The spider and the fly
The question
An ant is at one corner of a 5 × 10 × 20 box and wants to reach the corner diagonally opposite. It can only crawl along the outside faces — it cannot cut through the box.What is the shortest distance across the surface?Why
Unfold the box, then go straight
How it works.
box:5 × 10 × 20
5+10=15
15×15=225
20×20=400
25×25=625
shortest:25
- 01
A bug crawls the outside of an 5 × 10 × 20 box, corner to the far opposite corner, and may not cut through.
- 02
A bent path over two faces becomes a straight line once those faces are unfolded flat. Pairing the two shorter edges lays them along one side: 5 + 10 = 15, with 20 the other side.
- 03
Pythagoras on that flat rectangle: 15 × 15 = 225 and 20 × 20 = 400, together 625; and 25 × 25 = 625, so this path is 25.
- 04
It is the shortest: the other two unfoldings come out longer (squared lengths 725 and 925). Shortest surface path: 25.