1. **Problem statement:** We need to find the minimum number of colors required to color the faces of two nets (A and B) so that no two faces that touch share the same color. 2. **Understanding the problem:** Each net consists of a vertical stack of 8 square faces forming a cuboid, with one octagonal face attached on each side (left and right). When folded, faces that share an edge must have different colors. 3. **Key concept:** This is a graph coloring problem where each face is a vertex, and edges represent adjacency (touching faces). The minimum number of colors needed is the chromatic number of the graph. 4. **Net A analysis:** - The 8 squares form a vertical chain, so each square touches the one above and below (except top and bottom). - Each square also touches the octagonal face on the left and right. - The octagonal faces touch all 8 squares on their respective sides. 5. **Coloring Net A:** - The 8 squares form a path graph $P_8$, which is 2-colorable (alternate colors). - The octagonal face on each side touches all 8 squares, so it must have a different color from all squares it touches. - Since squares use 2 colors, the octagonal face must be a third color. - Both octagonal faces are on opposite sides and do not touch each other, so they can share the same third color. 6. **Minimum colors for Net A:** - 2 colors for squares + 1 color for octagons = 3 colors. 7. **Net B analysis:** - The description is the same as Net A (vertical stack of 8 squares with octagonal faces on left and right). - The coloring logic is identical. 8. **Minimum colors for Net B:** - Same as Net A, 3 colors. **Final answers:** - a) Net A requires minimum 3 colors. - b) Net B requires minimum 3 colors.