Industrious Dice

Or a d6 with the 4, 5 and 6 faces blank. When you roll, if the face isn't blank, that's your number. If it is then flip the die over and subtract that number from 7.

Only uses 6 pips.

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

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

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.

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

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

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.

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.