A new sequential one-dimensional cutting problem is studied in this paper. For a number of pieces of a tape or a bale has to be decided sequentially whether a piece is to be cut to smaller pieces of minimal/maximal length a and b, respectively, or if some part x is to be combined with some part y of the next piece. These parts must have a minimal length c and their sum should not be less than a nor greater than b. A cutting solution is needed which minimizes the unusable rest. A dynamic programming approach is offered in the paper to this problem as well as a modification. Real-life problems of cutting curtain bales have been solved by a corresponding PASCAL program on a personal computer.