(Absolutely final version?): As always (at least that part was right) I need to correct what I wrote. This time I want to delete (what was) rule 5.: ----- There may exist additional intersections between pairs of squares of X, as long as these are only along common vertices ----- because, my role isn't to tell anyone what may or may not happen, and *besides* with the new rule 5. (essentially forbidding triple intersections) I don't even know if they are possible globally, though I strongly suspect so. The point is, for any 3 squares P, Q, R in order, they are always congruent to what I shall call the squiggle: 3 unit grid squares of which A \int B is one edge of B, and B \int C is an *adjacent* edge of B, such that A, B, and C together *do not* form a "cube corner configuration": the SSS has no CCCs. —Dan ----- 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. 6. No three squares share a common vertex. 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 ----- -----