During the course of your career you may have taken a strip of paper, twisted the end, and taped it together to make a Moebius band. In this talk I will discuss the question, which goes back to B. Halpern and C. Weaver in the 1970s, about how short a strip of paper (which is assumed to have width 1) you can use. Halpern and Weaver show that the answer lies between $\Pi/2$ and $\sqrt{3}$, and conjecture that the true lower bound is $\sqrt{3}$. I'll explain these bounds, then sketch my proof that the lower bound is actually at least $\sqrt{3}-1/26$. Finally, I will explain how to reduce the $\sqrt{3}$ conjecture to the statement that a certain 10 semi-algebraic expressions in finitely many variables are non-negative. These expressions encode properties of tensegrity-like objects.