Welcome to Probistis Mu
by Foggy Brume
Teams are told for this puzzle, they'll need a map of nearby planet Probistis Mu. What they get is a mobius strip version of the image below, folded in half along the gray line, and then joined at the sides.
This table shows the distances between each pair of cities. The first number is using roads on the strip; the second number uses the switchbacks that go over the edge.
Since there are only two cities that are 2 miles apart, there are only two places that Sign11 can be: it's either alpha or eta, and Tukd is either alpha or eta. If Tukd is eta, Nelp is theta. If Tukd is alpha, Nelp is either theta or beta, but the introduction says Nelp is not beta. Either way, Nelp is theta.
On Sign12, Nelp is 7 miles away roadwise from another city. This can only be gamma. Bofj is therefore 9 miles away switchbackwise from gamma, and can only be alpha or eta. Tukd and Bofj are therefore alpha and eta in some order. On Sign9, Tukd is 13 miles away roadwise from that sign. This can only be gamma or delta, but gamma has a sign, so Sign9 is delta. The other city, Jasc, is 4 miles away switchbackwise from delta. This makes Jasc epsilon. Jasc is 9 miles away roadwise from Sign8, so this makes Sign8 eta. Sign11 must be alpha, Tukd eta, and Bofj alpha.
Kibl is 4 miles away switchbackwise from Sign8, making it either iota or pi. Kibl is 9 miles away roadwise from Sign10. If Kibl is iota, then Sign10 is either gamma or delta, but those signs are already allocated. Kibl is therefore pi and Sign10 is mu. Gicq is 10 miles away switchbackwise from Sign10, making Gicq either gamma or epsilon. Epsilon is already assigned, so Gicq is gamma.
Gicq is 13 miles away roadwise from Sign2, and the only cities left that are 13 miles from anything are kappa or lambda. Sign2 is 4 miles away switchbackwise from Sexy. Kappa is not 4 miles away from anything, so Sign2 is lambda. Sexy can only be iota. Sexy is 3 miles away roadwise from Sign3. Sign3 is either mu or nu, and mu is already assigned, so Sign3 is nu. Pudj is 6 miles away switchbackwise from Sign3, so Pudj is either epsilon or mu. Epsilon is already assigned, so Pudj is mu. Sign7 is 7 miles away roadwise from Pudj. This makes Sign7 either beta, eta, or lambda. Eta and lambda already have signs, so Sign7 is beta. Qapi is 7 miles away switchbackwise from beta, so it's epsilon, mu, nu, or omicron. Epsilon and mu are already assigned, so it's nu or omicron. If Qapi is nu, then Sign6 is delta, but delta already has a sign. Qapi is omicron. At 8 miles roadwise away from omicron, Sign6 is either epsilon or mu; but mu is already assigned, so Sign6 is epsilon.
Vanh is 7 miles switchbackwise away from epsilon. This makes Vanh beta, iota, kappa, or pi. Iota and pi already have signs. Vanh must also be 9 miles away roadwise from Sign5. If Vanh is beta, then Sign5 can only be nu, but nu already has a sign. So Vanh is kappa, and Sign5 can only be omicron. Foqw is 7 miles away switchbackwise from Sign5, making Foqw beta.
Sign4 is 9 miles roadwise away from Zomg and 6 miles switchbackwise away from Dujl. Five Greek letters do not have signs assigned: delta, zeta, kappa, lambda, nu. Four Greek letters do not have towns assigned: delta, zeta, lambda, and nu. Only zeta is 9 miles roadwise away and 6 miles switchbackwise away from greek letters that do not have a town assigned. Zomg is therefore nu, and Dujl lambda. Zomg is 9 miles away switchbackwise from Sign1. Cigz is 10 miles away roadwise from Sign1. Sign1 can only be kappa, and Cigz must be zeta. That leaves Mekr to be delta.
There are three spots that don't have signs: theta, iota, and pi. Sign13 cannot be theta or pi (since those towns are listed on the sign), so it's iota. Sign14 is 11 miles away from a town, so it's theta, and the missing towns are Bofj and Qapi. Sign15 is pi. The missing towns must be Vanh and Tukd.
In the names of the missing towns are a diagonal word: BandBandBandBand..., the name of the missing city. Since we can't enter an infinitely long answer, the answer to the puzzle is BAND.
All of the signs: