3. (7 points) Rubik made a net of a $3 \times 3 \times 3$ cube and marked two points on it - see the image. What will be the distance between these points after Rubik glues the net into a cube?

![](https://cdn.mathpix.com/cropped/2024_05_06_06b9f4bfa48aaf5de920g-2.jpg?height=388&width=511&top_left_y=523&top_left_x=1318)
Answer: $2 \sqrt{3}$.

Solution. Let's draw the completed cube (the image is chosen so that the highlighted face of the net is on top). The given points are two opposite vertices of the cube $2 \times 2 \times 2$. In a cube $2 \times 2 \times 2$, the diagonal has a length of $2 \sqrt{3}$.
![](https://cdn.mathpix.com/cropped/2024_05_06_06b9f4bfa48aaf5de920g-2.jpg?height=504&width=1082&top_left_y=1218&top_left_x=454)

Remark. It is not necessary to use the fact that the points are the endpoints of the cube's diagonal. One can simply draw the resulting picture and find the length of the required segment by applying the Pythagorean theorem a couple of times (or using the coordinate method, etc.)

## Grading Criteria.

- Correct solution (a correct picture and explanation of how the distance between the points is found is sufficient) - 7 points.
- The picture is drawn correctly (possibly not from the same perspective as in the provided solution), but the distance is found incorrectly - 3 points.
- Only the correct answer is provided - 0 points.
- An error is made in determining the location of the points on the cube - 0 points.