Brain teasers are meant to confuse and frustrate you. Often, what interviewers are trying to do is emulate a test of how you think, using the concept of Type I and II thinking from the book Thinking Fast and Slow. The general gist of Type I and II is that Type I is the intuitive thinking process and is quick, but may lead to mistakes, and Type II thinking is more thorough, uses logic and math foundations, and gives you more accurate answers, although they can take longer to formulate. In general, traders like to see people think in Type II, and to be able to do it quickly.
A way to prepare for these type of questions is to develop good problem solving skills. Another way is just to memorize enough and hope that you get asked one of them.
As with any problem solving skill, doing brain teasers is mostly valuable not so you can memorize the solution, but exercise your brain's ability to reason, think, experiment, and challenge itself with difficult problems. If you really think through problems and give an earnest effort, trying to unblock yourself constantly, you will develop much stronger problem solving skills than someone who just looks up the answer. There are just things that can't be taught when you attempt a problem earnestly.
This website has a great selection of brainteasers. Quant Questions also offers a large selection of puzzles.
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A rope is tied around the circumference of the Earth very tightly, so that it is tightly taut along the surface of the Earth. Now suppose that you add 1 feet of length to this rope, so that it is a bit looser. How high above the surface of the Earth will this new rope be?
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You have 1000 bottles of water, one of which is poisoned. You have a bunch of lab rats that you can use to test the water and try to figure out which bottle is poisoned. Unfortunately, since the poison takes 24 hours to activate, and you must figure out the results in 24 hrs, you must feed all your rats now, and then wait the next day to see the results. How many rats do you have to use to figure out which bottle of water is poisoned?
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You have a bag of 100 strings. You reach in the bag, grab two ends of strings, and tie them in a loop. You keep on doing this until there are no ends left. How many loops do you expect to end up with on average?
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There are 100 prisoners. There are 100 boxes with random numbers from 1-100 in them, one each. Each prisoner has a chance to go in the room and try to find their number. If they do so, they will make it out alive. Otherwise, they are killed on the spot. The prisoners are allowed to plan a strategy before they go in, but surviving prisoners will not be able to tell the other prisoners what they have seen. What is the best strategy for going about saving as many prisoners as possible, and how many prisoners can you save?
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Suppose we have some critter on the number line. It has two starting params, x, which is some integer that represents where the critter starts at time t = 0, and y, which is how far, and in what direction the critter moves each \delta t = 1. So for example, if a critter has x=4, y=-1, the critter will have positions 4, 3, 2, 1, ... The question is, can we devise an algorithm to find where this critter is, eventually? NOTE: you do not know what x and y are, so you don't know where the critter starts, and what direction / amount of movement on each turn. However, y is constant throughout this game, and in every turn, you can check any one square on this integer number line. You want to devise an algorithm that will always terminate.