Henry Ernest Dudeney/Puzzles and Curious Problems/148 - A Menagerie/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $148$

A Menagerie

A travelling menagerie contained two freaks of nature -- a four-footed bird and a six-footed calf.

An attendant was asked how many birds and beasts there were in the show, and he said:

"Well, there are $36$ heads and $100$ feet altogether.

You can work it out for yourself."

Solution

There are $12$ beasts and $24$ birds.

Proof

Let $a$ and $b$ be the number of beasts and birds respectively.

If the anomalies had the usual number of legs, there would be $96$ feet altogether.

Hence we have:

\(\text {(1)}: \quad\)

\(\ds a + b\)

\(=\)

\(\ds 36\)

as they each have one head

\(\text {(2)}: \quad\)

\(\ds 4 a + 2 b\)

\(=\)

\(\ds 96\)

as (normal) beasts have $4$ legs and birds $2$

\(\text {(3)}: \quad\)

\(\ds \leadsto \ \ \)

\(\ds 2 a + 2 b\)

\(=\)

\(\ds 72\)

$(1) \times 2$

\(\ds \leadsto \ \ \)

\(\ds 2 a\)

\(=\)

\(\ds 24\)

$(2) - (3)$

\(\ds \leadsto \ \ \)

\(\ds a\)

\(=\)

\(\ds 12\)

\(\ds \leadsto \ \ \)

\(\ds b\)

\(=\)

\(\ds 24\)

$\blacksquare$

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $148$. -- The Menagerie
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $220$. The Menagerie