  An ordinary hypersphere of a set of points in real $d$-space, where no $d+1$
points lie on a $(d-2)$-sphere or a $(d-2)$-flat, is a hypersphere (including
the degenerate case of a hyperplane) that contains exactly $d+1$ points of the
set. Similarly, a $(d+2)$-point hypersphere of such a set is one that contains
exactly $d+2$ points of the set. We find the minimum number of ordinary
hyperspheres, solving the $d$-dimensional spherical analogue of the
Dirac--Motzkin conjecture for $d \geqslant 3$. We also find the maximum number
of $(d+2)$-point hyperspheres in even dimensions, solving the $d$-dimensional
spherical analogue of the orchard problem for even $d \geqslant 4$.
