Suppose a 1 , . . . a n an are vectors of length at least 1 in m -dimensional real space, not necessarily distinct. Suppose S is an m -dimensional open sphere of diameter d . Let f m ( n , d ) denote the maximum, over all choices of the a i and S , of the number of sums {ie225-1} which lie in S , where each ɛ i is 0 or 1. For example, it is known that{fx225-1} for all m and n . Here we find the tightest packing of all 2 n sums in 2 dimensions: f 2 ( n , d ) =2 n if d > cosec (π/2 n ). The extremal configuration is to let the a i be n consecutive sides of a regular 2 n -gon with unit sides. Bounds on the tightest packing are given for m > 2. A different approach yields an upper bound on f m ( n , d ) for all m , n , d of the form {ie225-2}, where c m is a constant depending on m . This follows from a new generalization of Sperner's theorem.