## Cracking the Code: Mastering Investment Banking Brain Teaser Interview Questions

Are you looking to land your dream job in investment banking? If so, you'll likely have to navigate the challenging world of brain teaser interview questions. These questions are designed to assess problem-solving skills and critical thinking abilities, and they can be a stumbling block for many candidates. But fear not! In this article, we will share the secrets to cracking the code of investment banking brain teaser interview questions. Brand Voice: Professional and Authoritative Introduction: If you're aspiring to secure a coveted position in investment banking, you'll undoubtedly face brain teaser interview questions that test both your analytical and critical thinking abilities. These questions are carefully crafted to gauge your problem-solving skills, and mastering them can significantly boost your chances of success in the interview. To unravel the enigma of these challenging questions, read on as we delve into the art of cracking the code when it comes to investment banking brain teaser interview questions. Whether you're a seasoned professional or a fresh graduate, these insights will equip you with the knowledge and confidence you need to ace your investment banking interview. So, let's get started and unravel the secrets that will elevate your interview performance to new heights.

In investment banking interviews, candidates often encounter brain teaser questions that test their math, creativity, and critical thinking skills. These questions are designed to assess problem-solving abilities, logical reasoning, and the ability to think under pressure. Here's an overview of what to expect and how to approach these brain teaser interview questions.

### Types of Brain Teaser Questions

Investment banking firms occasionally ask brainteaser questions, which fall into three main categories: math-based, real-world deals, and industry-related questions. Math-based questions often involve probability calculations, mental math, and logical reasoning. Real-world deal questions assess a candidate's understanding of the industry and the important considerations when evaluating a deal. Industry questions require candidates to demonstrate their knowledge of the investment banking landscape and firm-specific information.

### Preparing for Brain Teaser Questions

While brain teasers are more common in sales & trading interviews or consulting interviews, they may still appear in investment banking interviews. It's important to be prepared for these questions by practicing math-based problem-solving, logical reasoning, and creative thinking. Candidates can benefit from familiarizing themselves with common types of brain teasers and developing strategies for approaching these questions effectively.

### Approaching Brain Teaser Questions

When faced with a brain teaser question, it's crucial to take the time to understand the question thoroughly. Candidates should avoid rushing to answer and instead approach the question calmly, considering it as an important problem to solve. Asking the interviewer clarifying questions to better understand the problem can demonstrate a willingness to put in effort to solve problems and can provide valuable information for formulating an answer.

### Tips for Handling Brain Teaser Questions

- Candidates should consider bringing a pencil and paper to the interview, as many finance interview brainteasers are math-based and may require calculations.

- It's advisable not to answer questions directly, especially if the candidate has encountered the question before. Instead, taking a moment to gather thoughts and asking for time to think can reflect positively on the candidate's application.

In conclusion, understanding investment banking brain teaser interview questions involves being prepared for math-based, real-world deal, and industry-related questions. By practicing problem-solving skills, logical reasoning, and creative thinking, candidates can approach these questions with confidence and demonstrate their ability to think critically under pressure.

### Here are some additional brain teaser interview questions for investment banking interviews, ranging from easy to hard:

**Q1. You have a deck of 52 cards. If you shuffle the deck and deal cards one at a time, what is the probability that the first king you deal will be the king of spades?**

**Suggested Answer: Probability of the First King Being the King of Spades**

To calculate the probability of the first king being the king of spades when dealing cards from a shuffled deck, we can consider the following:

**Total Number of Cards**: There are 52 cards in a standard deck.**Number of Kings**: There are 4 kings in a deck, one for each suit.**Probability**: We want to find the probability of dealing the king of spades first.

The probability can be calculated as follows: **Probability = (Number of favorable outcomes) / (Total number of outcomes)** In this case, the number of favorable outcomes is 1 (since we want to deal the king of spades first), and the total number of outcomes is 52 (since there are 52 cards in the deck). Therefore, the probability can be calculated as: **Probability = 1 / 52** So, the probability that the first king dealt will be the king of spades is **1/52**.

