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P90 · Packing and covering · Classic · Hard

Dominoes in a square · n = 19

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

Instancen = 19
ObjectiveMinimize container square side
Best known, unproven6.71428Source-precision interval: 6.71428–6.71429 (values inside match)exhibited site certificate 6.714285905Compiled by Erich Friedman; Found by Maurizio Morandi in June 2026. Source expression: s = 47/7 = 6.71428+.

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 = 19, 6.714285905
Current leader

6.714285905

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": "0.149003143",
      "y": "0.925531914"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.074468086",
      "y": "0.702063169"
    },
    {
      "quarter": 1,
      "turn": "-0.000000006",
      "x": "0.223404258",
      "y": "0.553191490"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.074468086",
      "y": "0.403662639"
    },
    {
      "quarter": 1,
      "turn": "-0.000000006",
      "x": "0.776595742",
      "y": "0.298207969"
    },
    {
      "turn": "-0.000000005",
      "x": "0.403861099",
      "y": "0.074468089"
    },
    {
      "turn": "0.000000000",
      "x": "0.702049132",
      "y": "0.074468086"
    },
    {
      "turn": "-0.000000008",
      "x": "0.447048233",
      "y": "0.925531911"
    },
    {
      "quarter": 1,
      "turn": "-0.000000004",
      "x": "0.776595743",
      "y": "0.596340407"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.925531914",
      "y": "0.750296645"
    },
    {
      "turn": "-0.000000005",
      "x": "0.297872340",
      "y": "0.776595742"
    },
    {
      "quarter": 1,
      "turn": "-0.000000008",
      "x": "0.627659570",
      "y": "0.446808511"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.925531914",
      "y": "0.447713807"
    },
    {
      "turn": "-0.000000004",
      "x": "0.553191488",
      "y": "0.223404257"
    },
    {
      "quarter": 1,
      "turn": "-0.000000002",
      "x": "0.925531913",
      "y": "0.149175907"
    },
    {
      "quarter": 1,
      "turn": "-0.403900599",
      "x": "0.163015951",
      "y": "0.161731086"
    },
    {
      "quarter": 2,
      "turn": "0.333333329",
      "x": "0.385165549",
      "y": "0.376640121"
    },
    {
      "quarter": 2,
      "turn": "0.333333332",
      "x": "0.469817522",
      "y": "0.626299311"
    },
    {
      "quarter": 1,
      "turn": "-0.404101844",
      "x": "0.690551403",
      "y": "0.841123403"
    }
  ],
  "radius": "0.148936166"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 19,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.149003143",
      "y": "0.925531914"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.074468086",
      "y": "0.702063169"
    },
    {
      "quarter": 1,
      "turn": "-0.000000006",
      "x": "0.223404258",
      "y": "0.553191490"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.074468086",
      "y": "0.403662639"
    },
    {
      "quarter": 1,
      "turn": "-0.000000006",
      "x": "0.776595742",
      "y": "0.298207969"
    },
    {
      "turn": "-0.000000005",
      "x": "0.403861099",
      "y": "0.074468089"
    },
    {
      "turn": "0.000000000",
      "x": "0.702049132",
      "y": "0.074468086"
    },
    {
      "turn": "-0.000000008",
      "x": "0.447048233",
      "y": "0.925531911"
    },
    {
      "quarter": 1,
      "turn": "-0.000000004",
      "x": "0.776595743",
      "y": "0.596340407"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.925531914",
      "y": "0.750296645"
    },
    {
      "turn": "-0.000000005",
      "x": "0.297872340",
      "y": "0.776595742"
    },
    {
      "quarter": 1,
      "turn": "-0.000000008",
      "x": "0.627659570",
      "y": "0.446808511"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.925531914",
      "y": "0.447713807"
    },
    {
      "turn": "-0.000000004",
      "x": "0.553191488",
      "y": "0.223404257"
    },
    {
      "quarter": 1,
      "turn": "-0.000000002",
      "x": "0.925531913",
      "y": "0.149175907"
    },
    {
      "quarter": 1,
      "turn": "-0.403900599",
      "x": "0.163015951",
      "y": "0.161731086"
    },
    {
      "quarter": 2,
      "turn": "0.333333329",
      "x": "0.385165549",
      "y": "0.376640121"
    },
    {
      "quarter": 2,
      "turn": "0.333333332",
      "x": "0.469817522",
      "y": "0.626299311"
    },
    {
      "quarter": 1,
      "turn": "-0.404101844",
      "x": "0.690551403",
      "y": "0.841123403"
    }
  ],
  "radius": "0.148936166"
}

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

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