Traveller whose Head goes Further than Feet
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Problem
- A traveller sets out on a journey,
- and eventually returns to the place where he started.
- During his journey, his head has travelled $12$ yards further than his feet,
- and yet his head remains attached to his body.
- How is this possible?
Solution
The traveller went around the world, at the equator.
Proof
Let the radius of Earth at the equator be $R$.
Thus, by Perimeter of Circle, the equator measures $2 \pi R$ all the way round.
Let the height of the traveller be $h$ yards.
Consider the equation:
- $12 = 2 \pi \paren {R + h} - 2 \pi R$
Thus we have:
- $h = \dfrac {12} {2 \pi} \approx 1.91$
So, if the traveller:
- were $1.91$ yards, that is, approximately $5$ feet $9$ inches tall
- and walked the entire way around the world, along the equator,
his head would have travelled $12$ yards further than his feet.
$\blacksquare$
Historical Note
When John Jackson included this puzzle in his Rational Amusement for Winter Evenings, he attributed the idea to William Whiston's The Elements of Euclid, With Select Theorems out of Archimedes.
Sources
- 1821: John Jackson: Rational Amusement for Winter Evenings: Geographical Paradoxes
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Rational Amusements for Winter Evenings: $164$