Henry Ernest Dudeney/Puzzles and Curious Problems/241 - A Hurdles Puzzle/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $241$

A Hurdles Puzzle

The answers given in the old books to some of the best-known puzzles are often clearly wrong.

Yet nobody ever seems to detect their faults.

Here is an example.

A farmer had a pen made of fifty hurdles, capable of holding a hundred sheep only.

Supposing he wanted to make it sufficiently large to hold double that number, how many additional hurdles must he have?

Solution

The answer completely depends on the arrangement of the hurdles to make the pen.

Consider the following diagrams:

The original answer that is frequently seen is that it can be done with $2$ hurdles.

The arrangement given is that of $A$, a long paddock $1$ hurdle wide and $24$ hurdles long, enclosing an area of $24$ square hurdles.

In $B$, we see we have added $2$ more hurdles to the short sides, making a paddock $2$ hurdles wide and $24$ hurdles long, enclosing an area of $48$ square hurdles.

But consider.

With $50$ hurdles you can make a paddock as in $C$, $156$ square hurdles by making it $12 \times 13$, which is $6 \tfrac 1 2$ times the size of the original $24 \times 1$ paddock.

Or you can make a $6 \times 8$ paddock with $28$ hurdles, as in $D$.

And so on.

Dudeney also manages to arrange $50$ hurdles in an elongated kite shape with sides $12$, $12$, $13$, $13$ explaining that this is also $48$ square hurdles in area, but his diagram is unconvincing.

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $241$. -- A Hurdles Puzzle
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $278$. A Hurdles Puzzle