In the picture, there is a structure composed of twelve identical cubes. To how many different places can we move the dark cube so that the surface area of the assembled body does not change?

![](https://cdn.mathpix.com/cropped/2024_04_17_22cb57ce78fb8d5dd5c3g-5.jpg?height=334&width=394&top_left_y=1758&top_left_x=837)

Just like in the original structure, the cubes in the new structure must touch each other with entire faces. The positions of the light cubes cannot be changed.

(D. Reichmann)

Hint. How will the surface area of the body change after removing the dark cube?
Originally, on the surface of the assembled body, there are three dark cube walls, and the other three walls touch light cubes. After removing the dark cube, the total surface area of the body does not change (three dark walls are replaced by three light ones).

In order for the surface area of the body to remain unchanged even after moving the dark cube, it must touch exactly three light cubes (three light walls will be replaced by three dark ones). By trial, we quickly find that the dark cube can only be moved to the following three places:
![](https://cdn.mathpix.com/cropped/2024_04_17_22cb57ce78fb8d5dd5c3g-6.jpg?height=336&width=1332&top_left_y=440&top_left_x=360)