# Interview Brain Teasers

Brain teasers you might encounter in a finance, consulting, or engineering interview

## Double the Amount You Need to Do

At the beginning of January, you set a goal to work every day, to reach a total of 5000 minutes of work by the end of the month. But to give yourself a better shot of achieving this, you decide to front-load it—at the beginning of each day, you figure out how much you’d need to work on average on each remaining day to achieve your goal, and then you work double the amount you need to do. For example, if you had 50 minutes left and 5 days left, you would need to work 10 minutes/day, so you would choose to work 20 minutes on that 5th-to-last day.

If you chose to work this way, how long does it take you to complete your goal of 5000 minutes?

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## Doubling Game

Three players gamble on a “doubling game.” In each round of the game, a single loser is determined, and this player has to double the money of the other two.

After three rounds of this game, each player has lost one round each, and each player now has \$24.

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The “boy or girl paradox” is a well-known brain teaser by famous puzzle-maker Martin Gardner. It’s popularly considered a “paradox” because (1) it has a highly unintuitive solution, and (2) its ambiguous wording meant either of two solutions could be valid solutions.

This is a rewording of that brain teaser to eliminate some ambiguity from that original question:

Out of all families with exactly two children, we randomly pick one family that has at least one boy. What is the probability that both children in this family are boys?

Assume only for the purposes of this puzzle that a child can only be a boy or a girl, and that either possibility is equally likely.

The two envelopes paradox is a famous brain teaser of sorts, and not a true paradox. The problem is generally posed like this:

You are given a choice between two identical envelopes. One envelope contains some amount of money, and the other contains twice that amount of money. There is no way to distinguish between the two. However, when you choose one of the envelopes, before opening it, you are given the option of switching to the other envelope. Should you switch?

Why is this sometimes called a paradox? Well, if you choose to switch, you have a 50% chance of doubling your money, and a 50% chance of halving your money. If the amount of money in the envelope you initially chose is M, this reasoning suggests the expected amount in the other envelope is (2M + 0.5M) / 2 = 1.25M. This is more than M, so you should always switch.

But that would suggest once you’ve switched, you’re in the same position you were before you switched, so you should switch again. What is the problem with this reasoning?

## 5 Quant Brain Teasers

Whether you’re preparing for an interview or just trying to keep your mind sharp, here are some quant brain teasers to test your skills.

Remember, if you get a brain teaser at an interview, you don’t necessarily need to get the correct final answer to do well, you just need to demonstrate your problem-solving skills, your ability to think and communicate, and your comfort with math, logic, and statistics. For more tips on tackling brain teaser interviews, check out our guide to brain teaser interview questions.

## 1. Friday the 13ths

Easy-medium difficulty

What is the minimum and maximum number of Friday the 13ths that can occur in a calendar year?

## Color of the Last Ball

There is a bag with 20 blue balls and 13 red balls. Randomly remove 2 balls from the bag:

• If they are the same color, replace them with a blue ball
• If they are different colors, replace them with a red ball

Repeat this process until there is just 1 ball remaining. What is the color of the last ball?

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## Prison Keys Strategy

A prison warden was feeling capricious and played a game with the prison keys:

1. Each prisoner is handed a key to another prisoner’s cell.
2. Each prisoner will know which other prisoner was initially given the key to their cell (but does not know whose key they were handed).
3. Each day, when all prisoners are out of their cells and no one is watching, each prisoner is allowed to place keys in another prisoner’s cell.
4. Each night, each prisoner can collect any keys placed in their cell.
5. The prisoners can summon the warden when they’re sure everyone has their own key – but if they are wrong, they’re immediately executed.
6. The prisoners can discuss a strategy beforehand but cannot communicate in any way after keys are handed out.

What is the fewest number of days it would take for the prisoners to be sure everyone has their key? What was the strategy to achieve this?

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## Sharing Game Theory Puzzle

In this sharing game theory puzzle, 3 friends take turns taking from a jar of 1000 candies to share. For example, the 1st friend could take 500 candies, then the 2nd friend could take 400, and the 3rd friend would take the remaining 100.

No one wants to be seen as greedy, but no one wants to end up with the fewest candies either. As such, their goals are (in order of preference):

1. Do not end up with the most candies, nor the fewest candies (a tie for most or fewest also fails this condition)
2. End up with as many candies as possible

All of them are logical, rational, know each other’s goals, but cannot communicate before or during sharing. How many candies should each friend end up with?

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