We are given a cube that has been painted on all its faces and then cut into 343 smaller cubes of equal volume. We need to determine how many of these smaller cubes have paint on at least one face.

Step 1: Determine the dimensions of the larger cube.

The cube is divided into 343 smaller cubes. Since 343 is a perfect cube, the number of smaller cubes along each side of the larger cube is: $\text{Side length of larger cube} = \sqrt[3]{343} = 7$ Thus, the larger cube has a side length of 7, meaning it is divided into a $7 \times 7 \times 7$ arrangement of smaller cubes.

Step 2: Consider which smaller cubes are painted.

• The smaller cubes that have paint on them are those that lie on the surface of the larger cube.
• Only the smaller cubes that are not completely inside the cube will have paint on at least one face.

The total number of cubes in the larger cube is $7 \times 7 \times 7 = 343$, and the smaller cubes that do not have any paint are the ones that are entirely inside, i.e., the inner $5 \times 5 \times 5$ cube (since the outermost layers on all sides are painted).

Step 3: Calculate the number of unpainted cubes.

The unpainted cubes are the ones that lie completely inside the larger cube. The side length of this inner cube is 5, so the number of unpainted cubes is: $5 \times 5 \times 5 = 125$

Step 4: Calculate the number of painted cubes.

The total number of smaller cubes is 343, and the number of unpainted cubes is 125. Therefore, the number of painted cubes is: $343 - 125 = 218$

Thus, the number of smaller cubes that have paint on at least one face is 218.

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Here are some related questions to consider:

1. What if the cube is divided into 1000 smaller cubes instead?
2. How many cubes have paint on exactly two faces?
3. How many smaller cubes have paint on exactly three faces?
4. What is the surface area of the larger cube if each smaller cube has a side length of 1 unit?
5. How many smaller cubes would have paint if the larger cube was divided into 27 smaller cubes?

Tip: For these types of problems, visualizing the layers of the cube can help clarify which smaller cubes are on the surface!