  Finite geometry is used to underpin finite, two d-dimensional particles
Hilbert space, d=prime 6= 2. A central role is allotted to states with mutual
unbiased bases (MUB) labeling. Dual affine plane geometry (DAPG) points
underpin single particle, MUB labeled, product states. The DAPG lines are shown
to underpin maximally entangled states which form an orthonormal basis spanning
the space. The relevance of mutually unbiased collective coordinates bases
(MUCB) for dealing with maximally entangled states is discussed and shown to
provide an economic alternative mode of study. These maximally entangled,
geometrically reasoned states, provide the resource to a transparent solution
to what may be termed tracking of the Mean King Problem (MKP): here Alice
prepares a state measured by King along some orientation which Alice succeed in
identifying with a subsequent measurement. Brief expositions of the topics
considered: MUB, DAPG, MUCB and the MKP are included, rendering the paper self
contained.
