2n kids are at a paintball party, and a mathematician parent is assigned the task of coming up with a sequence of 2n-1 ways of dividing the kids into two equal-sized teams, with the property that each kid gets to play on the same team with each other kid exactly n-1 times. For what values of n is this possible? For n=2, there is an obvious solution: {1,2 | 3,4}, {1,3 | 2,4}, {1,4 | 2,3} (here I am using notation that I hope is transparent). For n=3, there is no solution, as a brute-force approach discloses. (Unless I made a mistake.) For n=4, there is a solution based on the vector space F_2^3, where F_2 is the field with two elements. Specifically, associate the eight players with the eight elements of F_2^3 = {(x,y,z): x,y,z in F_2} in some arbitrary fashion, and then use the seven linear functions f(x,y,z) = x, y, z, x+y, x+z, y+z, and x+y+z to divide F_2^3 into two subsets of size 4 according to whether f(x,y,z) is 0 or 1. So I ask, for what values of n can a satisfactory rotation pattern of length 2n-1 be constructed? (This may be a known question in design theory, but I thought I'd ask you-all before I post to MathOverflow.) The case n=4 arose yesterday at my son's paintball birthday party, and I came up with the F_2^3 solution in the nick of time. Some extra issues arose that compromised the success of the design (as measured by my son's satisfaction), and it turns out that a different ordering of the seven linear functions would have led to a better rotation pattern, but I'll get into that later. Jim Propp