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

Puzzles and Curious Problems by Henry Ernest Dudeney: $221$

A fly, starting from point $A$, can crawl around the four sides of the base of this cubical block in $4$ minutes.

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

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

The shortest route is that indicated by the dotted line:

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 $l$.

$\blacksquare$