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Programming Enigma Puzzles

18 January 2014

Posted by on **From New Scientist #1307, 27th May 1982** [link]

The diagram is a map of a quoit (that is, of what topologists call a torus or a doughnut). The map can be made into a quoit by rolling it up so that the edges

ABandDCcoincide, and then bending it around — the map is printed on stretchy material — so that the rolled upAXDcoincides withBYC. Thus the twoE‘s coalesce, and so do the two 1’s.As you know, you can’t connect three utilities,

E,WandG, to each of three houses, 1, 2 and 3, on the surface of a sphere, without any pipe crossing another. But, as the map shows, you can do so on the surface of a quoit. In fact, you can do even better. You can connectE,W,GandVto each of the four houses. What I ask you for is a map showing how. The map must be a rectangle of squared material, cut at any angle so long as the edges match when the map is made into a quoit.E,W,G,V, 1, 2, 3 and 4 are to be at intersections of the material, and all the pipes are to run along the grid lines without touching or crossing.Finally, I want a map with the smallest possible area. You will find you can manage with a lot less than 60, which is the area of the map in the diagram.

[enigma162]

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I’ve come up with a solution on the back of an envelope that agrees with the published solution. But I haven’t come up with a programmed solution (yet?), so I’ve added this problem to the list of puzzles that remain unsolved programatically.

Solution:It is possible to connect all utilities on a grid of area 8.In this diagram I have labelled the utilities A, B, C and D.

I would like to program a solution to verify that this is indeed the smallest possible area.