[ejddhbrbrrnrn]

Pip position encoding can get that down to 3. One pip on 3 adjacent sides.

Centre pip = 1, Edge = 2, Corner = 3

[quirino]

If we're gonna go that route, you can just put a single pip on a corner and derive all of the other positions from that.

[gcr]

Nope, that’s rotationally symmetric around the (pip, center of dice) axis.

Put the pip on the face, but near the corner.

[ejddhbrbrrnrn]

Or the edge so, the pipped face is 1, then go over that edge for 2, keep going around for 3 and 4. If we consider that "going east" then 5 is on the north pole.

[quirino]

That's what I mean by "pip on the corner" :P

The comment above mine was using this terminology to refer to the corner of a face.