Fibonacci and his birds

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.

---