Solution to puzzle 6: Ant on a box
One obvious route is for the ant to crawl along the line of contact between the box and the floor.
This is clearly 25 + 36 = 61 cm in length.
Can the ant find a shorter path by crawling over the top of the box?
The key insight is to flatten the box. Having done this, it's clear that the shortest path must be one of the straight line paths from one corner to its opposite. There are four such paths. They are the diagonals of:
- a 36 by 12+25+12 cm rectangle
- a 25 by 12+36+12 cm rectangle
- two 12+25 by 12+36 cm rectangles
Using Pythagoras' Theorem, the lengths of these diagonals are, respectively, the square roots of 3697, 4225, and 3673. Note that 612 = 3721, and so the path along the line of contact between the box and the floor is longer than all but one of these diagonals.
The shortest path over the box is therefore 
cm, or a little over 60.6 cm.
Generalization
Find a condition, in terms of the dimensions of the box, that there exists a shorter path over the box than along the line of contact between the box and the floor.
Source: Traditional