Problem

An ant is at one corner of a \( 10 \times 10 \times 10 \) cube-shaped room and wants to reach the opposite corner, traveling only along the walls (not diagonally through space). What is the shortest path length it can take?

Key Insight

Though the room is 3D, the ant walks only along the surfaces of the cube. We “unfold” the cube’s walls onto a 2D plane, where the ant’s path becomes a straight line.

Unfolding the Cube

Imagine unfolding three adjacent walls into a flat rectangle of size:

\[ 10 \times (10 + 10) = 10 \times 20 \]

Now the ant moves from one corner to the diagonally opposite corner of this rectangle.

Apply Pythagoras

The shortest path is the diagonal:

\[ \text{Distance} = \sqrt{10^2 + 20^2} = \sqrt{100 + 400} = \sqrt{500} = \boxed{10\sqrt{5}} \approx 22.36 \]

Conclusion

The shortest path the ant can take along the surfaces of a \( 10 \times 10 \times 10 \) room is:

\[ \boxed{10\sqrt{5}} \]

Reference