How many windows are in the United States? (and other large estimation problems)

I’ve recently been preparing for some interviews, and came across a series of interesting estimation questions that are (somewhat) related to math. I apologize for the delay in writing about the second half of the Nash Equilibrium, but I promise that it will be up soon! But for now, enjoy a bit of math in a less rigorous and proof intensive manner. (Disclaimer: Some of these estimations have no real data basis.)

  1. How many windows are in New York City (Manhattan)?

    To first start off, consider the answer given here. However, I have a slightly different approach than the one given (assuming that we only consider building windows). Given there are approximately 150 streets and 10 avenues in Manhattan, we have 1,500 blocks. Then we have 5 buildings per block and 40 windows per floor. I would assume the average height of a building in NYC to be 20 stories. Given these assumptions, I estimate that the number of windows in NYC is 6 million. I do not subtract out Central Park because I am taking the average height of the buildings which include 0 (Central Park) and 50 (Wall Street).

  2. How many bottles of shampoo are produced in the United States in a year?

    (Go here to see one possible answer). There’s approximately 300 million people in the US. The average person takes a shower once every other day. Therefore, a person can go through one bottle of shampoo in 3 months and uses 4 bottles of shampoo in a year. Since there are 300 million people in the US, the number of shampoo bottles sold is 1.2 billion. Consequently, one can also ask how many shampoo bottles are produced in the world. There are 6 billion people in the world which is 20 times as many people as the US. Therefore, the number of shampoo bottles sold around the world would be 24 billion.

  3. Estimate the number of students who are college seniors, attend four-year schools, and graduate with a job in the United States every year.

    There are approximately 3,000 4-year colleges in the US. The average university has about 2,000 graduating seniors per year. The nationwide unemployment rate for new college grads is about 6%. Therefore, 94% of the new grads are employed. Multiplying all these numbers together gives 5,640,000 college seniors who attend four-year colleges and are employed after they graduate.

  4. How many quarters would you need to reach the height of the Empire State building?

    The first question you should ask your interviewer is whether the quarters are standing up or lying flat (the most probable answer is laying flat, but worth giving the question a shot since I personally think the former is easier to calculate). I’ll do a quick calculation for both. Without looking up how tall the Empire State building is and the radius (or thickness) of a quarter, this question seems like a very difficult question to answer. But speaking from personal experience, I’ve been to the top of the Rockefeller Center which is 70 stories, so I can estimate that the Empire State building is 100 stories.

    Laying flat: There are about 10 quarters in an inch, so 120 quarters in a foot. Assuming a story is 10 feet, the number of quarters is 120*10*100 = 120,000. The actual thickness of a quarter is 0.069 inches and the empire state building is 1454 feet, the number of quarters (actually) is \frac{1454*12}{0.069} \approx 252,000.

    Standing up: The diameter of a quarter is about an inch so 12 quarters make a foot. Assuming a story is 10 feet, the number of quarters needed to reach of the height of the Empire State building is 100 * 10 * 12 = 12,000 quarters. Now, looking up the diameter of a quarter to be 0.955 inches and the empire state building to be 1,454 feet, so the number of quarters is \frac{1454*12}{0.955} \approx 18,000.

  5. If I give you a traditional two-sided scale along with nine, similarly-sized balls—eight of which are of equal weight, one of which weighs less than the rest—how many times do you need to weigh the balls to determine which is the lighter one?

    What is not mentioned in the question is that they probably want you to find the least number of weighings. Take out 6 balls and weigh them with 3 on each side. If the scale balances, then you know the unique ball is one of the three balls you haven’t weighed. If the scale is unbalanced, then the unique ball is on the side of the scale that is tipping up. Then, taking the pile of 3 balls and choosing two balls to weigh would give you the ball that weighs less than all the rest since if the two balls you chose are evenly balanced, then the ball you have not weighed is the correct one. If the scale is unbalanced in this weighing, then the ball is on the side that weighs less. This gives a total of 2 weighings.

  6. If you have seven blue socks and nine yellow socks in a drawer, how many socks do you have to pull out blindly in order to ensure that you have a matching pair?

    This is a classic problem that is meant to trick you with all the numbers thrown at you (7 and 9 in this case). If you know some discreet math, this might sound like a pigeonhole problem. The point is that you only have to take out a total of 3 socks. The first sock you take out will be one color, and the second sock you take out is either the same color or different. If it’s the same color, then you already have a pair. If not, the third sock must match with one of the two since you only have 2 different colors in the drawer.

  7. How much of your day do you smile? (This is was a real question asked by Amazon.)

    This question should be a no-brainer and absolutely no math involved. For me, 80%.

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2 Responses to How many windows are in the United States? (and other large estimation problems)

  1. David Zhang says:

    I like this post – I remember in high school, there was a Fermi event for Science Olympiad that did questions like these. In that, though, answers were simply powers of ten – so you might put “6” if you thought the answer was on the order of 10^6. Still, it was tricky.

    80%? Nice =D
    I feel like I smile a lot more internally then externally, but yes, *you* certainly smile a lot. =)

    • Quanquan Liu says:

      Fermi event, interesting! Unfortunately, I didn’t do enough Science Olympiad in high school to know a lot about what these competitions are like. But you are more than welcome to provide a guest post on high school olympiad problems and such! 😀

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