hihi, all - well, barring a few power failures and such, i've been running these programs on one or more machines (up to five here at home)continuously since last december, and i have some preliminary results to report (more complete values will be ready in a few more weeks) minimum area convex lattice polygons with n sides i separated them into odd and even n, since the even case is much simpler (for even n, minimum area polygons are centrally symmetric, so the search range is much smaller) i ran systematic searches and random searches, all with a strict a priori bound on the maximum edge length (to make the search range finite) because the systematic searches were run with an a priori bound on edge length, they do not necessarily find the actual minimum area for a given n (a larger bound could find a smaller area polygon, which happens occasionally for lower values of n, and also for n=27 and n=29 in going from bound r=7 to bound r=8) the random search programs found all best known values for even n<=50, but there are low maximum edge bounds for n=48 and n=50 (marked with '??') - the random searches were run for a fixed amount of time (24hrs for larger n) the random search programs found most best known values for odd n<=51, but there are several low maximum edge bounds (marked with '??'), and there was no systematic search for odd n>=47 (there aren't enough edges possible for edge bound r=7 for any n>=48), so the random search program cg produced the best known values because the edge length bound (columns marked 'max edge bound') for smaller n is much larger than the smallest maximum edge length found for the minimum area polygons found, i am pretty confident that these best known values are the best values (except for the ones marked with '??') more later, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal@aero.org even n systematic max random (if different) min min longest edge min min longest n area perim edge bound area perim edge 04 1 4 1 50 06 3 8 2 50 08 7 12 2 50 10 14 18 3 50 12 24 24 3 50 14 40 30 3 50 16 59 36 3 50 18 87 44 4 50 20 121 52 4 50 22 164 62 5 50 24 210 72 5 40 26 274 80 5 30 28 345 90 5 20 30 430 98 5 14 32 523 108 5 14 34 632 118 5 14 36 749 128 5 12 38 890 138 5 12 40 1039 148 5 10 42 1222 162 7 10 44 1412 176 7 10 46 1620 192 7 10 48 1838 208 8 09 ?? 50 2088 218 8 09 ?? odd n systematic max random (if different) min min longest edge min min longest n area perim edge bound area perim edge 03 0.5 4 2 50 05 2.5 8 2 50 07 6.5 12 3 50 09 10.5 16 3 50 11 21.5 22 3 50 13 32.5 28 4 50 15 51.5 34 4 40 17 75.5 42 5 30 19 106.5 50 5 20 21 144.5 60 5 10 23 193.5 66 5 10 25 248.5 76 5 09 27 312.5 88 7 08 ?? 29 391.5 96 7 08 ?? 31 485.5 104 5 07 33 589.5 114 5 07 35 704.5 124 6 07 ?? 37 843.5 136 6 07 ?? 39 994.5 144 6 07 ?? 41 1170.5 156 6 07 ?? 43 1369.5 168 6 07 ?? 45 1605.5 180 6 07 1596.5 192 7 (cg, r10) 47 -- 07 1833.5 204 7 (cg, r08) 49 -- 07 2062.5 214 7 (cg, r08) 51 2315.5 224 7 (cg, r08) (cg is one of the random search programs, and r is the edge length bound)