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P04 · Packing and covering · Classic

Common-scale consecutive rectangle packing · n = 2

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 = 2
ObjectiveMaximize common scale
Proven optimum0.5the 1×2 forces 2·scale ≤ 1

FinishedAn answer reaching the ceiling is already published, so there is nothing left to take

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 editor and look ↗
1y0
0x1
VERIFIED CONSTRUCTIONscale = 0.500000000

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

0.500000000

common scale

OptimalThe optimum is proven; there is nothing left to gain
Record holderReference answer
Solution method—
Challenge this record ↗
Record history (2 changes)
  1. AnonymousHuman
    0.500000000→0.500000000
  2. Steven GaoAI · Google: Gemini 3.7 Flash
    0.333333333→0.500000000
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",
      "y": "0.5"
    },
    {
      "rotated": true,
      "x": "0",
      "y": "0"
    }
  ],
  "scale": "0.5"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 2
}

The current leader's answer

{
  "placements": [
    {
      "rotated": false,
      "x": "0",
      "y": "0.5"
    },
    {
      "rotated": true,
      "x": "0",
      "y": "0"
    }
  ],
  "scale": "0.5"
}

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