Fun From Fibonacci (via Keith Devlin)

Keith Devlin’s new volume, “Finding Fibonacci” is more historiography than math, but there is some simple fun math sprinkled along the way deriving from Fibonacci’s writings. One old problem (translated from Fibonacci’s “Liber abbaci”) that Keith quotes runs as follows [I’m re-wording it in updated English]:

A man buys 30 birds composed of partridges, pigeons, and sparrows, for 30 denari. A partridge costs 3 denari, one pigeon costs 2 denari, and 2 sparrows cost 1 denaro, or 1/2 denaro/each. How many birds of each kind has the man purchased?

Of course this looks like a classic multi-equation problem, except there are only 2 equations, yet 3 unknowns:

1)  x + y + z = 30  (number of birds; x partridges, y pigeons, z sparrows)

2)  3x + 2y + z/2 = 30 (total price)

You can solve it, or look at the answer below…

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Keith notes, there is a third hidden piece of knowledge buried in the problem: 

namely, that x, y, and z must be positive integers (because birds don’t come in fractions!)

Thus, the two equations above are easily reduced to:

5x + 3y = 30 (where both x and y must be whole positive numbers)

Keith notes the first and third terms are divisible by 5 and so the second term (3y) must also be divisible by 5. In turn, that means y must equal 5, 10, 15 etc… but 10 or more is too large to work in the equation, so only 5 can be the correct answer (and x = 3, and z = 22).

As I indicated previously, the book may be more appealing to math history buffs than for mathematicians themselves (I found the last few chapters, containing more math bits, the most interesting of the book), though, with warm weather approaching, I'm tempted to call it a good beach read for the nerdier among us. In it, Dr. Devlin makes a case that "Liber abbaci," from the under-appreciated Leonardo Bonacci ("Fibonacci") is "a book that changed the course of Western civilization," and seeing him build that case (including making an analogy between Fibonacci and Steve Jobs) is interesting in its own right.