When different treatments are applied In succession to the same unit of experimental material, it is necessary to consider the residual effects of preceding treatments on the present treatment. To facilitate the interpretation of the results of such experiments, balanced designs have been developed. In the designs discussed in this paper, each treatment is preceded by each pair of other treatments, so that the designs are balanced for the effects of pairs of residual effects. It is shown that balanced designs are possible for n treatments, using n(n-1) replications. Solutions of the combinatorial problem involved can be derived from solutions of the Round Table problem, discussed m many books of mathematical puzzles. The method of analysis of designs of this type is described, and exemplified by application to a set of experimental data