Lessons from teaching gifted elementary school students (Part 5d)

Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprisingly depth of mathematical knowledge.

A bright young student of mine noticed that multiplication is repeated addition:

$x \cdot y = x + x + x \dots + x$ ,

Also, exponentiation is repeated addition:

$x \uparrow y = x^y = x \cdot x \cdot x \dots \cdot x$ ,

The notation $x \uparrow y$ is unorthodox, but it leads to the natural extensions

$x \upuparrows y = x \uparrow x \uparrow x \dots \uparrow x$ ,

$x \uparrow^3 y = x \upuparrows x \upuparrows x \dots \upuparrows x$ ,

and so. I’ll refer the interested reader to Wikipedia and Mathworld (and references therein) for more information about Knuth’s up-arrow notation. As we saw in yesterday’s post, these numbers get very, very large… and very, very quickly.

When I was in elementary school myself, I remember reading in the 1980 Guiness Book of World Records about Graham’s number, which was reported to be the largest number ever used in a serious mathematical proof. Obviously, it’s not the largest number — there is no such thing — but the largest number that actually had some known usefulness. And this number is only expressible using Knuth’s up-arrow notation. (Again, see Wikipedia and Mathworld for details.)

From Mathworld, here’s a description of the problem that Graham’s number solves:

Stated colloquially, [consider] every possible committee from some number of people $n$ and enumerating every pair of committees. Now assign each pair of committees to one of two groups, and find $N$ , the smallest $n$ that will guarantee that there are four committees in which all pairs fall in the same group and all the people belong to an even number of committees.