hihi, all - well, the systematic search program i wrote to find minimum area convex lattice polygons took 20 hours for n=20 and 80 for n=21, with results as follows min min min area perim longest n found found edge 10 14 18 3 11 21.5 22 3 12 24 24 3 13 32.5 28 4 14 40 30 3 15 51.5 34 4 16 59 36 3 17 75.5 42 5 18 87 44 4 19 106.5 50 5 20 121 52 4 21 144.5 60 5 so i wrote a random search program that divides each polygon into four ``corners'', and only does the last corner systematically - here, the program runs for as long as i let it, and though it often finds smaller area polygons after a lot of time, it usually finds the best one it finds at all within a few minutes - the results are as follows (after some days on many machines) The random program did find the same smallest area polygons (also smallest perimeters and maximum edge lengths) as the systematic search program did for n<=21 It should be noted that the minimum perimeter found and minimum longest edge length found are not necessarily from the same polygon. However, they are from polygons that have the same minimum area min min min area perim longest n found found edge 22 164 62 5 23 193.5 66 5 24 210 72 5 25 248.5 76 5 26 274 80 5 27 315.5 86 5 28 345 90 5 29 394.5 94 5 30 430 98 5 31 485.5 104 5 32 527 108 5 33 594.5 114 5 34 661 120 6 35 715.5 126 6 36 764 128 5 37 845.5 134 6 38 912 140 6 39 994.5 144 6 40 1055 150 6 41 1170.5 156 6 42 1299 164 6 43 1439.5 186 7 44 1553 182 7 45 1677.5 190 7 46 1753 204 7 47 1867.5 202 7 48 1969 212 7 49 2105.5 216 7 50 2199 222 7 i will put the details of the algorithms, discussion of the results, and the source code on the web this week sometime, but i've been at this problem for a month now (even dreaming about arrangements of edges on graph paper), and it's time to stop for a while more later, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal@aero.org