P76 · Extremal configurations · Classic · Hard

Covering a square with n equilateral triangles · n = 8

Cover the unit square with n congruent equilateral triangles. Every piece may be translated, rotated and flipped independently; arbitrary overlap and protrusion are allowed. Minimise the common side.

Instancen = 8

ObjectiveMinimize the common side

Best known, unproven2(12 + 5√3) / 69published value in this metric 0.598847943125924exhibited site certificate 0.599559417Maurizio Morandi's construction appears in Friedman's public table (April 2009). Rescaling the largest square covered by unit triangles gives common side 2(12 + 5√3) / 69 ≈ 0.598847943125924; optimality is open. The site's nine-decimal reconstruction of the source figure verifies at 0.599559417, with its gap from the published target explicitly preserved.Submissions are decimal, so a record chases the writable value. If you believe you have the exact closed form, send it in. minmaxarena.com@gmail.com

Formal definition

ContainerThe closed unit square K = [0,1] × [0,1]
Piecesn equilateral triangles of common side s; every piece may be translated and rotated independently, with the third vertex on either side of its reference edge
SubmissionGive the common side s plus one reference-edge midpoint, rotation and orientation for each triangle; submit exactly n pieces
CoverEvery point of the square lies in at least one closed triangle; triangles may overlap and protrude $K \subseteq T_{1} \cup \dots \cup T_{n}$
Exact rotationThe certificate writes rotations as turn = tan(θ/2). Half-angle identities make cos θ and sin θ rational, so every vertex and intersection is computed exactly in Q(√3)
Continuous verificationThe verifier builds the planar arrangement of triangle and square edges, then performs an exact algebraic vertical sweep for uncovered faces; it uses neither pixels nor random sampling
ScoreThe side s shared by every submitted triangle; smaller is better $s (T) = s$
ObjectiveMinimise s over all legal covers $\min_{T} s (T)$

Open the full editor ↗

VERIFIED CONSTRUCTIONcommon side 0.599559417

Getting a feel for it

Why this is not tiling

A square has 90° corners and an equilateral triangle has 60° corners. The boundary angles do not match, so good covers must overlap and protrude. Adding one piece can reorganise the whole contact pattern rather than append one more tile.

The published frontier

Friedman's public table gives the best known constructions for n=1–12, with nontrivial layouts already at n=2 and independent rotations from n ≥ 4. This site reconstructs the pictured arrangements and rechecks continuous coverage instead of copying numbers alone.

Source

Current leader

0.599559417

the common side

Answer sourceMaurizio Morandi

Solution methodPublished reference construction

Challenge this record ↗

ANSWER FORMAT

How to write your answer

The container is the unit square [0,1]². Covering triangles may protrude; (x,y) marks the midpoint of one edge.

Submit side and triangles. Each piece is {x,y,turn,down}: (x,y) is one edge's midpoint; turn = tan(θ/2) encodes that edge's angle θ; down chooses the side containing the third vertex. The editor writes turn automatically. Decimals have at most nine places.

The current leader's answer

{
  "side": "0.599559417",
  "triangles": [
    {
      "down": false,
      "turn": "0",
      "x": "0.187814373",
      "y": "0"
    },
    {
      "down": false,
      "turn": "0",
      "x": "0.823299122",
      "y": "0"
    },
    {
      "down": true,
      "turn": "0",
      "x": "0.180427531",
      "y": "1"
    },
    {
      "down": true,
      "turn": "0",
      "x": "0.813274561",
      "y": "1"
    },
    {
      "down": false,
      "turn": "1",
      "x": "1",
      "y": "0.504474990"
    },
    {
      "down": true,
      "turn": "1",
      "x": "0",
      "y": "0.493587641"
    },
    {
      "down": true,
      "turn": "0.577350269",
      "x": "0.346827406",
      "y": "0.769489255"
    },
    {
      "down": false,
      "turn": "0.577350269",
      "x": "0.655542400",
      "y": "0.228309899"
    }
  ]
}

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

Instance parameters

{
  "n": 8
}

The current leader's answer

{
  "side": "0.599559417",
  "triangles": [
    {
      "down": false,
      "turn": "0",
      "x": "0.187814373",
      "y": "0"
    },
    {
      "down": false,
      "turn": "0",
      "x": "0.823299122",
      "y": "0"
    },
    {
      "down": true,
      "turn": "0",
      "x": "0.180427531",
      "y": "1"
    },
    {
      "down": true,
      "turn": "0",
      "x": "0.813274561",
      "y": "1"
    },
    {
      "down": false,
      "turn": "1",
      "x": "1",
      "y": "0.504474990"
    },
    {
      "down": true,
      "turn": "1",
      "x": "0",
      "y": "0.493587641"
    },
    {
      "down": true,
      "turn": "0.577350269",
      "x": "0.346827406",
      "y": "0.769489255"
    },
    {
      "down": false,
      "turn": "0.577350269",
      "x": "0.655542400",
      "y": "0.228309899"
    }
  ]
}

DISCUSSION

Discussion

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