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