The mathematician Leonardo of Pisa (better known as Fibonacci) is famous for his rabbits, but I was recently reminded of his “problem of the birds” or “the problem of the man who buys thirty birds of three kinds for 30 *denari*.” This problem appears in his influential book, the *Liber Abaci*.

The “problem of the birds” is expressed in terms of Italian currency of the time – 12 *denari* (singular: *denaro*) made up a *soldo*, and 20 *soldi* made up a *lira*. In the original Latin, the problem reads:

“Quidam emit aves 30 pro denariis 30. In quibus fuerunt perdices, columbe, et passeres: perdices vero emit denariis 3, columba denariis 2, et passeres 2 pro denario 1, scilicet passer 1 pro denariis ½. Queritur quot aves emit de unoquoque genere.”

In English, that translates to:

“A man buys 30 birds of three kinds (partridges, doves, and sparrows) for 30 *denari*. He buys a partridge for 3 *denari*, a dove for 2 *denari*, and 2 sparrows for 1 *denaro*, that is, 1 sparrow for ½ *denaro*. How many birds of each kind does he buy?”

How many birds of each kind **does** the man buy? It may help to cut out and play with the bird tokens below (click image to zoom). In a similar vein, what if the man buys birds as follows (still purchasing birds of all three kinds, and at the same price)?

- 4 birds for 6
*denari*
- between 6 and 10 birds for twice as many
*denari* as birds
- 8, 11, 13–14, 16–22, 24–25, or 27 birds for the same number of
*denari* as birds
- 8 birds for 12
*denari*
- 12 birds for 18
*denari*
- 16 birds for 12
*denari*
- 28 birds for 21
*denari*
- 6, 8–9, or 14 birds for 11
*denari*
- 7–10, 12, 15, or 18 birds for 13
*denari*

Solution to the main problem here.

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I really like this one. I like the way the true solution is bounded by two false solutions (20 sparrows, 10 doves; and 24 sparrows, 6 partridges).

Good point.

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