**Q2. There are five apples in a basket. If you eat one apple every day, how many days will it take to eat all the apples? (This one is a trick question!)**

#### Suggested Answer: Eating Apples from the Basket

This question seems straightforward at first, but there's a twist to it!

If there are five apples in a basket and you eat one apple every day, it would take **4 days** to eat all the apples.

However, the twist is that on the fifth day, you would have already eaten the fourth apple, leaving no apples left to eat on the fifth day. So, it takes **4 days** to eat all the apples, not 5!

**Q3. A train is 100 meters long and traveling at 100 kilometers per hour. How many seconds does it take for the entire train to pass a tunnel that is 1 kilometer long?**

#### Suggested Answer: Time for the Train to Pass the Tunnel

To calculate the time it takes for the entire train to pass a 1-kilometer-long tunnel while traveling at 100 kilometers per hour, we can follow these steps:

**Convert Speed to Meters per Second**: First, we need to convert the speed from kilometers per hour to meters per second. This can be done by multiplying the speed in kilometers per hour by (5/18) to convert it to meters per second.Speed in meters per second = (100 km/hr) * (5/18) = (500/18) m/s**Calculate Relative Speed**: The train, when passing through the tunnel, will have a relative speed equal to the speed of the train plus the speed of the tunnel. Since both the train and the tunnel are moving in the same direction, we can add their speeds together.Relative speed = Speed of train + Speed of tunnel = (500/18) + 0 = (500/18) m/s**Calculate Time**: The time it takes for the train to pass the tunnel can be calculated using the formula:Time = Distance / SpeedSince the train and the tunnel are moving in the same direction, the distance the train needs to cover is the length of the train plus the length of the tunnel.Time = (1000 + 100) / (500/18) = 1100 / (500/18) = (1100 * 18) / 500 = 19800 / 500 = 39.6 seconds

So, it will take approximately **39.6 seconds** for the entire train to pass through the 1-kilometer-long tunnel.

**Q4. You have 12 red balls and 13 blue balls in a bag. You blindly pick balls one at a time, without replacement. What is the minimum number of balls you need to pick to guarantee at least 3 balls of the same color?**

**Suggested Answer: Guaranteeing at Least 3 Balls of the Same Color**

To guarantee that at least 3 balls of the same color are picked from a bag containing 12 red balls and 13 blue balls, we can consider the worst-case scenario for each color. The worst-case scenario for picking balls of the same color is when you pick all the balls of one color before reaching the minimum of 3 balls of that color.

In this case, the minimum number of balls required to guarantee at least 3 balls of the same color can be calculated as follows:

**Picking All Balls of One Color**: In the worst-case scenario, you could pick all 12 red balls or all 13 blue balls without getting 3 balls of the same color.**Next Ball Guarantees 3 of the Same Color**: The next ball you pick after picking all the balls of one color will guarantee that you have at least 3 balls of that color.

Therefore, the minimum number of balls you need to pick to guarantee at least 3 balls of the same color is **14**.

**Q5. If a clock strikes once at 1:00, twice at 2:00, and so on, how many times will it strike throughout the day (12:00 AM to 11:59 PM)?**

Suggested Answer: Throughout the day, a clock will strike the same number of times as the total number of hours that have passed. This is because the clock strikes once at 1:00, twice at 2:00, and so on. So, from 1:00 AM to 12:00 PM, the clock will strike a total of 12 times, and from 1:00 PM to 12:00 AM, the clock will strike another 12 times. Therefore, throughout the entire day (12:00 AM to 11:59 PM), the clock will strike a total of **24 times**.

**Q6. You are on a floor that is 100 stories high. If you drop a glass ball from the top floor, it shatters. However, you have two eggs. You need to figure out the minimum number of drops it would take to determine from which floor you can safely drop the glass ball without it breaking.**

**Suggested Answer: **This is a classic problem involving minimizing the number of drops needed to determine the safe floor from which to drop the glass ball. The solution involves a strategy that minimizes the worst-case scenario.

To solve this problem, we can use a technique known as dynamic programming.

By considering the possible outcomes at each floor, we can determine the minimum number of drops needed. The optimal strategy involves dropping the first egg at intervals, and then using the second egg to narrow down the range of floors where the glass ball will not break.

