How many feet of rope?

Elmer got a new job working for the procurement department of a very large company. boss called him in on his second day, and said that he needs a rope long enough to go all the way around the Earth at the equator. Elmer’s job was to figure out how long the rope has to be, and to get quotes from suppliers: X feet of rope at Y dollars per foot.

Assume for the sake of this example that the Earth is perfectly spherical and smooth.

Elmer went back to his desk, did some calculation to figure out how much rope he’d need, and started burning up the phone talking to suppliers. A few hours later, quotes in hand, he entered the boss’s office and placed the quotes on his desk. Boss glanced at them and then said, “Good work. Before we place the order, this is enough rope to stretch all the way around the Earth, leaving a foot of space between the surface and the rope, right?”

Elmer, surprised, said, “No, sir. You asked for a rope to go all the way around the Earth. You didn’t say anything about a foot of slack.”

Boss pushed the quotes across the desk towards Elmer: “Well, you’ll have to re-do these quotes. I need the rope a foot away from the surface.”

Elmer, being the smart guy he is, replied “That’s okay, we’ll just add XX feet of rope to the original quote.”

What is the value of XX? How did Elmer figure that out so quickly?

My first games programming job was a huge learning experience. It was the first time I’d ever worked with people who were a lot smarter than I was. At least, smarter in terms of computer science. The people I’d worked with before understood specific things about programming, but they didn’t love programming like I did. To most of the people I’d worked with before, programming was just a job. I lived programming! I studied problems at home, solved problems at work, always had a few side projects, etc. Programming was my job and my most engaging hobby. Debra says that she shares me with my first wife: the computer.

One of the great things about working there was that we were always posing little brain teasers to each other. These could vary from math questions such as the above, algorithms for things like finding if a linked list has a loop, or sometimes larger design questions that don’t really have a “right” answer, but rather were designed to make us think about how to approach a problem. In many ways it was like an ongoing job interview. The puzzle that I presented above was one of them. This is one that I got after just a few seconds’ thought, but then had to check my work on the white board.

I’ve posed this question in interviews over the years, when a candidate says he’s good with math. The reason is that it’s a relatively simple problem that somebody who’s “good at math” should be able to solve relatively quickly. Whereas I’m interested in the solution, I’m more interested in how the candidate approaches the problem. Does he know how to set up the equation? Does he ask me questions to clarify the problem? Does he just sit there looking like he has no idea where to start?

Some people think the problem is too specialized, but I think it’s more than reasonable to expect a programmer, especially one who says he’s good at math, to know the equation for the circumference of a circle. This goes double for games programmers!

It’s surprising how many candidates utterly fail when it comes to this question. Some just sit there, dumbfounded. Others get up at the white board and start doodling, but can’t formulate the problem. It’s embarrassing.

The circumference of a circle is two times Pi, times the radius. That is:

C = 2πr

The original quote was for C feet of rope–enough rope to go around the Earth with radius r. The new rope has to be a foot away from the surface of the Earth. That radius is (r+1). So you have the original rope with length 2πr, and the new rope with length 2π(r+1). We want to know the difference. Showing my work, as my instructors always insisted:

C1 = 2πr
C2 = 2π(r+1)
C2-C1 = 2π(r+1) – 2πr
= 2πr + 2π – 2πr
= 2π

Elmer’s answer was 2π, or about 6.28 feet. It only takes about 6.28 feet to move that rope from being against the surface of the Earth to being a foot away.

I’ve had candidates reach that solution and say, “but that can’t be right,” and go back to check their work. I consider that a good thing; being willing to admit that one might have made a mistake is something I look for in the people I work with.

I’ll admit that this fits squarely in the category of “math that just don’t feel right,” but it is right.

I’ve had other candidates argue that the answer is just flat wrong. Even after I show them the math. That’s bad. Not only is arguing with the interviewer bad form, but unsupported statements like, “I know that can’t be the right answer” show an unwillingness to learn and an inability to form reasoned, coherent arguments.

Note that I have nothing against polite, informed, disagreement from a candidate. That shows backbone and an ability to support one’s argument.

It’s a fun interview question, and can tell you a lot about the candidate’s approach to problems, and his reaction to things that are non-intuitive. Will he accept the math, or will he try to argue that the math is somehow wrong?

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