Unconventional dice are to be designed such that the six faces are marked with numbers from $$1$$ to $$6$$ with $$1$$ and $$2$$ appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.

The cube (standard dice) has $$24$$ possible rotational symmetries. Two dice are considered identical if one symmetry converts the first into the second.

Step 1 Fix an axis using the condition “1 opposite 2”
Because faces $$1$$ and $$2$$ must be on opposite sides, they determine one complete axis of the cube. Any dice can be rotated so that $$1$$ comes to the top and $$2$$ to the bottom. After this choice, only rotations about this vertical axis are still possible, and that residual symmetry is the cyclic group $$C_4$$ of order $$4$$ (turns by $$0^\circ,\,90^\circ,\,180^\circ,\,270^\circ$$).

Step 2 Arrange the numbers 3, 4, 5, 6 on the four side faces
Without considering further symmetry, there are $$4! = 24$$ ways to place $$3,4,5,6$$ around the belt. To eliminate the over-count coming from the surviving rotations of $$C_4$$ we apply Burnside’s lemma.

• Identity rotation fixes all $$24$$ placements.
• A $$90^\circ$$ or $$270^\circ$$ turn would force all four positions to be the same number, impossible because the numbers are distinct ⇒ $$0$$ fixed.
• A $$180^\circ$$ turn pairs front↔back and right↔left; it would require the two numbers in each pair to coincide, again impossible ⇒ $$0$$ fixed.

The number of distinct arrangements is therefore $$\frac{24 + 0 + 0 + 0}{4} = 6.$$

Step 3 Choose colours
Opposite faces must share the same colour, and there are two colours (red, yellow). With three opposite pairs $$(1,2),\;( \text{front,back}),\;( \text{right,left}),$$ each pair can independently be coloured red or yellow, giving $$2^3 = 8$$ possible colourings for every fixed numerical arrangement.

Step 4 Combine numbers and colours
Total distinct dice designs $$= 6 \text{ (numberings)} \times 8 \text{ (colourings)} = 48.$$

Answer: $$48$$ distinct dice can be designed.