I wonder how different is the reflectivity of front and side surfaces?
Perhaps using polarized light...

I seem to recall that when I've pointed my camera at a house with a slate
roof a polarizing filter made quite a difference. It might have been wet
at the time.

To expose some of the front surface of each, they could perhaps be stacked
in a "hopper" that is not quite square, and leans backwards. As the
dimension of each slate is not fixed, you'd have to look at the rear
portion against the back, with a slot cut in the back of the hopper. Aim a
light source at the "staircase" of slate edges, at an angle, and measure
the reflected light with a light sensor behind a polarizing filter.

The above is totally untested. Apologies if it duplicates anything
already said.

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

Dr. Simon J. Melhuish
N.R.A.L., Jodrell Bank.
http://www.jb.man.ac.uk/~sjm/