John Conway wrote: << This is a problem that's worried me for quite some time : if and when there are 51 United States, then how should the resulting 51 stars be arranged on the "Stars and Stripes" flag? There are nice arrangements for 48, 49, 50 stars: * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
Good question. When I was in 4th grade my teacher asked the class to come up with arrangements for 49 and 50 stars, and I'm proud of coming up with the above patterns, especially the one for 50 (rotated 90 degrees). (Though as it happens the gov't chose for 49 a pattern of 7 staggered rows of 7 stars, possibly because it looked better filling the rectangular field available for it.) But (ignoring the rectangular field condition) this is actually a fascinating problem for all lowish numbers: What is the nicest arrangement of n points in the plane, for each n up to 100? Further, can you define explicitly what "nicest" means? --Dan