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This image presents a problem about a painted wooden cube, sized 3x3x3, that has been cut into smaller cubes of size 1x1x1. The task is to determine:

a) How many small cubes are obtained.
b) How many small cubes have exactly two faces painted.
c) How many small cubes have exactly three faces painted.
d) How many small cubes have exactly one face painted.
e) How many small cubes have no faces painted.

Let's go through each question step by step.

1. **Total number of small cubes:**
   Since the original cube is 3x3x3, the total number of 1x1x1 cubes will be \(3^3 = 27\).

2. **Cubes with two painted faces:**
   These cubes are located on the edges, but not at the corners. A 3x3x3 cube has 12 edges, and along each edge, only the middle cube (not the corner cubes) has two painted faces. Therefore, there are 12 such cubes.

3. **Cubes with three painted faces:**
   These are the corner cubes. A cube has 8 corners, and at each corner, the small cube will have three painted faces. Thus, there are 8 such cubes.

4. **Cubes with one painted face:**
   These cubes are located on the faces but not on the edges. Each face of the cube has a \(3 \times 3\) grid, so there are 9 small cubes per face. However, we exclude the cubes on the edges, leaving only the middle cube of each face. Since the cube has 6 faces, there are 6 such cubes.

5. **Cubes with no painted faces:**
   These cubes are completely inside the large cube and not exposed to any face. For a 3x3x3 cube, there is exactly 1 cube in the center that has no painted faces.

### Summary of answers:
- a) There are 27 small cubes in total.
- b) There are 12 small cubes with two painted faces.
- c) There are 8 small cubes with three painted faces.
- d) There are 6 small cubes with one painted face.
- e) There is 1 cube with no painted faces.

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Here are five additional related questions to explore:

1. How would the number of cubes with no painted faces change for a 4x4x4 cube?
2. How many cubes would have two painted faces if the original cube was 5x5x5?
3. How many cubes would have one painted face in a 6x6x6 cube?
4. How would you calculate the number of cubes with exactly three painted faces for any \( n \times n \times n \) cube?
5. What is the formula for determining how many cubes with no painted faces exist in an \( n \times n \times n \) cube?

**Tip:** The total number of small cubes in a larger cube is always given by \( n^3 \), where \( n \) is the length of the cube’s edge.

Дерев'яний пофарбований куб розмірами 3×3×3 розпиляли на кубики розмірами 1×1×1. Визначити: a) скільки всього отримано кубиків; б) у скількох маленьких кубиків пофарбовано лише дві грані; в) у скількох маленьких кубиків пофарбовано лише три грані; г) у скількох маленьких кубиків пофарбовано лише одну грань.

Solve the problem of determining the number of small cubes with painted faces when a wooden cube of size 3x3x3 is cut into 1x1x1 cubes. This solution covers geometry and combinatorics concepts, including how to find cubes with one, two, or three painted faces.

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