P90 · Packing and covering · Classic · Hard

Dominoes in a square · n = 8

Pack n 1 × 2 dominoes into a square. Contact is allowed; interiors cannot overlap. Minimise the container square side.

Instancen = 8

ObjectiveMinimize container square side

Site reconstruction of published figure4Compiled by Erich Friedman; Trivial. Source expression: s = 4.

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

4

container square side

Answer sourceTrivial.

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": [
    {
      "turn": "0.000000000",
      "x": "0.250000000",
      "y": "0.125000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.750000000",
      "y": "0.125000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.250000000",
      "y": "0.375000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.750000000",
      "y": "0.375000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.250000000",
      "y": "0.625000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.750000000",
      "y": "0.625000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.250000000",
      "y": "0.875000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.750000000",
      "y": "0.875000000"
    }
  ],
  "radius": "0.250000000"
}

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

Instance parameters

{
  "n": 8,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.250000000",
      "y": "0.125000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.750000000",
      "y": "0.125000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.250000000",
      "y": "0.375000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.750000000",
      "y": "0.375000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.250000000",
      "y": "0.625000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.750000000",
      "y": "0.625000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.250000000",
      "y": "0.875000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.750000000",
      "y": "0.875000000"
    }
  ],
  "radius": "0.250000000"
}

DISCUSSION

Discussion

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