Editing Three utilities problem (section)

==References==
{{reflist|refs=

<ref name=3wells>{{citation|title=Puzzle|url=https://books.google.com/books?id=yLSTwH0pINIC&q=%22three+houses+and+three+wells%22|magazine=Successful Farming|year=1914|volume=13|page=50}}; {{citation|url=https://books.google.com/books?id=w8tPAQAAMAAJ&pg=PA392|year=1916|magazine=The Youth's Companion|page=392|volume=90|issue=2|title=A well and house puzzle}}.</ref>

<ref name=ayres>{{citation
 | last = Ayres | first = W. L.
 | doi = 10.1080/00029890.1938.11990773
 | issue = 2
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2304276
 | mr = 1524194
 | pages = 88–92
 | title = Some elementary aspects of topology
 | volume = 45
 | year = 1938}}</ref>

<ref name=bollobas>{{citation
 | last = Bollobás | first = Béla | author-link = Béla Bollobás
 | doi = 10.1007/978-1-4612-0619-4
 | isbn = 0-387-98488-7
 | mr = 1633290
 | page = 23
 | publisher = Springer-Verlag, New York
 | series = Graduate Texts in Mathematics
 | title = Modern Graph Theory
 | url = https://books.google.com/books?id=JeIlBQAAQBAJ&pg=PA23
 | volume = 184
 | year = 1998}}</ref>

<ref name=bona>{{citation
 | last = Bóna | first = Miklós | author-link = Miklós Bóna
 | isbn = 9789814335232
 | pages = 275–277
 | publisher = World Scientific
 | title = A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory
 | year = 2011}}. Bóna introduces the puzzle (in the form of three houses to be connected to three wells) on p.&nbsp;275, and writes on p.&nbsp;277 that it "is equivalent to the problem of drawing <math>K_{3,3}</math> on a plane surface without crossings".</ref>

<ref name=cer93>{{citation
 | last1 = Campbell | first1 = S. R.
 | last2 = Ellingham | first2 = M. N. | author2-link = Mark Ellingham
 | last3 = Royle | first3 = Gordon F. | author3-link = Gordon Royle
 | journal = Journal of Combinatorial Mathematics and Combinatorial Computing
 | mr = 1220613
 | pages = 193–212
 | title = A characterisation of well-covered cubic graphs
 | volume = 13
 | year = 1993}}</ref>

<ref name=dixon>{{citation
 | last = Dixon | first = A. C. | author-link = Alfred Cardew Dixon
 | jfm = 30.0622.02
 | journal = [[Messenger of Mathematics]]
 | pages = 1–21
 | title = On certain deformable frameworks
 | url = https://gdz.sub.uni-goettingen.de/id/PPN599484047_0029?tify=%7B%22pages%22%3A%5B5%5D%7D
 | volume = 29
 | year = 1899}}</ref>

<ref name=dud13>{{citation
 | last = Dudeney | first = Henry | author-link = Henry Dudeney
 | magazine = [[The Strand Magazine]]
 | page = 110
 | title = Perplexities, with some easy puzzles for beginners
 | url = https://archive.org/stream/TheStrandMagazineAnIllustratedMonthly/TheStrandMagazine1913bVol.XlviJul-dec#page/n119/mode/2up
 | volume = 46
 | year = 1913}}</ref>

<ref name=dud17>{{citation
 | last = Dudeney | first = Henry | author-link = Henry Dudeney
 | contribution = Problem 251 – Water, Gas, and Electricity
 | contribution-url = https://archive.org/stream/amusementsinmath00dude#page/72
 | page = 73
 | publisher = Thomas Nelson
 | title = Amusements in mathematics
 | year = 1917| volume = 100 | issue = 2512 | doi = 10.1038/100302a0 | bibcode = 1917Natur.100..302. | s2cid = 10245524 }}. The solution given on [https://archive.org/details/amusementsinmath00dude/page/200 pp. 200–201] involves passing a line through one of the other houses.</ref>

<ref name=early>{{citation
 | last1 = Beineke | first1 = Lowell | author1-link = L. W. Beineke
 | last2 = Wilson | first2 = Robin | author2-link = Robin Wilson (mathematician)
 | doi = 10.1007/s00283-009-9120-4
 | issue = 2
 | journal = [[The Mathematical Intelligencer]]
 | mr = 2657999
 | pages = 41–48
 | s2cid = 122588849
 | title = The early history of the brick factory problem
 | volume = 32
 | year = 2010}}</ref>

<ref name=fountains>{{citation
 | contribution = 32. The fountain puzzle
 | contribution-url = https://books.google.com/books?id=kbI_AQAAMAAJ&pg=PA276
 | location = New York
 | page = 276
 | publisher = Dick & Fitzgerald
 | title = The Magician's Own Book, Or, The Whole Art of Conjuring
 | year = 1857}}</ref>

