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P90 · Packing and covering · Classic · Hard

Dominoes in a square · n = 5

Pack n 1 × 2 dominoes into a square container with boundary contact allowed and disjoint interiors, minimising the side length of the container square.

Instancen = 5
ObjectiveMinimize container square side

Record comparison

container square side · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[3.76776, 3.76777)Jonathan ViqueratErich Friedman ↗
Site record3.767766962今天Record details ↓

Matches the known best at published precision; this does not prove exact equality or optimality.

Formal definition

  • ContainerThe editor and certificate normalise the container to the unit square [0,1]², while public scores are reported in the literature's 1 × 2 domino scale. Boundary contact is allowed.
  • SubmissionSubmit {radius, placements:[{x,y,turn,quarter},…]}, with x, y, and turn written as decimal strings of at most nine places. Here radius is the short side of each domino inside the unit-square container, so the container side length is 1/radius. The coordinates x, y specify the domino centre; quarter is an integer in 0–3 specifying an exact multiple of 90° applied before the rational rotation turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.
  • ConstraintsExactly n congruent dominoes lie wholly inside the container with pairwise disjoint interiors. Dominoes may rotate independently by arbitrary angles.
  • PrecisionInput decimals are parsed as exact rationals, with zero floating-point tolerance in feasibility checks.
  • ObjectiveContainer square side = 1/r (where r is the domino's short side inside the unit square), where smaller is better. Pages, leaderboards, and literature records share this unit with automatic conversion.
Open full editor ↗

Layout shown: site record

VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Literature targets and coordinate certificates

Literature targets are converted at their published precision, while nine-decimal coordinate certificates are stored and displayed alongside the source values.

Source
The record-holding arrangement for Dominoes in a square n = 5, 3.767766962
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holder今天
Solution methodHuman
Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (2 changes)
  1. 今天Human
    3.767766976→3.767766962
  2. CJerryHuman
    3.767766991→3.767766976
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 written as decimal strings of at most nine places. Here radius is the short side of each domino inside the unit-square container, so the container side length is 1/radius. The coordinates x, y specify the domino centre; quarter is an integer in 0–3 specifying an exact multiple of 90° applied before the rational rotation turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.398113794",
      "y": "0.265409196"
    },
    {
      "turn": "0.414213562",
      "x": "0.718491035",
      "y": "0.718491035"
    },
    {
      "turn": "0.006112809",
      "x": "0.267011701",
      "y": "0.864060645"
    },
    {
      "quarter": 1,
      "turn": "0.008949139",
      "x": "0.862566665",
      "y": "0.268913665"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.132704598",
      "y": "0.265490916"
    }
  ],
  "radius": "0.265409196"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

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

The current leader's answer

{
  "placements": [
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.398113794",
      "y": "0.265409196"
    },
    {
      "turn": "0.414213562",
      "x": "0.718491035",
      "y": "0.718491035"
    },
    {
      "turn": "0.006112809",
      "x": "0.267011701",
      "y": "0.864060645"
    },
    {
      "quarter": 1,
      "turn": "0.008949139",
      "x": "0.862566665",
      "y": "0.268913665"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.132704598",
      "y": "0.265490916"
    }
  ],
  "radius": "0.265409196"
}

Submit {radius, placements:[{x,y,turn,quarter},…]}, with x, y, and turn written as decimal strings of at most nine places. Here radius is the short side of each domino inside the unit-square container, so the container side length is 1/radius. The coordinates x, y specify the domino centre; quarter is an integer in 0–3 specifying an exact multiple of 90° applied before the rational rotation turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4. · Verifier v1.0.0

DISCUSSION

Discussion

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本题n=2，4，8，12，18，24，32，40时的证明NUE_13#313 replies外部记录更新（这次不想抄数字了）NUE_13#310 replies外部记录更新（9.24更新）NUE_13#310 repliesn=1的证明LittleChasa#1352 replies