Three utilities problem: Difference between revisions

Revision as of 08:03, 1 November 2021

Revision as of 16:28, 28 November 2021

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A review of the history of the three utilities problem is given by {{harvtxt|Kullman|1979}}. He states that most published references to the problem characterize it as "very ancient".{{r|kullman1979}} In the earliest publication found by Kullman, {{harvs|first=Henry|last=Dudeney|authorlink=Henry Dudeney|year=1917|txt}} names it "water, gas, and electricity". However, Dudeney states that the problem is "as old as the hills...much older than [[electric lighting]], or even [[town gas|gas]]".{{r|dud17}} Dudeney also published the same puzzle previously, in ''[[The Strand Magazine]]'' in 1913.{{r|dud13}} A competing claim of priority goes to [[Sam Loyd]], who was quoted by his son in a posthumous biography as having published the problem in 1900.{{r|early}}

Another early version of the problem involves connecting three houses to three wells.{{r|3wells}} It is stated similarly to a different (and solvable) puzzle that also involves three houses and three fountains, with all three fountains and one house touching a rectangular wall; the puzzle again involves making non-crossing connections, but only between three designated pairs of houses and wells or fountains, as in modern [[numberlink]] puzzles.{{r|fountains}} Sam Loyd's puzzle "The Quarrelsome Neighbors" similarly involves connecting three houses to three gates by three non-crossing paths (rather than nine as in the utilities problem); one house and the three gates are on the wall of a rectangular yard, which contains the other two houses within it.{{r|quarrelsome}}

As well as in the three utilities problem, the graph <math>K_{3,3}</math> appears in late 19th-century and early 20th-century publications both in early studies of [[structural rigidity]]{{r|dixon|henneberg}} and in [[chemical graph theory]], where [[Hans Peter Jørgen Julius Thomsen|Julius Thomsen]] proposed it in 1886 for the then-uncertain structure of [[benzene]].{{r|thomsen}} In honor of Thomsen's work, <math>K_{3,3}</math> is sometimes called the Thomsen graph.{{r|bollobas}}