These are not business questions, but soft questions that should make any PhD candidate relaxed, even intrigued, and open to talk freely. There is no wrong answer, these are open questions, but some answers could hint that the candidate is still in his/her PhD bubble, feeling superior, not flexible, and unable to see the big picture behind the apparently innocent question. These questions were asked discretely, none of the responders knew about my PhD mathematical background.
Author of this article (myself)
I’ve posted them on Quora, Reddit, and other platforms attracting PhD scientists. Below is a summary of what I experienced.
Are all numbers created equal? Are prime numbers superior to other numbers?
While this looks like an innocent question (mostly a philosophical question that appeals to math PhD’s), answers that are a red flag — and I have seen many of them — include:
- Numbers are not equal: 2 is different from 3. Pi is different from SQRT(2).
- The number 3 is superior to 2, and 2.00001 is superior to 2.00000. Again, same problem here, and you don’t want to hire a PhD candidate providing this kind of answer. The candidate understands “superior” in his own framework, but can not relate to what the interviewer (usually not a mathematician) has in mind.
This mindset is typical of someone not flexible to think out of the box, and can be an indicator of tunnel vision. One interesting answer was:
- Prime numbers are to numbers what atoms are to molecules — the building blocks.
Even a better answer would be about how do you measure “superiority”, for instance by computing how many times a number such as 343 appears in social networks and search results, compared to a number such as Pi or e. Then the discussion could shift to how do you count occurrences of Pi, 3.14, or e = 2.71828182…, on the Internet, since the first one is a Greek character, and the last one an English character. How do you go about identifying the popularity of such numbers? This is indeed a great interview question for data scientists, especially in the context of natural language processing.
Are the digits of Pi random?
A typical answer that would make me reluctant to hire the candidate, is:
- The digits of Pi come from a deterministic algorithm. They are absolutely not random. You can then ask: Which numbers do you think have digits distributed as if it was random? Or how do you simulate randomness? In one case, a math PhD went as far as to say that you can not have any kind of distribution on an infinite set of digits. You have to wonder why the layman understands this concept, but some PhD statisticians do not. Again, it is a tunnel vision issue.
Some numbers indeed do not have any kind of digit distribution in their decimal representation, but they are hard to find and all known cases are artificially manufactured, see here. The immense majority of numbers have a uniform (“random”) distribution for their digits, with exceptions including numbers such as 1/7 and all fractions in general. To this day, no one has been able to prove or disprove that Pi or any other popular mathematical constant, has “random” digits. Pi is highly suspected of having “random-like” digits.
However, this was not the purpose of the question. Very few people know this fact, even among PhD statisticians. I was hoping for an answer that is philosophical, but not technical in nature (after all, PhD stands for Doctorate in Philosophy.) Maybe an answer about how you define randomness, and ideas to test it. Even an answer such as “probably” or “probably not” or “don’t know” would have been much better, especially if accompanied by insightful reasons for being unsure, or for believing or not in that hypothesis. Discussing the implications of the randomness (or lack of) in the context of business applications (cryptography), would have been interesting too.
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