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Enter the good citizens of Reddit, who tackled this question with their typical mixture of hilarity and random insight.
Among their most useful suggestions:
1. Consider why the interviewer is asking the question.
Sure, maybe they're just messing with you, but as one Redditor points out, maybe they're really trying to get a fix on who you are:
"I would assume that the point of the question is to indicate how well you know yourself and whether you're mindful enough to manage your weaknesses/know your limitations. Pick a weakness that isn't a trivial cop-out and make sure you've prepared a mitigation strategy, e.g., I tend to X, but I've found if I A, B and C, it's all good."
2. Don't be too funny.
"As someone who interviews others and knows this question well, I think this is the most useful answer. Being funny in a job interview is great, but you have to actually answer or you look like you avoid tough problems. So make your joke, then actually answer the question," writes ArgueOnTheInternet. "I always list a weakness I have overcome first: e.g. 'I used to be disorganized and miss deadlines from time to time, but now I make and check a daily to do list and haven't missed a deadline since.'"
3. Be prepared.
"The point is actually to see how you react to a predictable situation," writes onthevergejoe. "This question is one of the top 10 questions at interviews. Hence, anybody unable to answer it shows that they are unprepared for an easily foreseeable question that could make or break landing a job."
Don't want to take the word of (mostly) anonymous internet commenters? Here's some expert spin.
"When you're asked what your greatest weakness is there are several different ways you can answer, including mentioning skills that aren't critical for the job, skills you have improved on, and turning a negative into a positive," writes Alison Doyle at About.com's Job Searching site.
Note that where all these folks agree is that you need to take the question seriously. It might be a interviewing cliche, but that doesn't mean it isn't important. Be prepared, be serious, and you'll be able to turn a hackneyed question into an opportunity for your skills to shine.
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