The numbers of the three types of cookies must have a sum of six. Possible sets of whole numbers whose sum is six are \[ 0,0,6;\ 0,1,5;\ 0,2,4;\ 0,3,3;\ 1,1,4;\ 1,2,3;\ \ \text{and}\ 2,2,2.\] Every ordering of each of these sets determines a different assortment of cookies. There are 3 orders for each of the sets \[ 0,0,6;\ 0,3,3;\ \text{and}\ 1,1,4.\] here are 6 orders for each of the sets \[ 0,1,5;\ 0,2,4;\ \text{and}\ 1,2,3.\] There is only one order for $ 2,2,2 $. Therefore the total number of assortments of six cookies is $ 3\cdot 3 + 3\cdot 6 + 1 = \boxed{28} $.