Mass Transit

A problem from the October 1964 issue of Eureka, the journal of the Cambridge University Mathematical Society:

The planet Kophikkup is in the shape of a torus or ring-doughnut. There is a direct mono-rail line from each of the four space-ports to each of the major cities. No lines join or cross. What is the greatest possible number of major cities? Draw a diagram for this case.

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Four. The map above shows that each of four spaceports can communicate directly with each of four cities on the surface of the torus, but there’s nowhere to “put” a fifth city that wouldn’t require some of the monorail lines to cross. (From the Eureka Digital Archive, published under a Creative Commons license.) 03/26/2024 Reader Drake Thomas sent a better diagram: (Thanks, Drake.)

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