The 1963 work of Kervaire-Milnor on the classification of exotic spheres in terms of the stable homotopy groups of spheres left one unanswered question in dimensions congruent to two modulo four. Subsequent work by Brown-Peterson and Browder gave the answer in all dimensions that are not two less than a power of two.  We solve the problem in all remaining cases except 126. Our answer is the opposite of the one that many sought in the 1970s.