The problem of psychological carry-over of effects in analgesic trials led us to a general method of constructing Latin Squares balanced for such residual effects. Rows of the Latin Squares refer to individuals and columns refer to order of treatment. The following procedure produces Latin Squares balanced for immediate residual effects as well as in other respects: (a) Number the treatments, i = 1,..., n. (b) Start with a cyclic n x n Latin Square. (c) Interlace each row of the square with its own reverse order sequence. (d) Slice the resulting figure down the middle, thus forming two n x n Latin Squares which have the desired properties. A modification of step (d) produces Graeco-Latin Squares when n is odd. Proof is offered in the appendix. The particular analysis, whether Latin Square, paired comparison, randomized blocks, or partially balanced blocks (when some of the columns are discarded), etc., depends on the particular experimental circumstances. Additional balancing features can be introduced easily, but the reader is cautioned that none of these designs can provide perfect protection against carry-over effects. Furthermore, since extra-balancing procedures may severely limit randomization and may produce complications in the analysis, they should be used only for sound practical reasons.