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

Modern Puzzles by Henry Ernest Dudeney: $162$

The Fly and the Honey

I have a cylindrical cup four inches high and six inches in circumference.

On the inside of the vessel, one inch from the top, is a drop of honey,

and on the opposite side of the vessel, one inch from the bottom on the outside, is a fly.

Can you tell exactly how far the fly must walk to reach the honey?

Solution

$5$ inches.

Proof

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

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.

$\blacksquare$

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $162$. -- The Fly and the Honey
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $325$. The Fly and the Honey