Henry Ernest Dudeney/Puzzles and Curious Problems/146 - The Swarm of Bees/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $146$

The Swarm of Bees
The square root of half the number of bees in a swarm has flown out upon a jessamine bush;
eight-ninths of the whole swarm as remained behind;
one female bee flies about a male that is buzzing within the lotus flower into which he was allured in the night by its sweet odour,
but is now imprisoned in it.
Tell me the number of bees.


Solution

There were $72$ bees.


Proof

Let $n$ be the number of bees.

We have that:

\(\ds \sqrt {\dfrac n 2} + \dfrac {8 n} 9 + 2\) \(=\) \(\ds n\)
\(\ds \leadsto \ \ \) \(\ds \sqrt {\dfrac n 2}\) \(=\) \(\ds n - \dfrac {8 n} 9 - 2\)
\(\ds \) \(=\) \(\ds \dfrac n 9 - 2\)
\(\ds \leadsto \ \ \) \(\ds \dfrac n 2\) \(=\) \(\ds \paren {\dfrac n 9 - 2}^2\)
\(\ds \) \(=\) \(\ds \dfrac {n^2} {81} - \dfrac {4 n} 9 + 4\)
\(\ds \leadsto \ \ \) \(\ds 2 n^2 - 153 n + 648\) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds n\) \(=\) \(\ds \dfrac {153 \pm \sqrt {153^2 - 4 \times 2 \times 648} } {2 \times 2}\)
\(\ds \) \(=\) \(\ds 72 \text { or } \dfrac 9 2\)

The result follows as (Monty Python notwithstanding) half a bee is not viable.

$\blacksquare$


Historical Note

Henry Ernest Dudeney reports:

Here is an example of the elegant way in which Bhaskara, in his great work Lilivati, in $1150$, dressed his little puzzles.

Note his continued misspelling of Lilavati.


Sources

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