Painted cubes

In your book, Maths Mela (Chapter 7, Shapes and Patterns), Nisha glues 27 small cubes together to make a large solid cube. It has 3 layers, and each layer is 3 rows of 3 small cubes. Then she paints the whole outside of the large cube red. How many small cubes have red on them, and how many red faces does each one have?

The corner cubes show 3 faces on the outside, so they have 3 red faces. A cube has 8 corners, so 8 small cubes have 3 red faces. Along each edge of the large cube, between two corners, there is 1 small cube. It shows 2 faces, so it has 2 red faces. The large cube has 12 edges, so 12 small cubes have 2 red faces.

In the very middle of each face of the large cube there is 1 small cube that shows only 1 face. The large cube has 6 faces, so 6 small cubes have 1 red face. Last, right in the centre of the large cube, hidden inside, is 1 small cube that the paint never touched: it has no red face. Check: 8 + 12 + 6 + 1 = 27. Every small cube is counted once.

The book also shows cube frames. Frame (a) starts as the same 3 by 3 by 3 cube, and a square hole, one small cube wide, goes straight through the middle of every face: front to back, left to right and top to bottom. The holes take away the 6 middle cubes of the faces and the 1 cube in the centre, where the holes meet. So 7 small cubes are removed and 20 are left. Frame (b) starts as a 4 by 4 by 4 cube of 64 small cubes with holes 2 cubes wide; only the edges are left, 32 small cubes, so 32 were removed.