Devlin's Angle

December 2009

Strictly for the Birds

With Christmas almost upon us, I thought I'd use this month's column to give you two of my favorite brain-teasers to challenge your relatives at that family gathering. What I like about these problems is that when you first meet them, you think you don't have enough information to solve them. But you do. You just have to look carefully at what the problem says. Both puzzles are about birds. I don't know the origin of the first one, but the second has a famous history.

Bird puzzle 1

A new assistant accidentally left open the cages at the pet shop, and over 100 birds escaped There were exactly 300 birds to begin with. The next morning, the local newspaper carried a report that gave the following figures:

"Of the birds that remained, a third were finches, a quarter were budgies, a fifth were canaries, a seventh were mynah birds, and a ninth were parrots."

However, the reporter got one of the fractions wrong. How many parrots were left?

Bird puzzle 2

The second puzzle appears in the famous book Liber abbaci, published in 1228 by Leonardo of Pisa, known today as Fibonacci. It is sometimes referred to as "Fibonacci's problem of the birds." I'll state it by copying verbatim from the English language translation of Leonardo's great work by Laurence Sigler (Springer-Verlag, 2002), who until his death in 1997 was a Professor of Mathematics at Bucknell University. (The "denaro" was a unit of currency in Medieval Italy.)

A certain man buys 30 birds which are partridges, pigeons, and sparrows, for 30 denari. A partridge he buys for 3 denari, a pigeon for 2 denari, and 2 sparrows for 1 denaro, namely 1 sparrow for 1/2 denaro. It is sought how many birds he buys of each kind.

As before, what makes this problem particularly intriguing is that it seems you don't have enough information to solve it. Specifically, it looks like you have two equations in three unknowns. In fact, in terms of equations, that is precisely what you do have. But the problem gives you additional information that turns out to be all you need to find the (unique) answer.

I'll give the answers to both puzzles in next month's column.

Devlin's Angle is updated at the beginning of each month. Find more columns here.

Mathematician Keith Devlin (email: [email protected]) is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR's Weekend Edition. His most recent book for a general reader is The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern, published by Basic Books.