Ask HN: FizzBuzz for mathematicians ...

[ColinWright]

We have enough mathematicians here to ask this: What would you suggest as an equivalent of FizzBuzz for mathematicians?

Some suggest proving sqrt(2) is irrational, but many technical types who aren't really mathematicians can do that because they've read enough popular math books. Similarly proving that there are infinitely many primes.

So, mathematicians, what would you suggest?

And for the non-mathematicians, what would you like to have explained, that you think every mathematician probably knows?

Added in edit: I thought this was a great discussion with some interesting suggestions, but I see it's been flagged off the front page.

http://hnrankings.info/4244266/

So that's that, another interesting discussion killed. Oh well, never mind.

[algorias]

>many technical types who aren't really mathematicians can do that because they've read enough popular math books.

Well, fizzbuzz doesn't exactly require a masters degree either. Proving sqrt(2) is irrational requires at least some understanding of how math problems are approached (unless you do it completely from memory, which is also possible with fizzbuzz), so I would say it's the equivalent.

[TillE]

Yeah, the point of fizzbuzz is that it's an extremely simple test that demonstrates basic competence, nothing more.

I have a CS degree and took a bunch of math courses and did lots of proofs, but now (several years later) I'd struggle to write any of the suggested proofs.

[SatvikBeri]

Literally any proof is on the level of FizzBuzz. There is a wide gap between the level of competence it takes to prove anything, vs. not understanding how proofs work-once you know any higher-level Math, you know proofs. That's even a slightly higher level of competence than FizzBuzz-anyone can do FizzBuzz if they know even a modicum of programming. You certainly don't have to be a programmer to do FizzBuzz, and you don't have to be a Mathematician to prove that sqrt(2) is irrational.

[lotharbot]

Derive the quadratic formula.

It's just about as simple as FizzBuzz, and requires primarily symbolic manipulation. Anyone from any branch of mathematics (including applied math and statistics) should be able to do it in their sleep.

[dllthomas]

But so should a high-school student...

[ColinWright]

Other recent suggestions include:

* Show that two irrationals can add up to a rational.

* Find three whole numbers in arithmetic progression whose product is a prime.

* Find a prime that is one less than a square. Now find another.

[nhaehnle]

> * Find three whole numbers in arithmetic progression whose product is a prime.

I don't think there's an agreed upon definition of what a "whole number" is. Then again, if you're just looking for how someone reacts to such a question, I suppose it's fine, if a bit silly.

[alexeismart]

For 1, suppose that x is irrational, that p is rational and q = p - x. Either q is rational or irrational. If q is rational then p - q is rational which would imply that x is rational. By contradiction, then p - x must be irrational. Therefore x + q = p, implying two irrational numbers can sum to a rational number.

For 2, how about -3, -1 and 1.

For 3, 3 is the only example. One less than a square is n ^ 2 - 1 = (n+1)(n-1). For n=2 the answer is 3 (3*1). For n > 2 the result is clearly factorable into two numbers > 1.

[dangoldin]

I'm not sure what you mean by the second point. Three numbers whose product is prime? Did you mean sum?

Edit: Nevermind - I just figured these out. Makes me feel a bit dumb now. That's what I get for not doing math for a few years. I really liked the last one - got me thinking.

[tzs]

For the middle one, would you accept the arithmetic progression 1, 1, 1?

I know that 1 is not considered to be a prime now, but it was considered to be a prime by most mathematicians up through the 19th century, and Lehmer included it in his lists of primes as late as 1956.

[dllthomas]

s/whole number/integer/

[kurumo]

I have been using this when interviewing people with math degrees:

Two players play a game with a single six-sided die. The player that starts can only win by rolling a 1. If he or she doesn't win, the other player gets to roll; he or she can only win by rolling a 6. The game continues until one player wins. What's the probability the first player wins (eventually)?

[debacle]

Let me see if I can work through this.

Player 1 wins based on the following series:

Chance to get a 1 (turn 1): 1/6 Chance to be allowed to roll: Previous chance to be allowed to roll * 5/6 (Chance player 1 didn't roll a 1) * 5/6 (Chance player 2 didn't roll a 6) Chance to get a 1 (turns 2+): Chance to be allowed to roll * Chance to get a 1

So... summation( 1/6 * (25/36)^(n-1) )

I don't know what that comes out to be.

[stephencanon]

I like this one. It can be brute-forced easily enough, but there are also a couple much more elegant approaches that I would expect a good mathematician to produce.

[hypeibole]

As a mathematician I'll be interested in knowing why you ask that question to people with math degrees.

What information does that give to you as an interviewer?

[prophetjohn]

I think I'm at least as competent at math as one would need to be at programming to solve fizzbuzz and I have no clue how to prove either of those things.

