# Books! Books! Books!

I’ve been a bit incommunicado lately. Nothing bad has happened! I’m not working harder than normal! I don’t know why but I haven’t been moved to post anything. And you know my feeling about blogging — it can’t be a chore, so don’t force it.

That being said, I wanted to share something we’ve been diverted by in multivariable calculus recently:

I gotta say, I love this class and I love working with these kids. They remind me how when you find the right thing, exploration is captivating.

This is the “book overhang problem.” The question we dealt with was: can you stack books at the edge of a table so that the top book is off the table completely (meaning if you’re looking down on the stack of the books, the top book doesn’t lie over the table at all)?

We haven’t yet found the optimal solution, but we’re going to be discussing our musings on Friday — what the best 3, 4, and 5 book configuration might be, and if we can generalize it.

1. The solution I’m familiar with has an overhang of 1/(n+1) for the nth book from the top. That allows an unbounded extension (see harmonic function H_n), but it gets difficult to construct after 5 or 6.

1. The wolfram analysis is correct. I misremembered the result slightly. You can do the analysis fairly easily by using the lever law, treating the stack above the current book as having weight (n-1) centered at the end of the current book.

You want to push the book out by d until the whole stack is balanced with its centroid above 0.

That is, $\int_0^d x dx + (n-1)d =$int_0^(1-d) x dx $, or $d^2/2 + (n-1)d = (1-d)^2/2$, or $2(n-1)d = 1 - 2d$, or$d = 1/(2n)\$.

2. That’s similar to what my kids did!

2. Yay, I love this! I did something similar for my first interview when I had to bring a lesson to get hired. I found using cd-cases works well to get things to not tilt, but I think the huge calc books pack more visual punch.

3. “The solution I’m familiar with has an overhang of 1/(n+1) for the nth book from the top. That allows an unbounded extension (see harmonic function H_n), but it gets difficult to construct after 5 or 6.”

That’s the solution I’m also familiar with, or at least I was until a student a few years ago questioned whether this was an optimal solution and found a counterexample for a few individual cases. Have fun exploring that one!

1. I’d be interested in seeing one of these counterexamples, as I thought there was a simple inductive proof that the sum of 1/n is optimal.

4. “I’d be interested in seeing one of these counterexamples”

Tried to think of a hint that won’t give it all away…best I could come up with is to consider a case where the books aren’t all stacked on top of one another.

1. If the books aren’t stacked, it’s a different problem. What is the new statement of the problem then?

5. Hey Sam, thanks for the illustration here. I was in a room at a table the day after you posted this with a couple of guys, one trying to explain the problem to the other, and I was able to pull up your photo. Which was awesome.

6. Andy Johnson says:

That looks like an ideal use for textbooks! Finally they are good for something!