What interview questions did D.E. Shaw ask Larry Summers?

As director of the president's National Economic Council, Larry Summers is currently facing the world's biggest math problem. It was encouraging, therefore, to read in Monday's New York Times that, when he applied for a job in 2006 with investment firm D.E. Shaw, "Mr. Summers was asked to solve math puzzles. He passed, and the job was his."

It's hard to imagine Summers being subjected to the same brainteasers that entry-level quants have to answer. And a White House spokesperson confirmed that it wasn't the same series of questions. But he did have to answer analytical reasoning problems asked by a member of the company's executive committee. What kinds of questions does D.E. Shaw ask?

The New York-based firm is known for its rigorous, numbers-heavy interview process. Most applicants have sterling academic backgrounds. The goal, therefore, is to see if the person can apply the concepts he learned in school to the real world. "The question is, 'Can they get past their white papers?' " says Richard Rusczyk, a former D.E. Shaw trader who conducted dozens of interviews over four years at the firm.

The type of questions most interviewers ask—and those D.E. Shaw is known for—are those with no right answers. Here's an example:

Ten people are bidding on a stock at 90, while 100 people are offering to sell it at 91. What price is the next trade?

Interviewees often say that since there are more sellers than buyers, the sellers get to determine the price. That logic usually yields an answer between 90 and 91. That's exactly wrong. "They're not thinking about what's going on in the real world," says Rubczyk. In reality, when there are more sellers than buyers, the price falls. So the next sale would probably be in the mid- to low 80s.

"Some candidates would say you can't answer that question, because there's no formula," says Rusczyk. "If that makes their heads explode, that's a problem."

The next level of difficulty is the type of question with no answer at all. One such question, which Rusczyk has asked, is the famous St. Petersburg Paradox:

There's a dollar on the table. I'm going to flip a coin. If it comes up heads, I'll double the money. If it comes up heads again, I'll double it again. Whenever it comes up tails, we stop.

But there's a catch: You have to pay a fee to play. How much are you willing to pay?