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In the magnificent country of Appelhaken there is a zoo. Not just any zoo, but a grand zoo, a magnificent zoo, called Appelhaken Grand Magnificent Magnificent Grand Grand Zoo.

In this zoo there are three food stalls and two exhibits. The Law of Appelhaken specifies that there must be a clearly marked path from each food stall to each exhibit, and that these paths may not intersect or share parts.

A third exhibit is built, so to comply with the Law, the head architect of Appelhaken constructs a bridge within the zoo (not so grand as the Appelhaken Monument Bridge, but still pretty magnificent) and a path is placed on top of it, while another path goes beneath it, for without the bridge, the Law could not be upheld.

For National Celebration Day, the Magnificent Leader of Appelhaken proclaims that Appelhaken Grand Magnificent Magnificent Grand Grand Zoo shall be expanded immediately, to a grand total of six food stalls. The additional food stalls are built within the hour. However, while the zoo has enough income to afford more bridges, all the country's architects have taken a day off (National Celebration Day is not a public holiday) and are unavailable, so there will be only one bridge in the zoo during National Celebration Day.

How can the Grand Zookeeper reroute the paths of Appelhaken Grand Magnificent Magnificent Grand Grand Zoo in order to preserve the Law?

(TN: The magnificent language of Appelhaken contains only two embellishing adjectives, which I translate as magnificent and grand here. For further embellishment, repetition is necessary.)

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The grounds of the zoo, with one bridge,

are kinda-sorta topologically equivalent to the surface of a torus.

So, with sufficient abstraction, the following diagram shows how to do it

where the north and south edges are to be thought of as glued together, and likewise the east and west edges. (Obviously the red blobs are the exhibits and the blue blobs are the food stalls.)
enter image description here

I appreciate that

this may not be instantly convincing to those who have not studied topology, not least since this is only a kinda-sorta equivalence (the plane is a sphere minus one point, which is harmless; to turn a sphere into a torus you add a cylindrical handle, whereas we have added a mere strip)

so here is how to transform that into something more obvious. First

we route all the "horizontal" across-the-edges things above the square:
enter image description here

and then

we run the "vertical" across-the-edges things in a single bundle, above those, over the bridge:
enter image description here
(The bridge is rather wide. Its footprint is the shaded grey rectangle. Green paths are over, purple ones at ground level.)

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  • $\begingroup$ Well done! Magnificent depictions. $\endgroup$ – Magma Nov 22 '19 at 13:23

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