We are given a hollow sugar cube with a side length of 1 cm, and the ant starts at one corner and needs to travel to the opposite diagonal corner. The goal is to find the least distance it needs to travel.

Cube Layout:

• A cube has 8 corners, and we are interested in two corners at the farthest distance across the cube — one at the origin (0,0,0) and the other at (1,1,1).

Direct 3D Diagonal:

The straight-line distance from one corner of the cube to the farthest diagonal corner (ignoring the hollow nature) is given by the 3D Euclidean distance formula:

$\text{Distance} = \sqrt{(1 - 0)^2 + (1 - 0)^2 + (1 - 0)^2} = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}$

However, this would be the distance if the ant could fly straight through the cube, but since it's hollow, the ant must walk on the surface.

Surface Travel:

To minimize the surface travel distance, the ant can take the following path:

1. The ant can unfold the cube in such a way that it walks across two adjacent faces. This minimizes the distance because the 2D diagonal across the unfolded cube is the shortest route on the surface.

When we unfold the cube, the path forms a diagonal on a rectangle of dimensions $1 \times 2$, which is the two adjacent faces laid flat. The length of this diagonal is:

$\text{Distance} = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}$

Thus, the minimum distance the ant must travel is $\sqrt{5} \approx 2.236$ cm.

Would you like any further details or clarifications?

Related Questions:

1. What would be the distance if the ant had to travel along the edges of the cube?
2. How many unique shortest paths could the ant take on the surface?
3. How does the surface distance change for a cube with side length $a$?
4. Could the ant take a shorter path if the cube were not hollow?
5. What is the difference between this shortest path and the straight-line diagonal path?

Tip:

When dealing with shortest paths on 3D objects, unfolding the surface into a 2D plane can help visualize the minimal distance better.