Here are some visualization challenges. See if you can answer the questions in your head without drawing anything (or if you must draw, draw in the air rather than on paper). Picture the usual tiling of the plane by unit squares, and picture a shifted version of that tiling whose vertices all lie in the interiors of the squares of the first tiling. How many vertices of tiling #2 lie in each square cell of tiling #1? Now try it with an infinite tiling of the plane by regular hexagons. And now try it with an infinite tiling of the plane by equilateral triangles. Bonus question: Is there an easy way to see, in each case, what the AVERAGE number of vertices-of-tiling-#2 per cell-of-tiling-#1 must be? (Warning: Do not drive, operate heavy machinery, or make love while attempting to solve these problems.) Jim Propp