P77 · Extremal configurations · Classic · Hard

Covering an equilateral triangle with n equilateral triangles · n = 15

Continuously cover the unit equilateral triangle with n congruent equilateral triangles. Overlap and protrusion are allowed; minimise the common side.

Instancen = 15

ObjectiveMinimize the common side

Record comparison

the common side · Lower is better

Record	Value / interval	Author / holder	Source
External best known	4/15	See source attribution	Erich Friedman ↗
Site record	0.266666667	zzzcy	Record details ↓

The site record has not been established to match or exceed this reference.

Formal definition

ContainerThe closed equilateral triangle K = conv{(0,0),(1,0),(0.5,√3/2)}
SubmissionA common side s and exactly n {x,y,turn,down} placements, each independently rotatable and flippable
CoverEvery point of K belongs to at least one closed covering triangle $K \subseteq T_{1} \cup \dots \cup T_{n}$
Exact continuous verificationThe verifier computes all container edges, piece edges, and intersections exactly in Q(√3), then performs a vertical sweep to verify full coverage
ScoreThe common side s shared by all covering triangles; smaller is better $s (T) = s$
ObjectiveMinimise s over all valid covers $\min_{T} s (T)$

Open full editor ↗

Layout shown: site record

VERIFIED CONSTRUCTIONcommon side 0.266666667

Getting a feel for it

Non-square piece counts

Only when n is a perfect square can a regular triangular grid tile the container without overlap or protrusion. Other values of n require overlapping, rotated, or protruding pieces, and adding one piece often alters the layered structure.

Known constructions and literature data

Friedman's table lists constructions for n = 6–9 and n = 11–15; at n = 9, the 3 × 3 triangular grid meets the area lower bound and is provably optimal. For n = 10, the table uses the n = 9 grid plus one redundant piece as an initial baseline. All other instances are reconstructed from the published diagrams, with literature values and nine-decimal certificate scores listed separately.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

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 (1 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.266666668→0.266666667
2026-10-07

ANSWER FORMAT

How to write your answer

The container has vertices (0,0), (1,0), and (0.5,√3/2). Coordinates mark one reference-edge midpoint; covering pieces may protrude.

Submit side and triangles, each as {x,y,turn,down}. The editor allows entering or dragging the angle θ° and automatically converts it to turn = tan(θ/2).

The current leader's answer

{
  "side": "0.266666667",
  "triangles": [
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.600000001",
      "y": "0.000000000"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.433333332",
      "y": "0.404145189"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.133333333",
      "y": "0.000000000"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.500000001",
      "y": "0.635085296"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.866666667",
      "y": "0.000000000"
    },
    {
      "down": true,
      "turn": "-0.577350269",
      "x": "0.800000000",
      "y": "0.346410162"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.366666667",
      "y": "0.404145189"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.500000000",
      "y": "0.635085296"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.400000001",
      "y": "0.000000000"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.266666666",
      "y": "0.230940107"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.266666667",
      "y": "0.230940107"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.633333334",
      "y": "0.404145188"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.566666666",
      "y": "0.404145188"
    },
    {
      "down": false,
      "turn": "-0.577350269",
      "x": "0.666666667",
      "y": "0.115470054"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.500000001",
      "y": "0.288675135"
    }
  ]
}

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

Instance parameters

{
  "n": 15
}

The current leader's answer

{
  "side": "0.266666667",
  "triangles": [
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.600000001",
      "y": "0.000000000"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.433333332",
      "y": "0.404145189"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.133333333",
      "y": "0.000000000"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.500000001",
      "y": "0.635085296"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.866666667",
      "y": "0.000000000"
    },
    {
      "down": true,
      "turn": "-0.577350269",
      "x": "0.800000000",
      "y": "0.346410162"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.366666667",
      "y": "0.404145189"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.500000000",
      "y": "0.635085296"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.400000001",
      "y": "0.000000000"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.266666666",
      "y": "0.230940107"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.266666667",
      "y": "0.230940107"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.633333334",
      "y": "0.404145188"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.566666666",
      "y": "0.404145188"
    },
    {
      "down": false,
      "turn": "-0.577350269",
      "x": "0.666666667",
      "y": "0.115470054"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.500000001",
      "y": "0.288675135"
    }
  ]
}

Submit side and triangles, each as {x,y,turn,down}. The editor allows entering or dragging the angle θ° and automatically converts it to turn = tan(θ/2). · Verifier v1.0.0

DISCUSSION

Discussion

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论n=10的最优解hadsajnc#1463 replies30d ago