Henry Ernest Dudeney/Modern Puzzles/137 - Hurdles and Sheep/Solution

by : $137$

 * Hurdles and Sheep

Solution
$12$ hurdles.

Proof
From the problem definition, an area of $1$ hurdle length squared holds $1$ sheep.

Hence the problem is to find the largest area enclosed by a given number of hurdles.

This is that enclosed by a regular polygon.

Let $A_n$ be the area enclosed by a regular $n$-gon.

From Area of Regular Polygon:
 * $A_n = \dfrac n 4 \cot \dfrac \pi n$

Hence:

Hence the result.