Fizzbuzz isn't trying to separate the professional programmers from the pretenders, its' trying to identify people who have no capacity for programming whatsoever. Fizzbuzz only requires basic knowledge of the syntax of any programming language and basic problem solving skills. I think a better equivalent would be something like "integrate x*e^x". Or maybe even some college-level algebra.

[Locke1689]

The idea behind the problem seems to be to separate the mathematicians from the engineers, which the problems seem to do reasonably well.

However, since both problems are fairly trivial proof-by-contradictions, I would like every computer scientist to be able to solve them as well.

That said, my masters degree is in programming languages, so I may be exposed to a bit more math than the average software developer. I do not consider myself competent enough to be a mathematician, though.

[kolbe]

I think proving the sqrt(2) is irrational is great. That was the very first question on my first college math homework assignment. It's a great way to weed out people aren't even close to the level they should be at math-wise. Don't worry that others can do it. FizzBuzz can also be cargo-culted. FizzBuzz tells you nothing about how good someone is, but can tell you a lot about how bad they are.

I would also suggest this: Prove exp(pi*i)+1=0.

[aviraldg]

Is it possible to prove that 0.99... = 1? If yes, how?

[riffraff]

This one is just too much geek knowledge I think. E.g. I have read about it a bunch of times and I completely suck at math.

Something less "cool" would be more apt imo (fizzbuzz itself is rather uninteresting).

[gjulianm]

It's a pretty easy proof. I'm on mobile so I can't elaborate a lot, but just look how do you construct the fraction corresponding to a periodic number. In this case, it's 9 divided by 9. That's 1.

Another proof (requires more knowledge) is based on the structure of the real numbers. Let a and b be real numbers where a ≠ b. Then, there are infinite numbers between them. It's a pretty obvious statement.

Now, try to find any number between 0.99... and 1. You can't. So, if there isn't any number between 0.99 and 1, they are equal.

[khyryk]

After reading some of the comments, it seems as though this has a tendency to turn into another one of those interviews where the interviewer gets to feel superior if you can't remember shit you learned 20 years ago verbatim.

[gcv]

Explain and demonstrate the use of Taylor series. Quite a bit easier than many of the other suggestions below, and will weed out total idiots. Kind of like asking a supposed programmer to explain a "for" loop.

[ColinWright]

Except that not every mathematician will actually have done Taylor Series. Algebraists, logicians, topologists and graph theorists, for example, may very well not have done analysis to that level.

[dangoldin]

What is the goal? If it's to test the mathematical ability of someone who you want to hire to code it might make sense to ask some probability questions.

[Locke1689]

Prove that the inverse of any element in a group G is unique.

Edit: If this is too easy, prove that g in G, g -> g^2 is a homomorphism iff G is abelian.

Note: I am not a mathematician.

[hypeibole]

Maybe I'm just a sucker for group theory but that strikes me as too trivial for any mathematician.

In that same vein, I'd suggest proving that a group G is abelian if and only if (ab)^2 = a^2.b^2, where a, b are in G.

It's not as straightforward as your question in the sense that proving it requires a little idea that you have to come up with.

[seagaia]

Perhaps it shouldn't be solving one problem, but maybe explaining a base concept in a field most mathematicians know, and how it's important - since there's no one right answer, it would also show their experience and so forth - most people probably know what they are, but the more experienced will be able to talk for quite a while. iono, Cauchy sequences, maybe. or groups

[klickverbot]

Well, depends on your definition of »mathematician« – I'd probably go for a question about the very basics of algebra. Something like: How many elements does the symmetric group S_3 have? (Optionally: Write down its Cayley table.) Is it abelian?

Not exactly relevant to practice, and utterly trivial, but I suppose nobody comfortable with formal math should any problem solving it on the spot.

[hypeibole]

In what sense are they to be equivalent?

In any case I'm going to suggest this:

Q. Give two different proofs of the fundamental theorem of algebra.

[bdittmer]

The Monty Hall problem? Like other suggestions, too many people might be able to recreate it from memory.

[caustic]

Or apply Bayes' theorem to a real world problem http://yudkowsky.net/rational/bayes/

[dangoldin]

To take as much from FizzBuzz as possible - how about "What is the sum of all integers between 1 and 1000 that are divisble by 3 or 5?"

It's simple to figure out with pen and paper but the goal is to just confirm a basic level of competence.

[ColinWright]

Scarcely requires math, seems only to require arithmetic.

[klickverbot]

Another very basic question, maybe analogous to FizzBuzz in that regard, could involve some real analysis, e.g.: Let f: [0,1]->[0,1] be a continuous function. Prove that there exists some x in [0,1] such that f(x) = x.

[S4M]

Compare e^Pi and Pi^e ?

EDIT: OK, that one is more for engineer. How about: find a bijection between R and P(N)?

[ColinWright]

Interesting. Can you produce a bijection between R and P(N)? It might be harder than you think.

[netvarun]

My suggestion: Prove the infinitude of prime numbers.

[ColinWright]

Hmm. Did you see that I already mentioned that in the question?