The goal is to minimize the maximum number of drops required in the worst-case scenario. The minimum number of drops required to determine the safe floor with the given constraints is a well-known puzzle in the field of computational mathematics and is often referred to as the "Two Egg Problem" or the "Egg Dropping Puzzle." The minimum number of drops required to determine the safe floor with 2 eggs and a building of 100 stories is **14 drops** using the optimal strategy.

**Q7. You are given a list of 10 numbers, and you know that the sum of all the numbers is 100. However, one number is incorrect. You are also given the incorrect sum of the first 5 numbers. Can you identify the incorrect number?**

**Suggested Answer: **To identify the incorrect number in the list of 10 numbers, we can use the given information to determine the correct sum of the first 5 numbers and then compare it to the incorrect sum provided. The difference between the two sums will point to the incorrect number.

Let's denote the correct sum of the first 5 numbers as S1 and the incorrect sum of the first 5 numbers as S2. We know that the correct sum of all 10 numbers is 100. We can calculate the correct sum of the first 5 numbers by subtracting the sum of the last 5 numbers from the total sum of all 10 numbers:

**Correct Sum of First 5 Numbers (S1) = Total Sum - Sum of Last 5 Numbers**

Now, the incorrect number is the difference between S1 and S2.

To summarize, we can identify the incorrect number by calculating the difference between the correct and incorrect sums of the first 5 numbers.

**Q8. You are on an island with two ropes, each 1 meter long. You need to measure a precise length of exactly 3 meters. How can you do this by only tying or cutting the ropes?**

**Suggested Answer: **To measure a precise length of 3 meters using only two 1-meter ropes and without any other measuring tools, you can use the following steps:

**Step 1: Tie the Ropes Together**

Tie the two 1-meter ropes together to create a single rope that is 2 meters long.

**Step 2: Create a Loop**

Hold one end of the 2-meter rope in your hand and create a loop by folding the rope back onto itself. This loop will effectively create a 1-meter section within the 2-meter rope.

**Step 3: Measure the 3-Meter Length**

With the loop in one hand and the loose ends in the other, you can now measure out the 3-meter length. By pulling the loop and the loose ends, you effectively create a 3-meter measurement (1 meter from the loop, 1 meter from the loose end, and 1 meter from the rest of the rope).

By following these steps, you can use the two 1-meter ropes to measure a precise length of exactly 3 meters.

**Q9. If a sheet of paper can be folded in half 10 times, how thick will the folded sheet be? (Assume the paper is very thin initially)**

**Suggested Answer: **If a sheet of paper can be folded in half 10 times, we can calculate the thickness of the folded sheet using a simple exponential growth formula.

When a piece of paper is folded in half, the thickness doubles. Therefore, if we fold the paper in half 10 times, the thickness will be 2 raised to the power of 10, which can be calculated as:

**Thickness = 2^10**

Calculating 2^10 results in a thickness of **1024 times the original thickness**.

This demonstrates the astonishing exponential increase in thickness as the paper is folded multiple times.

**Q10. Imagine you have a cylindrical can filled with water. You have two measuring cups, one holding 5 liters and another holding 3 liters. How can you accurately measure exactly 4 liters of water using only these two cups?**

**Suggested Answer: **To accurately measure exactly 4 liters of water using only a 5-liter measuring cup and a 3-liter measuring cup, you can follow these steps:

**Step 1: Fill the 3-Liter Cup**

Fill the 3-liter measuring cup to its full capacity with water.

**Step 2: Pour Water into the 5-Liter Cup**

Pour the water from the 3-liter measuring cup into the 5-liter measuring cup.

**Step 3: Refill the 3-Liter Cup**

Fill the 3-liter measuring cup again with water.

**Step 4: Add Water to the 5-Liter Cup**

Pour the water from the 3-liter measuring cup into the 5-liter measuring cup until the 5-liter cup is full. This will leave exactly 1 liter of space remaining in the 5-liter cup.

By following these steps, you will have precisely 4 liters of water in the 5-liter measuring cup, as you poured 3 liters into it and there was 1 liter of space remaining.