<ref name=gs93>{{citation
 | last1 = Gries | first1 = David | author1-link = David Gries
 | last2 = Schneider | first2 = Fred B. | author2-link = Fred B. Schneider
 | contribution = Chapter 19: A theory of graphs
 | doi = 10.1007/978-1-4757-3837-7
 | pages = 423–460
 | publisher = Springer | location = New York
 | title = A Logical Approach to Discrete Math
 | year = 1993| isbn = 978-1-4419-2835-1 | s2cid = 206657798 }}. See p. 437: "<math>K_{3,3}</math> is known as the ''utility graph''".</ref>

<ref name=harary>{{citation
 | last = Harary | first = F. | author-link = Frank Harary
 | doi = 10.1007/BF01897149
 | journal = Acta Mathematica
 | mr = 166775
 | pages = 405–411
 | title = Recent results in topological graph theory
 | volume = 15
 | year = 1964| issue = 3–4 | hdl = 2027.42/41775 | s2cid = 123170864 | hdl-access = free
 }}; see p. 409.</ref>

<ref name=henneberg>{{citation
 | last = Henneberg | first = L.
 | contribution = Die graphische Statik der starren Körper
 | contribution-url = https://archive.org/details/encyklomath104encyrich/page/n367
 | pages = 345–434
 | title = Encyklopädie der Mathematischen Wissenschaften
 | volume = 4
 | year = 1908| issue = 1
 }}. See in particular [https://archive.org/stream/encyklomath104encyrich/#page/n425 p.&nbsp;403].</ref>

<ref name=intuitive>{{citation
 | last = Harary | first = Frank | author-link = Frank Harary
 | doi = 10.1137/1002023
 | journal = [[SIAM Review]]
 | mr = 111698
 | pages = 123–131
 | title = Some historical and intuitive aspects of graph theory
 | volume = 2
 | year = 1960| issue = 2 | bibcode = 1960SIAMR...2..123H }}</ref>

<ref name=kappraff>{{citation
 | last = Kappraff | first = Jay | author-link = Jay Kappraff
 | isbn = 9789810245863
 | page = 128
 | publisher = World Scientific
 | series = K & E Series on Knots and Everything
 | title = Connections: The Geometric Bridge Between Art and Science
 | url = https://books.google.com/books?id=twF7pOYXSTcC&pg=PA128
 | volume = 25
 | year = 2001}}</ref>

<ref name=kullman1979>{{citation
 | last = Kullman | first = David
 | issue = 5
 | journal = [[Mathematics Magazine]]
 | jstor = 2689782
 | pages = 299–302
 | title = The utilities problem
 | volume = 52
 | year = 1979| doi = 10.1080/0025570X.1979.11976807
 }}</ref>

<ref name=kuratowski>{{citation
 | last = Kuratowski | first = Kazimierz | author-link = Kazimierz Kuratowski
 | journal = Fundamenta Mathematicae
 | language = fr
 | pages = 271–283
 | title = Sur le problème des courbes gauches en topologie
 | url = http://matwbn.icm.edu.pl/ksiazki/fm/fm15/fm15126.pdf
 | volume = 15
 | year = 1930
 | doi = 10.4064/fm-15-1-271-283|doi-access=free }}</ref>

<ref name=larsen>{{citation
 | last = Larsen | first = Mogens Esrom
 | editor1-last = Guy | editor1-first = Richard K. | editor1-link = Richard K. Guy
 | editor2-last = Woodrow | editor2-first = Robert E.
 | contribution = Misunderstanding my mazy mazes may make me miserable
 | isbn = 0-88385-516-X
 | mr = 1303141
 | pages = 289–293
 | publisher = Mathematical Association of America | location = Washington, DC
 | series = MAA Spectrum
 | title = Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History held at the University of Calgary, Calgary, Alberta, August 1986
 | year = 1994}}. See [https://books.google.com/books?id=FsH2DwAAQBAJ&pg=PA292 Figure 7, p. 292].</ref>

<ref name=little>{{citation
 | last = Little | first = Charles H. C.
 | editor1-last = Casse | editor1-first = Louis R. A.
 | editor2-last = Wallis | editor2-first = Walter D.
 | contribution = A theorem on planar graphs
 | doi = 10.1007/BFb0097375
 | mr = 0427121
 | pages = 136–141
 | publisher = Springer
 | series = Lecture Notes in Mathematics
 | title = Combinatorial Mathematics IV: Proceedings of the Fourth Australian Conference Held at the University of Adelaide August 27–29, 1975
 | volume = 560
 | year = 1976}}</ref>

