Musings on mathematics and teaching.

Month: December, 2010

Still Counting with Integrals

Mathematical proofs can often seem like magic tricks, because the authors do not explain how they discovered the ideas that led to their solutions. We endeavor to guide the reader along the shortest path from Point A to Point B, but sometimes it is better to take the scenic route.

The great mathematician Carl Gauss was known for writing clever solutions that concealed the original method of solution. His contemporary, Niels Abel, said of Gauss that “He is like the fox, who effaces his tracks in the sand with his tail.”

In my previous post, I pulled a rabbit out of a hat. I solved a recurrence by “guessing” the solution and plugging it in. In this post, I will explain how the solution could be discovered without guessing, and then I will reveal how I actually solved it.

Read the rest of this entry »


Counting with Integrals

Integrals are not usually associated with counting, but there are many interesting quantities that can be counted using integrals. A fundamental example is the number of permutations of a set of n elements.

\displaystyle a_n = \int_0^\infty x^n e^{-x}\, dx

Read the rest of this entry »

Stacking cannonballs

James Tanton posed a wordless puzzle about stacking cannonballs. In this post, I will explain how I solved this puzzle, but I will ask you to visit the link and think about the puzzle on your own before you read further.

Read the rest of this entry »