My interviewer this week randomly asked me the square root of 10,000,000 and it totally flustered me. I took a couple minutes to get to an answer - how bad do we think that is?

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I feel like that's a "how well do you respond" type of question. Of course you don't know it then and there on the spot, but letting them know that you'd be happy to get back to them with the correct answer after you've had time to find the right answer, I believe that is the right answer to their question.

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I think this is correct. Those are variations of the most basic case interview questions. The purpose is to see how you can structure an approach to figure out something you’ve never encountered before. The answer doesn’t have to be correct. But you have to verbalize your thought process and identify assumptions you make to get an answer. And you would probably be expected to talk it out in real time. At least describe your approach, then calculate.

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I've had an interviewer ask me to go up to a whiteboard draw a square and put a dot in the center and then asked me to cut that into five equal pieces as if it was a square pie. And they all had to come from that Center Dot. The main purpose of this was to see how you think not necessarily to get the right answer. I broke each side down into five segments and counted four sections and then made a line from the center. I later on ended up figuring out that that is correct. I even 3D modeled it.

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I'm sure that your response won't be the deciding factor in determining whether or not you'll move on to the next round, if that's what you're worried about.

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this is probably more of a thought process question to see how you break down a complex issue/problem into manageable steps

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It’s almost like you are their entertainment. 

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Boy unless your interviewing to be a math professor they dont wxpect a correct answer its about your thought process. I was once asked how I would go about painting a 747 aircraft in an interview where the job had no relationship to either painting air aircraft maintainence.

Why would you need to know the square root of 10,000,000 if your not taking a position that has nothing to do with math?

Wow… I never heard a question so complex as this one. The world is collapsing from here onwards

What an idiotic question. Not sure if you are applying for something math-related, but my answer would be that I would never devote any of my limited cognitive space or energy to completing or preparing to complete a task that we have tools to complete much more efficiently for us. It’s called comparative advantage and my comparative advantage is doing a lot of other things that computers CANT do. Computing tasks are for computers, calculating tasks are for calculators, I am excellent at human tasks if you have any for me.

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