David the Gnome

My hat is conical!

When I was very young I watched a show called David the Gnome. As you might guess, the show was about David and he was a gnome. Not a terribly creative title, but when I was young I was captivated.

The Wikipedia article on David the Gnome says that the American version was actually a dubbed Spanish version that was based on a book written by a Dutch author. So apparently, in addition to being a gnome, David was an ambassador to the UN.

In one of David's adventures, he came across a chicken with six chicks. Tragically, the chicken could only count to three, and the chicks kept getting lost without the mother hen even knowing. Although a negligent parent, you have to give the bird bonus points for being able to count at all. [footnote 1]

David, recognizing the problem as a serious one, taught the mother that she should arrange her chickens in two groups of three. The mother could count to three, and she could do it twice, thus she could keep track of all her baby chicks. [footnote 2]

I think that the simple brilliance of this might have been lost on even the creators of the show. David’s poultry grouping insight alludes to many fundamental concepts in mathematics. The fact that every whole number can be represented with a unique product of its prime factors (like 6 = 2×3) is called the fundamental theorem of arithmetic. Had the mother had a prime number of chicks (like five or seven) David’s solution wouldn’t have been so simple because neither 5 nor 7 can be written as a product of smaller numbers.

The chicken could even keep track of 12 chicks, though it's a little trickier. You couldn't put the chicks in 2 rows of 6 (mama chicken can't count to 6), nor could you put them in 3 rows of 4 (4 is still too big), but you could put the chicks in three 2 by 2 groups:

You could also organize a peep of 18, or 27. If you're clever, you can do 16 too.

In addition to the fundamental theorem of arithmetic, David alluded to a tool often employed by mathematicians: when confronted with a difficult problem, reduce it to a problem that’s already been solved. Rather than teach  the mama chicken how to count higher, most likely an impossible task, he had the chicken do the something simple twice.

David applied simple and elegant mathematical thinking to a life threatening situation and his solution is profound, yet straightforward enough for an animated chicken to understand. He's my hero.

-Nick

Footnotes
[1] There is actually some anecdotal evidence that some birds can count as high as three. (Chapter 1, fourth paragraph)
[2] I like to imagine that before David came across the chicken, there were at least a dozen more chicks, all dead now due to negligence.