**Q11. A company has 100 employees, and each employee knows exactly 30 other employees. However, no two employees know exactly the same set of 30 people. Can you prove that there must be at least one group of three employees who all know each other?**

**Suggested Answer: **To prove that there must be at least one group of three employees who all know each other, we can apply the pigeonhole principle, which is a fundamental concept in combinatorics and discrete mathematics.

The pigeonhole principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item.

In the context of the given scenario, we can apply the pigeonhole principle as follows:

Each employee knows exactly 30 other employees. If we consider the employees as "containers" and the employees they know as "items," then we have 100 "containers" (employees) and 30 "items" (other employees) for each "container."

Since each employee knows exactly 30 other employees, we can think of this as placing 30 "items" into each of the 100 "containers." By the pigeonhole principle, if we have more items than containers, at least one container must contain more than one item.

Therefore, there must be at least one group of three employees who all know each other, as the pigeonhole principle guarantees that there are at least three employees who share the same set of 30 people they know.

This application of the pigeonhole principle demonstrates that in a scenario where each of 100 employees knows exactly 30 other employees, there must be at least one group of three employees who all know each other.

**Q12. You are at a party with 100 people. Everyone shakes everyone else's hand once (except for themselves). How many total handshakes occur? (Hint: This can be solved without extensive calculation)**

**Suggested Answer:**

To determine the total number of handshakes that occur at the party with 100 people, we can use a simple formula based on combinations.

When two people shake hands, it can be viewed as a combination of 2 out of the total number of people. Since each pair of people only needs to shake hands once, we need to count the number of unique combinations of 2 people from the total group.

The formula to calculate the total number of handshakes is given by:

**Total Handshakes = (n *** (n - 1)) / 2** *

*Where n represents the total number of people at the party. Using this formula, we can calculate the total number of handshakes for 100 people: *

*Total Handshakes = (100 *** (100 - 1)) / 2 = 4950**

Therefore, a total of 4950 handshakes occur at the party with 100 people.

**Q13. You are given a scrambled chessboard where all the pieces are randomly placed on the squares (except for the two kings, which are not on the board). Is it always possible to arrange the remaining pieces such that the kings can be placed on any two opposing squares and still be unable to capture each other?**

**Suggested Answer: **Yes, it is always possible to arrange the remaining pieces on a scrambled chessboard such that the kings can be placed on any two opposing squares and still be unable to capture each other.

This is due to a property known as "undermining," where certain pieces can be positioned in a way that prevents the kings from capturing each other, regardless of their initial placement.

By strategically positioning the remaining pieces, such as bishops and knights, it is possible to create a scenario where the kings are unable to capture each other, regardless of their starting positions. This can be achieved through careful arrangement of the pieces to create a barrier that prevents direct capture.

Therefore, regardless of how the pieces are initially scrambled on the chessboard, it is always possible to arrange the remaining pieces in a way that allows the kings to be placed on any two opposing squares and still be unable to capture each other.

**Q14. Imagine a company that sells two products, A and B. Product A has a profit margin of 20%, and product B has a profit margin of 30%. The company needs to sell a total of 100 units to break even. What is the minimum number of units of product A that the company needs to sell to achieve this, assuming they sell some units of product B as well?**

**Suggested Answer:**

To find the minimum number of units of product A that the company needs to sell to achieve the break-even point of 100 units, we can set up and solve a simple equation based on the profit margins of the two products.

Let's denote the number of units of product A sold as "x" and the number of units of product B sold as "y". Given that the company needs to sell a total of 100 units to break even, we have the equation:

**x + y = 100**

Additionally, we can calculate the total profit from the sales of products A and B using their respective profit margins. The total profit should be zero at the break-even point, so we have the equation:

**0.20x + 0.30y = 0**

Now, to find the minimum number of units of product A that the company needs to sell, we can solve these equations simultaneously.

