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Three utilities problem: History & Companies

The three utilities problem, also known as water, gas and electricity, is a mathematical puzzle that asks for non-crossing connections to be drawn between three houses and three utility companies on a plane. When posing it in the early 20th century, Henry Dudeney wrote that it was already an old problem. It is an impossible puzzle: it is not possible to…

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Three utilities problem topic overview

The analysis highlights History and Companies as prominent areas in the source structure around Three utilities problem.

Related topics
50
Source areas
5
Connected nodes
55
Extracted relationships
19
Concept neighborhoods
26
Bridge connections
55

What this topic covers Research coverage

Source areas are shown by the number of related topics found in each part of the analysis. Use smaller areas too: they can reveal specialized angles and content gaps.

Overview · 17 topics
Properties of the utility graph · 13 topics
History · 9 topics
Puzzle solutions · 7 topics
Statement · 4 topics

Smaller areas are not necessarily less important. They contain fewer connections in this analysis and can be useful for finding specialized angles or coverage gaps.

Explore all related topics Closing gaps

Browse the complete topic structure, not only the most central items. Less prominent entities and concepts can reveal missing angles, specialized context and useful research gaps. Each item opens a new analysis centered on that subject.

Overview

History

Statement

Puzzle solutions

Properties of the utility graph

Advanced semantic analysis

Deeper signals for content research, entity SEO and topical coverage. The plain-language headings explain what each technical view is useful for.

How Three utilities problem connects Entity context

The extracted context around Three utilities problem shows recurring relationship patterns in the source. For example, Three utilities problem → Another, Dudeney, He, HenryDudeney, However, In, It, Kullman, Loyd's, Sam Loyd, The Quarrelsome Neighbors, The Strand Magazine Another extracted example is Three utilities problem → An, Its, The. Use these groups to spot repeated connection types before inspecting the individual relationships.

Three utilities problem

Top relations

related to history · 12
Three utilities problem → Another, Dudeney, He, HenryDudeney, However, In, It, Kullman, Loyd's, Sam Loyd, The Quarrelsome Neighbors, The Strand Magazine
related to Statement · 3
Three utilities problem → An, Its, The

Important terminology

Use these terms to understand the vocabulary surrounding the topic, not as a checklist for keyword stuffing.

Important terminology

graph displaystyle problem puzzle utilities utility houses one graphs three planar plane vertices connections bipartite two nine complete crossings embedding

Three utilities problem relationships Subject–Predicate–Object triples

TTTA extracted 19 structured relationships around Three utilities problem. Examples in this analysis include a torus or Möbius strip → instance of → Versions of the problem on nonplanar surfaces and Three utilities problem → related to history → Kullman. The table shows each extracted connection, where it came from and its confidence.

SubjectPredicateObjectConfidenceSrc
a torus or Möbius stripinstance ofVersions of the problem on nonplanar surfaces0.80text
or that allow connections to pass through other houses or utilitiesinstance ofVersions of the problem on nonplanar surfaces0.80text
can be solved.This puzzle can be formalized as a problem in topological graph theory by asking whether the complete bipartite graph K 3instance ofVersions of the problem on nonplanar surfaces0.80text
3instance ofVersions of the problem on nonplanar surfaces0.80text
Three utilities problemrelated to historyKullman0.60section
Three utilities problemrelated to historyHe0.60section
Three utilities problemrelated to historyIn0.60section
Three utilities problemrelated to historyHenryDudeney0.60section
Three utilities problemrelated to historyHowever0.60section
Three utilities problemrelated to historyDudeney0.60section
Three utilities problemrelated to historyThe Strand Magazine0.60section
Three utilities problemrelated to historySam Loyd0.60section

Related concept clusters Concept neighborhoods

The concept neighborhoods around Three utilities problem bring nearby vocabulary together. In this analysis, examples include Utilities, House and Houses. Use the clusters to find adjacent concepts and terminology that may deserve separate research.

  • Three utilities problem
    • Utilities
    • House
    • Houses
    • Problem
    • Puzzle
    • Three
    • Also
    • Companies
    • Gas
    • One
    • Rigidity
    • Graph
  • three utilities problem
    • Utilities
    • House
    • Houses
    • Problem
    • Puzzle
    • Three
    • Bipartite
    • Complete
    • Utility
    • Displaystyle
    • Gas
    • Graph
  • mathematical puzzle
    • Utilities
    • Houses
    • Theory
    • Plane
    • Connections
    • Three
    • Utility
    • Drawn
    • Lines
    • Graphs
    • Companies
    • Puzzle
  • topological graph theory
    • Displaystyle
    • Rigidity
    • Vertices
    • Utility
    • Planar
    • Graphs
    • Bipartite
    • Complete
    • Two
    • Problem
    • Well-covered
    • Three
  • complete bipartite graph
    • Bipartite
    • Complete
    • Displaystyle
    • Number
    • Vertices
    • Utility
    • Edges
    • Graphs
    • Embedding
    • Planar
    • Crossings
    • Problem
  • graph embedding
    • Displaystyle
    • Vertices
    • Theory
    • Utility
    • Planar
    • Bipartite
    • Complete
    • Two
    • Problem
    • Impossibility
    • Three
    • Rigidity
  • planar graph
    • Displaystyle
    • Vertices
    • Utility
    • Planar
    • Two
    • Graphs
    • Bipartite
    • Complete
    • One
    • Problem
    • Three
    • Rigidity
  • turán's brick factory problem
    • Utilities
    • Houses
    • Three
    • Bipartite
    • Complete
    • Displaystyle
    • Puzzle
    • Gas
    • Graph
    • Connections
    • Vertices
    • Utility

Connections between topic areas Semantic bridges

For Three utilities problem, one of the stronger structural bridges in this analysis connects Three utilities problem with Overview. Bridges highlight paths between different parts of the map and can reveal research angles that are easy to miss in a flat list.

Min side: 3
Three utilities problem — Overview · splits 38 ⟂ 18
Three utilities problem — Properties of the utility graph · splits 42 ⟂ 14
Three utilities problem — History · splits 46 ⟂ 10
Three utilities problem — Puzzle solutions · splits 48 ⟂ 8
Three utilities problem — Statement · splits 51 ⟂ 5

Map overview Semantic statistics

Three utilities problem

Nodes56
Edges55
Triples19
Avg. degree1.96
Density0.035714
Components1

Source & methodology

TTTA analyzes the structure around Three utilities problem to surface related topics, entities, relationships, concept neighborhoods and bridge connections. Use the map to explore areas such as History & Companies, including less central topics that may reveal useful research gaps. Automatically extracted connections are research leads rather than rewritten encyclopedia content.

Source: Wikipedia — Three utilities problem · EN edition · Analysis: TopicsToTalkAbout

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