Brainteasers are like standardized tests: of little relevance to the actual subject but designed to compare all people equally. As for what these tricky interview questions entail, they typically encompass probability and/or statistics. Both hedge funds and private equity firms use brainteasers during the interview process. But hedge funds use them more often because these questions test mental math skills, which are perhaps more important for hedge fund jobs than PE positions.
Accuracy, of course, is key to answering brainteasers. But so is your thinking process. You should expect to explain your reasoning; a logical verbal breakout followed by a wrong answer is better than just a plain wrong answer. Don’t forget: Pay attention to your interviewer’s face for clues as to whether you're going in the right or wrong direction. And, since brainteasers are meant to be tough, it’s fine to take a minute to collect your thoughts and outline the steps toward calculating your answer.
That said, here are some example questions (along with their answers) to give you an idea of what to expect if you’re interviewing with a PE firm or hedge fund.
1. What is the probability of drawing two sevens in a card deck?
You can multiply the individual probabilities to get the cumulative probability. There are four 7s in a deck of 52 cards. Therefore, the probability of drawing the first 7 is 4/52 or 1/13. On the second draw, there are only three 7s in a deck of 51 cards, yielding a probability of 3/51 or 1/17. So 1/13 multiplied by 1/17 equals a cumulative probability of 1/221. (Don’t expect to be able to use paper or a calculator for 13 times 17. You can just simplify the math in your head by saying 17 times 10 is 170, plus 3 times 17 which is 51, and yields 221.)
2. You play a game of dice where you are paid the equivalent amount of dollars to the number you roll (i.e., if a 4 is rolled then you get $4). You roll one fair six-sided die. How much are you willing to pay for this roll?
The expected return is every possibility multiplied by the probability of the possibility. The average of all the potential die rolls, which each have equal probabilities, is $3.50, the midpoint between 1 and 6.
How much would you pay to play the same game, but with the option to roll again? If you only roll once you get that score; if you choose to roll again, you get the score of the second roll.
Intuitively, you know the price should be higher since you’re given the option to roll again if you’re dissatisfied with your first roll. You should only roll a second time if the first roll is less than 3.5, the expected value. Thus, you have the following six scenarios: rolling 4, 5, 6 and stopping, or rolling 1, 2, 3, and rolling again. Again, the expected roll is 3.5, so the latter three outcomes have expected returns of 3.5. Therefore, a game of two rolls’ expected return is (4 + 5 + 6 + 3.5 + 3.5 + 3.5)/6 = $4.25.
Again, same games, option for a third roll now. How much will you pay?
Follow the same logic as before; two rolls have an expected return of 4.25 so you will only roll a third time if you get above that. You have an expected return of (4.25 + 4.25 + 4.25 + 4.25 + 5 + 6)/6 = $4.67. As the number of rolls approaches infinity, the price you pay gets closer to $6.00.
3. You have stacks of quarters, dimes, nickels, and pennies. The number of coins in the stacks is irrelevant. You can take coins from a stack in any amount, any order, and place them in your hand. What is the greatest dollar value in coins you can have in your hands without being able to make change for a dollar with the coins in your hand?
Start adding the highest coin to your hand, the quarter. Four quarters make a dollar, so you can only have three quarters: $0.75. Five dimes would bring it to a dollar, so you can only have four dimes: $1.15 = 0.75 + .40. You can’t add a nickel because three quarters, two dimes, and the additional nickel would create a dollar. But you can add four pennies for a maximum total of $1.19 = 1.15 + .04.
4. A closet has three light bulbs inside. Next to the door (outside) are three switches for each light bulb. If you can only enter the closet one time, how do you determine which switch controls which light bulb?
Turn on two switches, A and B, and leave them on for a few minutes. Then turn off switch B and enter the room. The bulb that is lit is controlled by switch A. Touch the other two bulbs, which are off. The one that is still warm is controlled by switch B. The third bulb, off and cold, is controlled by switch C.
5. What is the square root of 7,000,000 (approximately)?
You know that 2 * 2 = 4 and that 3 * 3 = 9, and that 1,000 * 1,000 = 1,000,000 so the answer has to be between 2,000 and 3,000. Edge closer in, 2.5 * 2.5 = 6.25 and 2.7 * 2.7 = 7.29 so the answer is approximately 2,600.
6. You are given a length of rope, which can be lit to burn for an hour. However, the rope burns unevenly (meaning half of it burnt does not indicate a half-hour has passed). How would you burn the rope to know that a half-hour has passed?
To measure a half-hour, burn both ends at once. One side will burn faster than the other, but the opposite side will burn slower such that when they meet, the equivalent of half the time has passed.
If you were given two ropes, how would you measure 45 minutes?
For two ropes, take one rope and burn both ends like the previous situation. At the same time, light the second rope on only one end. When the first rope burns out, a half hour has passed. The second rope only has 30 more minutes on it. Immediately burn the opposite end of the second rope. The fire will meet at both ends again, which is fifteen minutes.
7. You’ve got a 10 x 10 x 10 cube made up of 1 x 1 x 1 smaller cubes. The outside of the larger cube is completely painted. On how many of the smaller cubes is there any paint?
First, note that the larger cube is made up of 1000 smaller cubes. Think about how many cubes are NOT painted. 8 x 8 x 8 inner cubes are not painted which equals 512 cubes. Therefore, 1,000 - 512 = 488 cubes that have some paint. Alternatively, you can calculate this by recognizing that two 10 x 10 sides are painted (200) plus two 10 x 8 sides (160) plus two 8 x 8 sides (128): 200 + 160 + 128 = 488.
The above was excerpted from the Vault Guide to Private Equity and Hedge Fund Interviews.
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