When I was in school, perhaps my favorite pet project was trying to find a formula for the number of digits in . For starters:

• : 1 digit
• : 1 digit
• : 1 digit
• : 1 digit
• : 2 digits
• : 3 digits
• : 3 digits
• : 4 digits
• : 5 digits

I owned what was then a top-of-the-line scientific calculator (with approximately the same computational capability as a modern TI-30), and I distinctly remember making a graph like the following on graph paper. The above calculations contribute the points , , , , , , , , and .

I had to stop (or, more accurately, I thought I had to stop) at because my calculator couldn’t handle numbers larger than .

I stared at this graph for weeks, if not months, trying to figure out an equation that would fit these points. And I never could figure it out.

And, to this day, I’m somewhat annoyed at my adolescent self that I wasn’t able to figure out this puzzle for myself… since I had all the tools in my possession needed to solve the puzzle, though I didn’t know how to use the tools.

In this series of posts, I’ll answer this question with the clever application of some concepts from calculus and precalculus.