A sequence m 1=m 2=¿;=m k of k positive integers isn-realizable if there is a partition X 1, X 2,..., X k of the integer interval [1, n] such that the sum of the elements in X i is m i for each i=1, 2,..., k. We consider the modular version of the problem and, by using the polynomial method by Alon (1999) [2], we prove that all sequences in Z/pZ of length k=(p-1)/2 are realizable for any prime p=3. The bound on k is best possible. An extension of this result is applied to give two results of p-...