<ref name=obeirne>{{citation
 | last = O’Beirne | first = T. H.
 | date = December 21, 1961
 | issue = 266
 | magazine = [[New Scientist]]
 | pages = 751–753
 | title = Christmas puzzles and paradoxes, 51: For boys, men and heroes
 | url = https://books.google.com/books?id=rykw9gx81GoC&pg=PA751
 | volume = 12}}</ref>

<ref name=parker>{{citation
 | last = Parker | first = Matt | author-link = Matt Parker
 | isbn = 978-0-374-53563-6
 | mr = 3753642
 | pages = 180–181, 191–192
 | publisher = Farrar, Straus and Giroux | location = New York
 | title = Things to Make and Do in the Fourth Dimension: A Mathematician's Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More
 | year = 2015}}</ref>

<ref name=ps09>{{citation
 | last1 = Pach | first1 = János | author1-link = János Pach
 | last2 = Sharir | first2 = Micha | author2-link = Micha Sharir
 | contribution = 5.1 Crossings—the Brick Factory Problem
 | pages = 126–127
 | publisher = [[American Mathematical Society]]
 | series = Mathematical Surveys and Monographs
 | title = Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures
 | volume = 152
 | year = 2009}}</ref>

<ref name=quarrelsome>{{citation|title=Mathematical Puzzles of Sam Loyd|first=Sam|last=Loyd|author-link=Sam Loyd|editor-first=Martin|editor-last=Gardner|editor-link=Martin Gardner|publisher=Dover Books|year=1959|isbn=((9780486204987))<!-- isbn ok, for later printing of same edition by same publisher -->|contribution=82: The Quarrelsome Neighbors|page=79|contribution-url=https://books.google.com/books?id=QCy6DzgqcI4C&pg=PA79}}</ref>

<ref name=streinu>{{citation
 | last = Streinu | first = Ileana | author-link = Ileana Streinu
 | doi = 10.1007/s00454-005-1184-0 | doi-access=free
 | issue = 4
 | journal = [[Discrete & Computational Geometry]]
 | mr = 2173930
 | pages = 587–635
 | title = Pseudo-triangulations, rigidity and motion planning<!-- Note: Journal website has incorrect title in metadata; do not change title to "Acute triangulations of polygons" -->
 | volume = 34
 | year = 2005| s2cid = 25281202 }}. See p. 600: "Not all generically minimally rigid graphs have embeddings as pseudo-triangulations, because not all are planar graphs. The smallest example {{nowrap|is <math>K_{3,3}</math>".}}</ref>

<ref name=thomsen>{{citation
 | last = Thomsen | first = Julius | author-link = Hans Peter Jørgen Julius Thomsen
 | date = July 1886
 | doi = 10.1002/cber.188601902285
 | issue = 2
 | journal = Berichte der Deutschen Chemischen Gesellschaft
 | pages = 2944–2950
 | title = Die Constitution des Benzols
 | url = https://archive.org/download/crossref-pre-1909-scholarly-works/10.1002%252Fcber.18860190141.zip/10.1002%252Fcber.188601902285.pdf
 | volume = 19}}</ref>

<ref name=trudeau>{{citation
 | last = Trudeau | first = Richard J.
 | isbn = 978-0-486-67870-2
 | location = New York
 | pages = 68–70
 | publisher = Dover Publications
 | series = Dover Books on Mathematics
 | title = Introduction to Graph Theory
 | year = 1993}}</ref>

<ref name=tutte>{{citation
 | last = Tutte | first = W. T. | author-link = W. T. Tutte
 | doi = 10.1017/s0305004100023720
 | journal = [[Proceedings of the Cambridge Philosophical Society]]
 | mr = 21678
 | pages = 459–474
 | title = A family of cubical graphs
 | volume = 43
 | year = 1947| issue = 4 | bibcode = 1947PCPS...43..459T | s2cid = 123505185 }}</ref>

<ref name=wh07>{{citation
 | last1 = Walter | first1 = D.
 | last2 = Husty | first2 = M. L.
 | editor1-first = Jean-Pierre | editor1-last = Merlet
 | editor2-first = Marc | editor2-last = Dahan
 | contribution = On a nine-bar linkage, its possible configurations and conditions for paradoxical mobility
 | contribution-url = https://geometrie.uibk.ac.at/obsolete/institutsangehoerige/husty/dld/A681.pdf
 | publisher = [[International Federation for the Promotion of Mechanism and Machine Science]]
 | title = 12th World Congress on Mechanism and Machine Science (IFToMM 2007)
 | year = 2007}}</ref>

}}