NCERT Solutions Ganita Prakash (Part 1) Chapter 7 Shortest Path in a Box! — Puzzle Time
Book page 172 Updated on2026-09-05
Q1.
There is a spider in a corner of a box. It wants to reach the farthest opposite corner (marked in the figure). Since it cannot fly, it can reach the opposite point only by walking on the surfaces of the box. What is the shortest path it can take? Take a cardboard box and mark the path that you think is the shortest from one corner to its opposite corner. Compare the length of this path with that of the paths made by your friends.
Answer
Flatten the box. Open out two adjoining faces so they lie in one plane — then the shortest path is the straight line joining the two corners on that flat sheet.
Unfold two faces into one flat sheet; the straight red line beats the path along the edges.
Take a cubical box of edge a. Unfolding two faces gives a rectangle that is a tall and 2a wide, so the straight path is the diagonal of that rectangle:
Path along the edges = a + a + a = 3a Straight path on the unfolded sheet = the diagonal of an a by 2a rectangle = √(a² + (2a)²) = √(5a²) = a√5 ≈ 2.24a 2.24a < 3a → the unfolded straight path is shorter
For a box of length l, breadth b and height h there are three different ways to unfold, and the shortest of the three is the answer:
Unfolding
Rectangle
Path length
over the breadth
l by (b + h)
√(l² + (b + h)²)
over the height
b by (l + h)
√(b² + (l + h)²)
over the length
h by (l + b)
√(h² + (l + b)²)
Why it happens: The spider is stuck on the surface, but the surface can be laid flat without stretching it — the distances painted on the cardboard do not change when you open it out. Once the two faces are flat, the old rule applies again: between two points the straight line is the shortest path. This is the same idea as the triangle inequality at the start of the chapter.
Try This: Stretch a thread between the two corners of a real cardboard box, keeping it on the surface. Then cut the box open along one edge and lay it flat — the thread will lie perfectly straight.