1. **Stating the problem:** We have a shape made from 4 identical cubes arranged with three cubes stacked vertically and one cube attached to the middle cube on the right. 2. **Vertices of a single cube:** A cube has 8 vertices. 3. **Part (a): Number of vertices of the 4-cube shape.** - When cubes are joined face-to-face, some vertices coincide and are not counted multiple times. - We need to find the total unique vertices of the combined shape. 4. **Analyzing the arrangement:** - Three cubes stacked vertically share faces, so each pair of adjacent cubes shares a face with 4 vertices overlapping. - The fourth cube is attached to the middle cube on the right, sharing one face with 4 vertices overlapping. 5. **Calculating vertices:** - Start with 4 cubes × 8 vertices = 32 vertices. - Subtract overlapping vertices: - Between the 3 vertical cubes, there are 2 overlaps, each removing 4 vertices: $2 \times 4 = 8$ vertices. - Between the middle cube and the right cube, 1 overlap removing 4 vertices. - Total overlapping vertices removed: $8 + 4 = 12$. 6. **Total unique vertices:** $$32 - 12 = 20$$ 7. **Part (b): Adding another identical cube adjacent to one of the cubes.** - We want the minimum number of vertices the new shape could have. - Adding a cube adjacent to an existing cube shares a face with 4 vertices overlapping. - The new cube adds 8 vertices but overlaps 4 vertices with the existing shape. 8. **Calculating minimum vertices after adding the new cube:** - Current vertices: 20 - Add new cube vertices: 8 - Subtract overlap vertices: 4 - Total vertices: $$20 + 8 - 4 = 24$$ **Final answers:** - (a) The shape has **20** vertices. - (b) After adding one cube adjacent to the shape, the minimum number of vertices is **24**.