You can make a paper Moebius band by starting with a 1 by L rectangle, giving it a twist, and then gluing the ends together. The question is: How short can you make L and still succeed in making the thing? This question goes back to B. Halpern and C. Weaver in 1977. The previously known lower bound was π/2 and the previously known upper bound, conjectured to be the true answer, is 3‾√. In my talk I will explain how to get the lower bound up to about 3‾√−1/26, which is an improvement on π/2, and I will also explain show to reduce the question about the sharp upper bound to showing, a finite number of times, that some explicit piecewise algebraic function is non-negative on the unit cube in 14 dimensions. (I'm working on the calculations...) These calculations have to do with geometrical properties of finite tensetrigies.