Henry Ernest Dudeney/Puzzles and Curious Problems/221 - The Fly's Journey/Solution

by : $221$

 * The Fly's Journey
 * A fly, starting from point $A$, can crawl around the four sides of the base of this cubical block in $4$ minutes.


 * Dudeney-Puzzles-and-Curious-Problems-221.png


 * Can you say how long it will take to crawl from $A$ to the opposite corner $B$?

Solution
The shortest route would take the fly approximately $2.236$ minutes.

Proof
The shortest route is that indicated by the dotted line:


 * Dudeney-Puzzles-and-Curious-Problems-221-solution.png

Let $d$ be the length of one edge of the cube.

Then the length $l$ of the dotted line is given by Pythagoras's Theorem:


 * $l = \sqrt {d^2 + \paren {2 d}^2} = d \sqrt 5$

As it takes $1$ minute to walk the length $d$, it takes $\sqrt 5$ minutes to walk the distance $l$.