Cubes & Dice

At a glance 10 rules

1. 1Painted cube, n per edge$\text{2 faces: } 12(n-2),\ \ \text{1 face: } 6(n-2)^2$$\text{3 faces: } 8,\ \ \text{none: } (n-2)^3$
2. 2Painted a × b × c block$\text{none: } (a-2)(b-2)(c-2)$$\text{2 faces: } 4(a + b + c - 6),\ \ \text{3 faces: } 8$
3. 3Bare or coloured facesWith some faces bare or in different colours, sort small cubes by place: a corner touches 3 big faces, an edge cube 2, a face cube 1. Then read off the paint place by place.
4. 4Cuts and pieces$\text{pieces} = (x+1)(y+1)(z+1)$$x, y, z = \text{cuts in each direction}$
5. 5Hidden cubesPeel one layer off each face that can be seen. For n per edge, $(n-1)(n-2)^2$ stay hidden on the floor and $(n-1)^3$ in a room's corner.
6. 6Standard dieOpposite faces add to 7: 1-6, 2-5, 3-4. If two numbers seen side by side add to 7, the die is not standard.
7. 7Seen together, never oppositeNumbers seen next to X in any view are never opposite X; once four neighbours are known, the sixth is opposite. Two views sharing two numbers: the two leftovers are opposite.
8. 8Two views, one sharedOne shared: the number in neither view is opposite it. Turn one view so the shared number sits as in the other, then faces in the same place are opposite pairs.
9. 9NetsIn a straight run of three squares the end squares are opposite, and so are the end squares of a Z-shaped run of four. Opposite squares never share an edge.
10. 10Rolling a dieA roll to the east sends the top face east, the east face down, the bottom face west and the west face up. Track every roll in order: rolls do not cancel.