313. Cardboard Box. The reader has probably noticed that there are many problems and questions whose answers, it seems, should have been known to many generations before us, but which, however, have apparently never even been considered. Here is an example of such a problem that came to my mind.

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-097.jpg?height=271&width=1008&top_left_y=521&top_left_x=521)

Suppose I have a closed cardboard box in the shape of a cube. By cutting it with a razor along 7 out of 12 edges (it must be 7), I can unfold the box onto a plane, and the unfolding can take different forms. For example, if I cut along the edges shown in the figure with a thick line and along the invisible edge marked with a dotted line, I will get the unfolding $A$. By cutting the box differently, I can obtain the unfolding $B$ or $C$. It is easy to notice that the unfolding $D$ is simply the flipped version of the unfolding $C$, so we consider these two unfoldings to be identical.

How many different unfoldings can be obtained in this way?
313. There are 11 different nets in total, if we do not distinguish between two nets that can be obtained from each other by flipping. If, however, the outer side of the box, for example, is 
![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-254.jpg?height=526&width=1352&top_left_y=1920&top_left_x=360)
any color, and the inner side is white, and the nets need to be laid out with the white side up, then this can be done in 20 different ways, since then, for each net, except for cases 1 and 5, an additional mirror-symmetric net will be added, which will now differ from it.