By rearranging the first equation to solve for y, we get:

**y = 100 - x**

Substituting this value of y into the second equation, we have:

**0.20x + 0.30(100 - x) = 0**

**Solving for x, we find: **

**0.20x + 30 - 0.30x = 0**

**-0.10x + 30 = 0**

**-0.10x = -30**

**x = 300**

Therefore, the company needs to sell a minimum of 300 units of product A to achieve the break-even point, assuming they sell some units of product B as well.

**Q15. A company is considering investing in a new project. The project has a 50% chance of success, resulting in a profit of $10 million. However, there is a 50% chance of failure, resulting in a loss of $5 million. What is the expected value of this project, and should the company invest based solely on this metric?**

**Suggested Answer:**

The expected value of a project is calculated by multiplying the probability of each possible outcome by the value of that outcome and then summing these values. In this scenario, the project has a 50% chance of success and a 50% chance of failure, resulting in different financial outcomes.

**The expected value (EV) of the project can be calculated as follows:**

**EV = (Probability of Success) *** (Profit from Success) + (Probability of Failure) *** (Loss from Failure)**

**Substituting the given values:**

** **

**EV = (0.50 *** $10 million) + (0.50 *** -$5 million)**

**EV = $5 million - $2.5 million**

**EV = $2.5 million**

**The expected value of the project is $2.5 million.**

However, while the expected value provides a useful metric for decision-making, it does not capture the full risk associated with the project. The company's decision to invest should not be based solely on the expected value, as it does not consider the company's risk tolerance or the potential consequences of a large loss.

**Q16. You are given a list of 1 million numbers between 1 and 1 million. You know that one number appears twice and all the others appear only once. How can you find the duplicate number in at most log(n) time (where n is the number of elements)?**

**Suggested Answer:**

To find the duplicate number in a list of 1 million numbers in at most log(n) time, you can use a binary search approach. The binary search algorithm allows for efficient searching in a sorted list, and by applying it to this scenario, you can find the duplicate number in logarithmic time complexity. Here's a step-by-step approach to finding the duplicate number using binary search:

Start by sorting the list of 1 million numbers. Sorting the list can be done with a time complexity of O(n log n).

Once the list is sorted, you can perform a binary search to find the duplicate number. The search process involves dividing the sorted list into smaller sublists and comparing the middle element of each sublist with the expected value based on the position.

If the count of numbers smaller than or equal to the middle element is greater than the element itself, then the duplicate number must be in the lower half of the list; otherwise, it's in the upper half.

Continue the binary search process, narrowing down the search range until the duplicate number is found.

By using the binary search algorithm on the sorted list, the duplicate number can be found in at most log(n) time, providing an efficient and effective solution to the problem.

**Q17. Imagine a square cake cut into a 4x4 grid. You can make cuts only along the grid lines (horizontal or vertical). What is the minimum number of cuts needed to divide the cake into 8 pieces?**

**Suggested Answer:**

To divide the square cake into 8 pieces using only cuts along the grid lines (horizontal or vertical), we can use a strategic approach to minimize the number of cuts required. Here's how we can achieve this:

**First Cut:**

Make 3 equally spaced parallel vertical cuts. This divides the cake into 4 strips.

**Second Cut:**

Make a single horizontal cut that is equally spaced between the top and bottom of the cake. This divides each strip into 2 pieces, resulting in a total of 8 pieces.

By following this approach, the minimum number of cuts needed to divide the square cake into 8 pieces is **4 cuts**.

**Q18. A company is considering acquiring another company. The target company's stock is currently trading at $10 per share. The acquiring company believes that the target company's stock will be worth $15 per share in one year. However, there is a 20% chance that the deal will fall through due to regulatory hurdles. Should the acquiring company proceed with the deal, assuming a cost of capital of 10%? Why or why why not?**

**Suggested Answer: **

To determine whether the acquiring company should proceed with the deal to acquire the target company, we can evaluate the expected value of the acquisition, taking into account the probability of the deal falling through and the cost of capital.

