  In this paper we study the existence of rational points for the family of K3
surfaces over $\mathbb{Q}$ given by $$w^2 = A_1x_1^6 + A_2x_2^6 + A_3x_3^6.$$
When the coefficients are ordered by height, we show that the Brauer group is
almost always trivial, and find the exact order of magnitude of surfaces for
which there is a Brauer-Manin obstruction to the Hasse principle. Our results
show definitively that K3 surfaces can have a Brauer-Manin obstruction to the
Hasse principle that is only explained by odd order torsion.
