You’ll never count the same after this week’s adventure, brought to us by Harold Reiter from the University of North Carolina at Charlotte. I met Professor Reiter at a Math Circle event over the summer and found him overflowing with curiosity and enthusiasm about anything math-related — including a number of terrific math puzzles. Fusing Dots is one of his favorites.

Here’s Professor Reiter:

Infinitely many empty boxes, each capable of holding six dots, are lined up from right to left. Each second a new dot appears in the rightmost box. Whenever six dots appear in the same box they fuse together to form a single dot in the next box to the left. How many dots will there be after 2012 seconds?

For example, after seven seconds we have just two dots — one in the rightmost box and one in the next box over.

SOLUTION
Professor Reiter will be leading this week’s collaborative inquiry; you’ll find his solution among the comments on Wednesday evening. The initial solution, of course, is just the launching pad for further exploration — so be sure to drop by on Thursday and Friday.

Harold Reiter and the Danger of Memorization
Professor Reiter describes his experience learning calculus as a senior in high school and then arriving at Louisiana State University to study math.

When I was in high school, as a typical college bound student, I took all the standard math and science courses available. After a good semester of trigonometry and three-dimensional geometry, I was offered the opportunity to take a half-year calculus course.

The course consisted mostly of computing limits of rational functions, f(x)/g(x), where both functions were factorable, and the limit was requested at a point where g(x) was zero. Later in the course, I learned about the power rule for differentiating polynomials, and a few other rules for taking derivatives.

It was not until the next year as a freshman at L.S.U. that I realized that I had not really learned calculus. There I saw that I had not the same understanding of ideas or reasoning skills of my peers in that (very strong) class.

What I had ‘learned’ in the high school class were formulas and procedures, which I had memorized without much real understanding. Not only that, I did not realize that calculus is a collection of concepts that one could learn to reason about and even figure out for oneself.

What a shock it was for me to realize that I did not even understand the idea of ‘function’, the main object of study in calculus.

Of course there is some memory work in mathematics, but every memorized fact must be reinforced by an understanding that itself is not memorizable. That understanding becomes a part of the learner in a way that no memorized fact ever can. It cannot be forgotten. Thus, there is great danger in memorizing without understanding.
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NOTES

1.I’d like to thank Professor Harold Reiter for the Fusing Dots puzzle and for guiding our exploration this week. Professor Reiter is one of five Americans to win the Paul Erdős International Award for significant contributions to mathematics learning in the United States. He teachers a variety of math courses including Mathematical Thinking, Discrete Math and Combinatorics at The University of North Carolina at Charlotte and is the founding director of the Charlotte Teachers’ Circle.

2.For more about finding (or creating) a Math Circle in your area, check out the National Association of Math Circles. Catch the passion.

3.This puzzle was originally created by research mathematician James Tanton. Check out Mr. Tanton’s site Thinking Mathematics! for an experience of joyous, accessible, true mathematics.

4.Ideas for a puzzle? Send them to numberplay@nytimes.com.

5.Pradeep Mutalik returns next week.