**The expected value (EV) of the acquisition can be calculated as follows:**

**EV = (Probability of Success) *** (Value of Success) - (Probability of Failure) *** (Cost of Capital)**

**Substituting the given values:**

**EV = (0.80 *** $15 million) - (0.20 *** $10 million) - (0.10 * $10 million)**

**EV = $12 million - $2 million - $1 million**

**EV = $9 million**

**The expected value of the acquisition is $9 million.**

Based on the expected value calculation, the acquiring company should proceed with the deal, as the expected value of the acquisition is positive. This means that, on average, the potential gain from the successful acquisition outweighs the potential loss due to the deal falling through and the cost of capital.

However, it's important to note that the decision to proceed with the acquisition should also consider other factors such as strategic fit, synergies, and potential risks beyond the financial aspects. Therefore, while the expected value analysis supports proceeding with the deal, a comprehensive evaluation of the acquisition should take into account a range of strategic and financial considerations.

**Q19. Estimate the total market size for the shoe-shining industry in the United States. Explain your thought process and the assumptions you make.**

**Suggested Answer:**

**Estimating the Market Size for the Shoe-Shining Industry in the United States**

To estimate the market size for the shoe-shining industry in the United States, we can make a few assumptions and use available data.

**Assumption of Shoe-Shining Frequency:**Let's assume that on average, a person shines their shoes once a month. This is a rough estimate and can vary based on the individual's profession, lifestyle, and personal preferences.**Assumption of Shoe-Shining Cost:**The cost of a shoe shine can vary, but let's assume an average cost of $10 per shine. This is a ballpark figure and can vary based on location and the type of service provided.**Population of Shoe-Wearing Individuals:**According to the U.S. Census Bureau, the population of the United States is approximately 330 million. Let's assume that 75% of the population wears shoes that might require shining. This gives us a total of 247.5 million individuals.

Using these assumptions, we can estimate the total market size for the shoe-shining industry in the United States as follows:

**Total Market Size = Shoe-Shining Frequency *** Shoe-Shining Cost *** Population of Shoe-Wearing Individuals**

Substituting the values: **Total Market Size = 12 times/year *** $10/shine *** 247.5 million individuals Total Market Size = $29.7 billion/year**

This is a rough estimate and the actual market size could be different based on the actual frequency of shoe shines, the cost of the service, and the percentage of the population that uses shoe-shining services. It's also important to note that this estimate does not take into account the potential impact of factors such as economic conditions, changes in fashion trends, and shifts in consumer behavior.

**Q20. Imagine a company that has a monopoly on the production of widgets. The company sells 1 million widgets per year at a price of $10 per widget. The company's cost of goods sold is $5 per widget, and all other operating expenses are negligible. The company has no debt and $10 million in cash on its balance sheet. If the company is expected to grow at a rate of 5% per year for the foreseeable future, and the required rate of return for this type of investment is 10%, what is the company's enterprise value?**

**Suggested Answer:**

To calculate the enterprise value (EV) of the company, we can use the Gordon Growth Model, which is a version of the Dividend Discount Model (DDM) that assumes dividends grow at a constant rate. In this case, we can treat the company's net income as its "dividends" since it has no debt and negligible operating expenses.

**The formula for the Gordon Growth Model is:**

**EV = D / (r - g)**

where:

**D is the annual "dividend" (or net income in this case)**

**r is the required rate of return**

**g is the growth rate**

**First, let's calculate the company's net income:**

The company sells 1 million widgets per year at a price of $10 per widget, so its annual revenue is $10 million. The cost of goods sold is $5 per widget, so its total cost is $5 million. Therefore, the net income (or "dividend") is $10 million - $5 million = $5 million.

**Substituting these values into the Gordon Growth Model formula:**

**EV = $5 million / (0.10 - 0.05)**

**EV = $5 million / 0.05**

**EV = $100 million**

This is the enterprise value of the company based on its current earnings, growth rate, and the required rate of return. However, this calculation assumes that the company's earnings will grow at a constant rate indefinitely, which may not be realistic. It also doesn't take into account the company's cash balance.

**To get the total firm value, we add the cash balance to the enterprise value:**

**Firm Value = EV + Cash**

**Firm Value = $100 million + $10 million**

**Firm Value = $110 million**

So, the total value of the firm, including its cash balance, is $110 million. This is a simplified calculation and actual valuation would require a more detailed analysis and potentially consider other factors.

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