As always, I need to adjust the wording of a previous post, the SSS puzzle (corrected version below). I neglected to add one essential condition. So I will restate the whole thing from scratch with better numbering. The only actual difference is the addition of 4., which I had accidentally left out — sorry! — and question C. Strip of squares in space puzzles (new & improved): --------------------------------------------------- Let a "strip of squares in space" (SSS) satisfy these conditions: 1. An SSS X is a union, of some collection of unit squares in R^3, with all vertices having integer coordinates. 2. Each square Q in X intersects exactly 2 other squares (Q-, Q+) in the strip along *entire edges*. 3. The edges in 2. are *adjacent edges* in Q. 4. The 3 squares Q, Q-, Q+ mentioned in 2. lie in mutually perpendicular planes. 5. There may exist additional intersections between pairs of squares of X, as long as these are only along common vertices. Let the "size" of X be how many squares are in X. PUZZLES: ------- A. What is the smallest number of squares in an SSS that is topologically a cylinder, if possible? B. What is the smallest number of squares in an SSS that is topologically a Möbius band, if possible? C. What is the smallest number of squares in an SSS that is knotted? (I still don't know the answers.) —Dan -----