Imran On Tech

Why logic puzzles make good interview questions January 10, 2007

Logic puzzles in interviews seem to be one of those things that everyone either loves or hates, but speaking as an interviewer I find logic questions critically important in deciding between candidates – far more than “behavioural” or “situational” type questions that HR peeps seem to favour.

My aim in this post is to convince those of you that hate them, that they are actually useful questions to ask (and be asked) and give everyone else an insight why interviewers especially at big tech companies seem to love this type of question.

To begin with I feel I should draw a separating line between logic puzzles and brain teasers. Logic puzzles are of the type where there’s no “trick” but rather puzzles which have answers that can be logically deduced. Often these are puzzles where there are a number of different valid solutions but you’re asked to find the optimal solution, examples of this type of question include the Rope Bridge and The Orb.

Brain teasers however often have a “trick” – something unusual you have to guess often about the assumptions of the problem. While these can provide some insight into the candidate I think their value is limited, and that they’re far weaker than pure logic puzzles.

I’d like to show what an interviewer can find out about a candidate from a logic puzzle via an example, walking through an interviewer asking the question and showing how various canidates respond:

Interviewer: Imagine you have eight coins, seven of which weigh the same and one that doesn’t (it’s heavier). You need to use a pair of scales to find out what’s the odd one out.

If this seems familiar it probably is, this is apparently one of the most asked logic puzzles in the known universe. It’s used in interviews by Microsoft, Amazon, Google, and probably hundreds of other firms. It’s also in pretty much every book on the topic and on dozens of websites. Now for some example responses:

Candidate Alice: What’s a pair of scales ?

Candidate Bob: How much heavier is the other coin ?

Candidate Carol: Well you could weigh them one-by-one until you find the pair which is uneven, and then….

All of the above responses are good, both Alice and Bob recognized that they had been given a problem they didn’t fully understand so they took the step of asking relevant questions to clarify the problem in their own mind before trying to solve it.

This is a situation that occurs frequently in intellectually demanding jobs, people are frequently asked to solve problems for others. Everyone has different experience and background and will understand the problem differently. This a major advantage, in a team diversity will help you come up with new solutions to a problem. However it also means that everyone will interpret the problem differently, and asking question to clarify what the problem is can be a key part of ensuring everyone’s on the same page.

Carol’s right too, she’s come up with a valid solution right-off-the-bat, it’s not the optimal solution but it is a correct one. We can discuss it and see if she can improve it.

Now a bad response:

Candidate Dan: I don’t know.

If the only purpose of asking logic questions was to catch people who gave this answer, then they’d still be worth asking. This is a candidate you never want in your team. I have three hypothesis about Dan-esqe candidates,

• They really can’t solve the problem at all
  • This indicates the candidate struggles with problems which intelligent 5 year-olds can solve, not a good sign.
• They can’t be bothered to try
  • Not a good sign of motivation
• They don’t want to answer it
  • If they can’t explain why the don’t want to answer the question then their communication skills are probably too weak. In most jobs refusing to do something without giving any reason is likely to be unconducive to a good working environment.

Luckily all three are grounds for rejecting a candidate so interviewers don’t have to think too hard about which the candidate falls into.

So lets look at the next stage of this response, where the candidate has understood the problem and has come up with the trivial solution that Carol has above.

Interviewer: Yes that would work, how many times would you have to use the scale to find the solution using that method in the average and worst case ?

Candidate Carol: Four and eight respectively

Interviewer: Are you sure about that ?

Candidate Carol: hmm, four and ten ?

Carol’s initial response is both a good and bad answer, Carol managed to correctly estimate the average and worst times, showing an understanding of how the speed of the solution depends quite a lot on luck. However her answers aren’t the exact correct values, and worse still when she’s challenged she come’s back with a worse answer indicating that maybe she just guessed.

At this point the interviewer might start to push her on how she got those numbers, if it turns out she starts wildly guessing answers while under pressure, she might not be suitable for a work environment which is pressure heavy and detail-critical.

But now lets move on:

Interviewer: So can you think of a better solution to this problem which lets you use the balance fewer times ?

Candidate Errol: You can split the coins in half. weigh them against each other, throw away the lighter ones, and repeat until you only have one left.

This is a good answer, I’d expect anyone coming from a Comp Sci or Maths background to get this far without any help. People from other backgrounds I might nudge in the right way and see if they can find the solution.

However if you’re from the first group and don’t get this solution I’d be a little worried, it wouldn’t be an instant reject but it’d definitely be a question-mark. Maths/Comp Sci people should have seen similar methods of tackling problems as part of their studies, and should be able to reason about this problem from their previous experiences.

If the candidate came straight to this solution skipping the obvious solution then I’d ask them about the average/worst times for this solution instead. But otherwise back to improving:

Interviewer: Do you think this is the best solution ?

Candidate Fred: I think it is

Interviewer: (smiling) I’ll give you a clue, it isn’t.

Interviewer: You’ve come up with solutions balancing one-against-one and four-against-four, what else can you do ?

Candidate Fred: Three-and-Three ?

Interviewer: What do you learn if you do that ?

Candidate Fred: Which three is heaviest

Interviewer: What if the scale is balanced ?

Candidate Fred: The heavy coin must be one of the other two — you can split the coins three ways every time.

Interviewer: Yep, that’s the optimal solution, do you know for N coins how long it’ll take ?

Candidate Fred: About Log N

Interviewer: What base ?

Candidate Fred: Base 3

That’s typically how the best solution is arrived at by a good candidate. How much help I give the candidate depends how much they’re struggling, some candidates just click about going through the other combinations and come up with the answer. Some like Fred manage to come through with a little help, some candidates just get stuck. I think you can guess which order I want to hire those candidates.

And for bonus points:

Interviewer; Can you prove it’s the optimal solution ?

Candidate Goyle: No I’m not sure how, but I’m fairly sure it is.

Candidate Hermione: Each weighing gives you three pieces of information (if the left side is heavier, if it’s balanced, if the right side is heavier), that is one trit (trinary bit) of information. As the heavy coin can be anyone of the N you need at least enough trits to give N outcomes. Which means you need at least log3 of N goes, which happens to be how many our solution takes. So there can’t be a solution better than the one we discussed.

I don’t really expect a candidate to be able to answer this, but it’s a good sign if a candidate can. People who are solid on the practical side but can excel on the theoretical side as well are rare indeed and it’s well worth identifying them.

So hopefully that’s shown you that there’s a lot of information about the candidate that an interviewer can get by asking a logic question. The interviewer doesn’t just want the candidate to show off their intellect by coming up with the correct answer. They want to see the process by which the candidate got to that answer, as that process is likely to be one that the successful candidate will have to go through frequently during their job.

If you’re interested in learning more about this sort of logic puzzle I’d recommend Moving Mount Fuji (the book; not the task) which gives a good history of how logic puzzles became a part of corporate interviewing. If you’re an (ex)student looking for your first job in the tech sector I’d recommend Programming Interviews Exposed which gives lots of logic and programming questions which are typical in technical interviews.