Jim [Propp] I have a copy of my book with me and can report the following on the circle packing problem. The first time an extra circle can be inserted is when 329 circles can pack into a 2 by 164 rectangle. For that rectangle there are 101 sets of three circles with a set of 7 circles at adjusted placements on each end, fitting into a 2 by 163.99958... rectangle. The smallest 2 by x rectangle admitting 329 circles has 101 sets of three circles with a set of 13 circles adjusted at each end to fit a 2 by 163.9973967...rectangle. I believe this can be proven minimal. The spacing between like circles of triangular elements is 2+sqrt((sqrt(3)-.75). Coordinates of circle centers is different for the two solutions but the set of adjusted circles can be described as follows. Circles 1, 3, 5 and 7 on the bottom have circles 3 and 7 resting on the bottom edge. Circles 2, 4 and 6 are near the top edge with 2 and 6 touching the top edge. Circle 6 then touches the first triangular group touching the top edge. At the right end of the rectangle circles 323-329 are in the same formation in reverse order. Circles 323 and 327 touch the bottom edge and circles 323 and 328 touch the top edge. I don't have the arrangement involving the 13 circle sets at each end with me. I hope this helps you. I had no intent of keeping me email address secret on math fun. How do I make it public? Happy puzzling Dick [Hess] http://gosper.org/pennies1.bmp and http://gosper.org/pennies2.bmp show respective ends of a 2×200 strip containing 133 triads, with ample room for two more pennies. --rwg