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P90 · Packing and covering · Classic · Hard

Dominoes in a square · n = 25

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

Instancen = 25
ObjectiveMinimize container square side
Best known, unproven7.70710Source-precision interval: 7.7071–7.70711 (values inside match)exhibited site certificate 7.707106998Compiled by Erich Friedman; Found by Maurizio Morandi in August 2026. Source expression: s = 7+1/√2 = 7.70710+.

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.
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 Dominoes in a square n = 25, 7.707106998
Current leader

7.707106998

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.
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 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": "0.610748876",
      "y": "0.935124811"
    },
    {
      "turn": "-0.000000004",
      "x": "0.870249624",
      "y": "0.935124810"
    },
    {
      "quarter": 1,
      "turn": "0.000000001",
      "x": "0.675624060",
      "y": "0.740499249"
    },
    {
      "turn": "-0.000000007",
      "x": "0.870249623",
      "y": "0.805374436"
    },
    {
      "quarter": 1,
      "turn": "0.000000005",
      "x": "0.324375939",
      "y": "0.648751885"
    },
    {
      "turn": "-0.000000008",
      "x": "0.870249623",
      "y": "0.675624062"
    },
    {
      "turn": "-0.000000004",
      "x": "0.870249623",
      "y": "0.545873688"
    },
    {
      "turn": "-0.000000012",
      "x": "0.389251125",
      "y": "0.454126319"
    },
    {
      "turn": "-0.000000010",
      "x": "0.389251126",
      "y": "0.324375943"
    },
    {
      "turn": "-0.000000006",
      "x": "0.389251123",
      "y": "0.064875191"
    },
    {
      "turn": "-0.000000001",
      "x": "0.648751871",
      "y": "0.064875190"
    },
    {
      "turn": "-0.000000002",
      "x": "0.129750376",
      "y": "0.722812726"
    },
    {
      "turn": "0.000000004",
      "x": "0.129750377",
      "y": "0.590898421"
    },
    {
      "turn": "-0.000000011",
      "x": "0.129750377",
      "y": "0.460613021"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.935124811",
      "y": "0.351248126"
    },
    {
      "turn": "0.000000002",
      "x": "0.129750376",
      "y": "0.330693500"
    },
    {
      "quarter": 1,
      "turn": "-0.000000013",
      "x": "0.583876688",
      "y": "0.259500753"
    },
    {
      "turn": "-0.000000006",
      "x": "0.129750376",
      "y": "0.066049839"
    },
    {
      "turn": "-0.000000004",
      "x": "0.389251125",
      "y": "0.194625566"
    },
    {
      "turn": "0.023475417",
      "x": "0.135843248",
      "y": "0.860149170"
    },
    {
      "turn": "0.000000002",
      "x": "0.129750376",
      "y": "0.197657110"
    },
    {
      "quarter": 2,
      "turn": "-0.413990188",
      "x": "0.861903911",
      "y": "0.138124699"
    },
    {
      "quarter": 2,
      "turn": "-0.414069522",
      "x": "0.731196832",
      "y": "0.398571526"
    },
    {
      "quarter": 2,
      "turn": "-0.414143754",
      "x": "0.411125590",
      "y": "0.848369445"
    },
    {
      "quarter": 1,
      "turn": "0.414089057",
      "x": "0.544146788",
      "y": "0.585612402"
    }
  ],
  "radius": "0.129750372"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 25,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.610748876",
      "y": "0.935124811"
    },
    {
      "turn": "-0.000000004",
      "x": "0.870249624",
      "y": "0.935124810"
    },
    {
      "quarter": 1,
      "turn": "0.000000001",
      "x": "0.675624060",
      "y": "0.740499249"
    },
    {
      "turn": "-0.000000007",
      "x": "0.870249623",
      "y": "0.805374436"
    },
    {
      "quarter": 1,
      "turn": "0.000000005",
      "x": "0.324375939",
      "y": "0.648751885"
    },
    {
      "turn": "-0.000000008",
      "x": "0.870249623",
      "y": "0.675624062"
    },
    {
      "turn": "-0.000000004",
      "x": "0.870249623",
      "y": "0.545873688"
    },
    {
      "turn": "-0.000000012",
      "x": "0.389251125",
      "y": "0.454126319"
    },
    {
      "turn": "-0.000000010",
      "x": "0.389251126",
      "y": "0.324375943"
    },
    {
      "turn": "-0.000000006",
      "x": "0.389251123",
      "y": "0.064875191"
    },
    {
      "turn": "-0.000000001",
      "x": "0.648751871",
      "y": "0.064875190"
    },
    {
      "turn": "-0.000000002",
      "x": "0.129750376",
      "y": "0.722812726"
    },
    {
      "turn": "0.000000004",
      "x": "0.129750377",
      "y": "0.590898421"
    },
    {
      "turn": "-0.000000011",
      "x": "0.129750377",
      "y": "0.460613021"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.935124811",
      "y": "0.351248126"
    },
    {
      "turn": "0.000000002",
      "x": "0.129750376",
      "y": "0.330693500"
    },
    {
      "quarter": 1,
      "turn": "-0.000000013",
      "x": "0.583876688",
      "y": "0.259500753"
    },
    {
      "turn": "-0.000000006",
      "x": "0.129750376",
      "y": "0.066049839"
    },
    {
      "turn": "-0.000000004",
      "x": "0.389251125",
      "y": "0.194625566"
    },
    {
      "turn": "0.023475417",
      "x": "0.135843248",
      "y": "0.860149170"
    },
    {
      "turn": "0.000000002",
      "x": "0.129750376",
      "y": "0.197657110"
    },
    {
      "quarter": 2,
      "turn": "-0.413990188",
      "x": "0.861903911",
      "y": "0.138124699"
    },
    {
      "quarter": 2,
      "turn": "-0.414069522",
      "x": "0.731196832",
      "y": "0.398571526"
    },
    {
      "quarter": 2,
      "turn": "-0.414143754",
      "x": "0.411125590",
      "y": "0.848369445"
    },
    {
      "quarter": 1,
      "turn": "0.414089057",
      "x": "0.544146788",
      "y": "0.585612402"
    }
  ],
  "radius": "0.129750372"
}

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. · Verifier v1.0.0

DISCUSSION

Discussion

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