.248842822"
    },
    {
      "turn": "-0.393220177",
      "x": "1.343257562",
      "y": "1.187552119"
    },
    {
      "turn": "-0.393220177",
      "x": "1.823613860",
      "y": "1.183157082"
    },
    {
      "turn": "-0.393220177",
      "x": "0.590893175",
      "y": "1.178206632"
    },
    {
      "turn": "0.150062122",
      "x": "0.221877674",
      "y": "1.134858977"
    },
    {
      "turn": "-0.393220177",
      "x": "0.997150788",
      "y": "1.084926069"
    },
    {
      "turn": "0.150062122",
      "x": "0.537037618",
      "y": "0.943653690"
    },
    {
      "turn": "0.150062122",
      "x": "1.489137527",
      "y": "0.907138026"
    },
    {
      "turn": "-0.393220177",
      "x": "0.306981032",
      "y": "0.873017500"
    },
    {
      "turn": "0.150062122",
      "x": "0.943295230",
      "y": "0.850373127"
    },
    {
      "turn": "-0.393220177",
      "x": "1.259080941",
      "y": "0.836501836"
    },
    {
      "turn": "-0.393220177",
      "x": "1.765055580",
      "y": "0.826224587"
    },
    {
      "turn": "-0.393220177",
      "x": "0.713238645",
      "y": "0.779736937"
    },
    {
      "turn": "0.150062122",
      "x": "0.250487394",
      "y": "0.639070286"
    },
    {
      "turn": "0.150062122",
      "x": "1.205225384",
      "y": "0.601948894"
    },
    {
      "turn": "0.150062122",
      "x": "1.711200023",
      "y": "0.591671645"
    },
    {
      "turn": "0.150062122",
      "x": "0.656745007",
      "y": "0.545789723"
    },
    {
      "turn": "-0.393220177",
      "x": "1.475559994",
      "y": "0.539220298"
    },
    {
      "turn": "-0.393220177",
      "x": "0.975168798",
      "y": "0.531312704"
    },
    {
      "turn": "-0.393220177",
      "x": "0.426688422",
      "y": "0.475153534"
    },
    {
      "turn": "0.150062122",
      "x": "1.421704436",
      "y": "0.304667356"
    },
    {
      "turn": "0.150062122",
      "x": "0.921313240",
      "y": "0.296759762"
    },
    {
      "turn": "-0.393220177",
      "x": "1.191647850",
      "y": "0.234031166"
    },
    {
      "turn": "-0.393220177",
      "x": "0.690462191",
      "y": "0.228711081"
    }
  ],
  "radius": "0.240656400"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 35,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.160740342",
      "x": "1.120792395",
      "y": "1.838063361"
    },
    {
      "turn": "0.150062122",
      "x": "0.769842796",
      "y": "1.749797091"
    },
    {
      "turn": "-0.393220177",
      "x": "0.539786211",
      "y": "1.679160901"
    },
    {
      "turn": "0.150062122",
      "x": "1.448837095",
      "y": "1.663600400"
    },
    {
      "turn": "-0.393220177",
      "x": "1.220912092",
      "y": "1.586021814"
    },
    {
      "turn": "-0.393220177",
      "x": "1.625038123",
      "y": "1.499683648"
    },
    {
      "turn": "-0.393220177",
      "x": "0.874805319",
      "y": "1.483395764"
    },
    {
      "turn": "0.150062122",
      "x": "0.505789817",
      "y": "1.440048109"
    },
    {
      "turn": "-0.393220177",
      "x": "0.275733231",
      "y": "1.369411919"
    },
    {
      "turn": "0.150062122",
      "x": "1.167056534",
      "y": "1.351468872"
    },
    {
      "turn": "0.150062122",
      "x": "1.571182565",
      "y": "1.265130706"
    },
    {
      "turn": "0.150062122",
      "x": "0.820949761",
      "y": "1.248842822"
    },
    {
      "turn": "-0.393220177",
      "x": "1.343257562",
      "y": "1.187552119"
    },
    {
      "turn": "-0.393220177",
      "x": "1.823613860",
      "y": "1.183157082"
    },
    {
      "turn": "-0.393220177",
      "x": "0.590893175",
      "y": "1.178206632"
    },
    {
      "turn": "0.150062122",
      "x": "0.221877674",
      "y": "1.134858977"
    },
    {
      "turn": "-0.393220177",
      "x": "0.997150788",
      "y": "1.084926069"
    },
    {
      "turn": "0.150062122",
      "x": "0.537037618",
      "y": "0.943653690"
    },
    {
      "turn": "0.150062122",
      "x": "1.489137527",
      "y": "0.907138026"
    },
    {
      "turn": "-0.393220177",
      "x": "0.306981032",
      "y": "0.873017500"
    },
    {
      "turn": "0.150062122",
      "x": "0.943295230",
      "y": "0.850373127"
    },
    {
      "turn": "-0.393220177",
      "x": "1.259080941",
      "y": "0.836501836"
    },
    {
      "turn": "-0.393220177",
      "x": "1.765055580",
      "y": "0.826224587"
    },
    {
      "turn": "-0.393220177",
      "x": "0.713238645",
      "y": "0.779736937"
    },
    {
      "turn": "0.150062122",
      "x": "0.250487394",
      "y": "0.639070286"
    },
    {
      "turn": "0.150062122",
      "x": "1.205225384",
      "y": "0.601948894"
    },
    {
      "turn": "0.150062122",
      "x": "1.711200023",
      "y": "0.591671645"
    },
    {
      "turn": "0.150062122",
      "x": "0.656745007",
      "y": "0.545789723"
    },
    {
      "turn": "-0.393220177",
      "x": "1.475559994",
      "y": "0.539220298"
    },
    {
      "turn": "-0.393220177",
      "x": "0.975168798",
      "y": "0.531312704"
    },
    {
      "turn": "-0.393220177",
      "x": "0.426688422",
      "y": "0.475153534"
    },
    {
      "turn": "0.150062122",
      "x": "1.421704436",
      "y": "0.304667356"
    },
    {
      "turn": "0.150062122",
      "x": "0.921313240",
      "y": "0.296759762"
    },
    {
      "turn": "-0.393220177",
      "x": "1.191647850",
      "y": "0.234031166"
    },
    {
      "turn": "-0.393220177",
      "x": "0.690462191",
      "y": "0.228711081"
    }
  ],
  "radius": "0.240656400"
}

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?