P04 · Packing and covering · Classic

Common-scale consecutive rectangle packing · n = 19

Scale the rectangles 1×1, 1×2, …, 1×n by one common factor and fit them all into the unit square, making that factor as large as possible.

Instancen = 19

ObjectiveMaximize common scale

Formal definition

ContainerThe unit square: the origin (0, 0) at its lower-left corner, (1, 1) at its upper right
SubmissionSubmit placements in order 1×1 through 1×n, each a position plus an optional 90° rotation, together with the shared scale
ConstraintsAll rectangles use the one scale; rotation only by 90°; none overlap and none crosses the boundary
ObjectiveMake the shared scale as large as possible

Open the full editor ↗

1y0

1×11×21×31×41×51×61×71×81×91×101×111×121×131×141×151×161×171×181×19

0x1

VERIFIED CONSTRUCTIONscale = 0.052631578

Getting a feel for it

Where the room for improvement is

Strips from 1×1 to 1×n grow together, the narrow ones ever longer: upright or flat, whose seam hugs whose, a jigsaw where every piece must fit. Raise the scale a hair and all n strips must find new homes.

Where the frontier is

n ≤ 2 is trivial. The close relative — consecutive rectangles in similar rectangles — is collected on Friedman's Packing Center (the minrect page); this into-a-square variant has no per-n literature, and every other n is open.

Source

Current leader

0.052631578

common scale

Record holderlird

Solution methodHuman

Challenge this record ↗

Record history (1 changes)

lirdHuman
0.050000000→0.052631578
2026-08-25

ANSWER FORMAT

How to write your answer

The container is a square of side 1. Its lower-left corner is the origin (0, 0) and its upper-right corner is (1, 1). Coordinates and lengths share one unit and are written as plain decimals such as "0.5", to at most nine decimal places.

Submit placements in 1×i order; 90° rotation is allowed.

The current leader's answer

{
  "placements": [
    {
      "rotated": false,
      "x": "0.000000000",
      "y": "0.819130425"
    },
    {
      "rotated": false,
      "x": "0.000000000",
      "y": "0.685217418"
    },
    {
      "rotated": false,
      "x": "0.000000000",
      "y": "0.504347826"
    },
    {
      "rotated": false,
      "x": "0.000000000",
      "y": "0.274782661"
    },
    {
      "rotated": false,
      "x": "0.000000000",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.095217365",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.100000000",
      "y": "0.342608841"
    },
    {
      "rotated": false,
      "x": "0.184782608",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.187826140",
      "y": "0.439999987"
    },
    {
      "rotated": false,
      "x": "0.279565297",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.362608802",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.445652173",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.549565324",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.634347800",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.689565244",
      "y": "0.003478274"
    },
    {
      "rotated": false,
      "x": "0.750000000",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.810434757",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.870869513",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.947368422",
      "y": "0.000000000"
    }
  ],
  "scale": "0.052631578"
}

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

Instance parameters

{
  "n": 19
}

The current leader's answer

{
  "placements": [
    {
      "rotated": false,
      "x": "0.000000000",
      "y": "0.819130425"
    },
    {
      "rotated": false,
      "x": "0.000000000",
      "y": "0.685217418"
    },
    {
      "rotated": false,
      "x": "0.000000000",
      "y": "0.504347826"
    },
    {
      "rotated": false,
      "x": "0.000000000",
      "y": "0.274782661"
    },
    {
      "rotated": false,
      "x": "0.000000000",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.095217365",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.100000000",
      "y": "0.342608841"
    },
    {
      "rotated": false,
      "x": "0.184782608",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.187826140",
      "y": "0.439999987"
    },
    {
      "rotated": false,
      "x": "0.279565297",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.362608802",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.445652173",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.549565324",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.634347800",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.689565244",
      "y": "0.003478274"
    },
    {
      "rotated": false,
      "x": "0.750000000",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.810434757",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.870869513",
      "y": "0.000000000"
    },
    {
      "rotated": false,
      "x": "0.947368422",
      "y": "0.000000000"
    }
  ],
  "scale": "0.052631578"
}

Submit placements in 1×i order; 90° rotation is allowed. · Verifier v1.0.0