P05 · Packing and covering · Classic

Tilted equal squares in a circle · n = 8

Place n freely rotatable squares of one common side inside a circle of radius 1, none overlapping, making that side as large as possible.

Instancen = 8

ObjectiveMaximize the smallest side

Best known, unproven0.505364222the packing is David W. Cantrell's, 2002; its radius 1.97877086533976321… was computed to 36 places by David Ellsworth in 2023; optimality open

Formal definition

ContainerA circle of radius 1 centred at (1, 1), so both coordinates run from 0 to 2
SubmissionEach square is {cx, cy, ux, uy}: a centre plus one half-edge vector, the other fixed as (−uy, ux)
ConstraintsEvery square tilts freely; all lie inside the container; no two overlap in their interiors, touching allowed. Only the smallest square is scored, so unequal sides gain nothing
ObjectiveMake the common side as large as possible; compared internally by its square, exactly

Open the full editor ↗

2y0

0x2

VERIFIED CONSTRUCTION8 squares

Getting a feel for it

Where the room for improvement is

A circular container has no corners: straight sides cannot hug the arc, so optima are almost always tilted, squares bracing corner against corner.

Where the frontier is

n = 4 is proven; the other best known values come from Friedman's squares-in-circles survey (many contributors since 1997), none of them proven.

Source

Current leader

0.248069464

the smallest side

Record holderFounding benchmark

Solution methodHuman

Challenge this record ↗

ANSWER FORMAT

How to write your answer

The container is a circle of radius 1 centred at (1, 1), so coordinates run from 0 to 2. Coordinates and vectors share one unit and are written as plain decimals such as "0.4", to at most nine decimal places.

Submit {cx,cy,ux,uy} per square: a centre plus one half-edge vector, the other half-edge being (-uy,ux). Write every number as a decimal string such as "0.4", every angle is free, and the smallest square is the one scored, so writing them equal is the winning move. The side is almost always irrational, so what you write is the half-edge vector and the side follows from it exactly.

The current leader's answer

{
  "squares": [
    {
      "cx": "0.131756876",
      "cy": "1",
      "ux": "0.124034732",
      "uy": "0"
    },
    {
      "cx": "0.37982634",
      "cy": "1",
      "ux": "0.124034732",
      "uy": "0"
    },
    {
      "cx": "0.627895804",
      "cy": "1",
      "ux": "0.124034732",
      "uy": "0"
    },
    {
      "cx": "0.875965268",
      "cy": "1",
      "ux": "0.124034732",
      "uy": "0"
    },
    {
      "cx": "1.124034732",
      "cy": "1",
      "ux": "0.124034732",
      "uy": "0"
    },
    {
      "cx": "1.372104196",
      "cy": "1",
      "ux": "0.124034732",
      "uy": "0"
    },
    {
      "cx": "1.62017366",
      "cy": "1",
      "ux": "0.124034732",
      "uy": "0"
    },
    {
      "cx": "1.868243124",
      "cy": "1",
      "ux": "0.124034732",
      "uy": "0"
    }
  ]
}

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

Instance parameters

{
  "n": 8
}

The current leader's answer

{
  "squares": [
    {
      "cx": "0.131756876",
      "cy": "1",
      "ux": "0.124034732",
      "uy": "0"
    },
    {
      "cx": "0.37982634",
      "cy": "1",
      "ux": "0.124034732",
      "uy": "0"
    },
    {
      "cx": "0.627895804",
      "cy": "1",
      "ux": "0.124034732",
      "uy": "0"
    },
    {
      "cx": "0.875965268",
      "cy": "1",
      "ux": "0.124034732",
      "uy": "0"
    },
    {
      "cx": "1.124034732",
      "cy": "1",
      "ux": "0.124034732",
      "uy": "0"
    },
    {
      "cx": "1.372104196",
      "cy": "1",
      "ux": "0.124034732",
      "uy": "0"
    },
    {
      "cx": "1.62017366",
      "cy": "1",
      "ux": "0.124034732",
      "uy": "0"
    },
    {
      "cx": "1.868243124",
      "cy": "1",
      "ux": "0.124034732",
      "uy": "0"
    }
  ]
}