Henry Ernest Dudeney/Modern Puzzles/162 - The Fly and the Honey/Solution

by : $162$

 * The Fly and the Honey

Solution

 * $5$ inches.

Proof
Let $F$ be the starting position of the fly and $H$ be the position of the honey.


 * Dudeney-Modern-Puzzles-162-solution.png

We unroll the mug and lay it flat so as to see the geometry easily.

The shortest distance between $F$ and $H$ is found by Heron's Principle of Reflection to be via the point $A$.

We extend $FA$ to $B$ where $B$ is the reflection of $H$ in the rim of the cup.

We see that the fly needs to move:
 * $3$ inches horizontally.
 * $3$ inches vertically and another inch vertically for a total of $4$ inches vertically.

The shortest distance from $F$ to $H$ is then found by Pythagoras's Theorem, where $FB$ is seen to be the hypotenuse of the $3$-$4$-$5$ triangle.