P89 · Packing and covering · Classic · Hard

Tans in a square · n = 6

Pack n tans (right isosceles triangles of leg 1) into a square. Contact is allowed; interiors cannot overlap. Minimise the container square side.

Instancen = 6

ObjectiveMinimize container square side

Best known, unproven1.937Source-precision interval: 1.937–1.938 (values inside match)exhibited site certificate 1.937914833Compiled by Erich Friedman; Found by Maurizio Morandi in February 2008. Source expression: s = 1.937+.

Formal definition

ContainerThe editor and certificate normalize to the unit square [0,1]². This is a coordinate convention only; scores use the literature's unit-piece scale. Boundary contact is allowed.
SubmissionSubmit {radius, placements:[{x,y,turn,quarter},…]}, with x, y and turn as decimal strings of at most nine places. radius is the piece's short side (the leg, the domino's short side, the cell side); the container side is 1/radius. x,y is the centre of the piece's bounding box; quarter is an integer 0–3, an exact number of right angles applied about the centre before the rational 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 square side = 1/r (r is the piece's short side inside the unit square), smaller is better. Pages, leaderboards and literature records share this unit; nothing is converted by hand.

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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

Current leader

1.937914833

container square side

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.

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ANSWER FORMAT

How to write your answer

the unit square [0,1]²

Submit {radius, placements:[{x,y,turn,quarter},…]}, with x, y and turn as decimal strings of at most nine places. radius is the piece's short side (the leg, the domino's short side, the cell side); the container side is 1/radius. x,y is the centre of the piece's bounding box; quarter is an integer 0–3, an exact number of right angles applied about the centre before the rational turn = tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "quarter": 2,
      "turn": "0.414213559",
      "x": "0.483981445",
      "y": "0.603082668"
    },
    {
      "quarter": 3,
      "turn": "-0.413423643",
      "x": "0.510104573",
      "y": "0.000730888"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.258009278",
      "y": "0.741990722"
    },
    {
      "quarter": 2,
      "turn": "0.058838192",
      "x": "0.713514129",
      "y": "0.225972168"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.741990722",
      "y": "0.709953612"
    },
    {
      "quarter": 3,
      "turn": "0.093928203",
      "x": "0.324552917",
      "y": "0.301541334"
    }
  ],
  "radius": "0.516018549"
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 6,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "quarter": 2,
      "turn": "0.414213559",
      "x": "0.483981445",
      "y": "0.603082668"
    },
    {
      "quarter": 3,
      "turn": "-0.413423643",
      "x": "0.510104573",
      "y": "0.000730888"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.258009278",
      "y": "0.741990722"
    },
    {
      "quarter": 2,
      "turn": "0.058838192",
      "x": "0.713514129",
      "y": "0.225972168"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.741990722",
      "y": "0.709953612"
    },
    {
      "quarter": 3,
      "turn": "0.093928203",
      "x": "0.324552917",
      "y": "0.301541334"
    }
  ],
  "radius": "0.516018549"
}

DISCUSSION

Discussion

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