Given a long strip of paper, it is very easy to tape the ends together with a half-twist to make a Mobius band. If your strip of paper is too short (such as a square), however, this is impossible. Two years ago, Schwartz proved a long-standing conjecture of Halpern and Weaver that the minimal aspect ratio for which a strip of paper can be made into a Mobius band is the square root of 3. One can ask the same minimal aspect ratio question for Mobius bands and annuli with more twists. For the first half of the talk, we will discuss key elements of Schwartz's proof of the Halpern-Weaver conjecture. For the second half, we will discuss the presenter's construction which shows that for any desired number of half-twists, the minimal aspect ratio stays bounded (by 6, in fact!).