How many possible nets does a cube have, and what are they? (Networks that can be transformed into each other are not considered different.)





Translating the question as requested, while maintaining the original formatting and structure.
1. 4 flaps are arranged in a row (1 edge is common) for example, flaps 1, 2, 3, 4 and flaps 5, 6 are placed on two sides of them. The possible nets are:
![](https://cdn.mathpix.com/cropped/2024_05_02_d9cddad81d2f5e9fe27cg-1.jpg?height=726&width=1440&top_left_y=1304&top_left_x=384)

There are no more. Flaps 5 and 6 cannot be on the same side.

2. 3 flaps are arranged in a row, for example, 1, 2, 3.

There are two cases:

a.) Flaps 5 and 6 are placed on two sides, and flap 4 connects to one of them—let's say to flap 6—because it cannot connect to the row of 1, 2, 3 (4 flaps would be in a row) or from the side (it would overlap with flap 5 or 6). The 6th flap, to which the 4th connects, can only be at the end of the row. Therefore, the possible cases are as shown in the diagram.
![](https://cdn.mathpix.com/cropped/2024_05_02_d9cddad81d2f5e9fe27cg-2.jpg?height=322&width=1442&top_left_y=125&top_left_x=408)

b.) Flaps 5 and 6 are placed on one side. In this case, flap 4 must be between them because 5 and 6 are not adjacent flaps. There are two such cases:
![](https://cdn.mathpix.com/cropped/2024_05_02_d9cddad81d2f5e9fe27cg-2.jpg?height=222&width=1544&top_left_y=629&top_left_x=363)

3. 2 flaps (for example, 1, 2) are arranged in a row. In this case, flaps 5 and 6 are on two sides, and the 4th flap connects to one of them (to the 6th), and the 3rd flap connects to the other (to the 5th). The only possible case is:

![](https://cdn.mathpix.com/cropped/2024_05_02_d9cddad81d2f5e9fe27cg-2.jpg?height=328&width=422&top_left_y=1164&top_left_x=884)

There are no more cases, because at least 2 flaps must always be connected.

Therefore, the cube has 11 possible nets.