# Brain teasers in case interviews are used to assess your ability to think quickly and out-of-the-box

Brainteasers are a special sub category of cases. In contrast to regular cases, they are less focused on multifaceted business problems and more on a single issue. The goal is to solve a problem that is usually not straightforward and requires **out-of-the-box thinking**.

Typical problems cover every-day life's topics and might even include unrealistic assumptions. All necessary information is usually included in the question, so that further assumptions are not necessary. Questions to the interviewer should focus on presenting solution approaches and checking if they will lead to the solution.

## What should you expect during brainteasers?

First of all, in consulting case interviews, brainteasers are rarely used. Regular cases are considered as more suitable as they allow the interviewer to better test key skills needed in a consultant (structured, analytical and conceptual thinking).

During business cases, interviewers can add value through their expertise and a more fruitful and realistic interaction may arise, which is more in alignment to that of a consultant's daily life. Nevertheless, brainteasers provide the interviewer with valuable insights about the candidate's potential.

Brainteasers are an **additional challenge** to the candidate. They require a solution path different from standard frameworks. The candidate’s ability to recognize, structure and solve problems is tested at a different level. The necessary insights are usually hidden in the starting question and need to be deduced. For some brainteasers, an information overload can cover the essential hints. As a result, brainteasers put candidates under **stress**.

For a candidate, practicing brainteasers helps to focus on certain aspects. You can **train your creativity, math skills and logical thinking** with different questions. Because brainteasers require out-of-the-box thinking, you become more **flexible in using and developing frameworks** and solution paths. They also help to avoid excessive usage of standard frameworks.

## How to solve brainteasers?

As with all case studies, the structure leading towards the solution of a consulting brainteaser is more important than actually finding the final answer.

**Write down**every piece of information you get. Listen very carefully because one specific word can convey the answer.**Repeat**the question to make sure you didn’t add or miss anything important.- Think of solutions or required information you might need
**without looking at the facts**. There are three possibilities:

1. You’ll find a solution without any information. Sometimes common sense brainteasers can be solved this way. In this case, the only purpose of the starting information is to confuse you.

2. Both needed and given information match.

3. You need further information in order to be able to solve the brainteaser. In this case, convey to the interviewer how you would have solved it if that information were available. This will demonstrate your creativity.

- Take notes of
**solutions that didn’t work**and communicate the reason to the interviewer. Clearly communicate the reasons that would prevent you from repeating similar analysis - Make the problem visible through
**schematics and graphs**. These habits foster creative thinking - Put yourself in the
**shoes of the person affected**by the issue addressed by the consulting brainteaser. If the problem were your own, you might tackle it in a more creative way

## Quantitative brain teasers are about translating verbal problems into variables

**Solve with equations**

When you’re confronted with numbers, such as rates, percentages, amounts or a time, try to set up an equation. While brainteasers don’t require sophisticated math skills, they do require the knowledge of translating a verbal problem into variables.

**Example:** A man's youth lasted one sixth of his life. He grew a beard after one twelfth more. After one seventh more of his life, he married. 5 years later, he and his wife had a son. The son lived exactly half as long as his father. The man died four years after his son.

How many years did the man live?

**Solution:** The brainteaser requires you to formulate the facts into an equation. First, define the missing value. The question asks for the number of years he lived, so represent the age of death with the variable x. Then sum up the different periods of his life, which are either given as fraction of his lifespan or as an absolute number.

x/6 + x/12 + x/7 + 5 + x/2 + 4 = x

The solution of the equation is x = 84. The man lived for 84 years.

**Brute Force**

When the problem asks for a specific number, go for the trial and error approach. It is an efficient method if you know the possible number of solutions is restricted to a just few.

**Example: **Replace all 'x' with all of the digits 1, 2, 3, 4, 5 and 6 to make the statement true: xx * x = xxx

**Solution:** Instead of thinking about a complicated arithmetic solution, just plug in numbers until you reach the solution.

54 * 3 = 162

## Qualitative brain teasers are about drawing the right conclusions out of the given information

**Solve with Logic**

Most brainteasers can be solved logically by inferring the right conclusions out of the given information. While that is, to a certain extent, necessary for every brainteaser, some will require special logical thinking. Make sure to prove that you draw the right logical conclusions from the given information.

**Example: **Five friends bought 10 cookies. To distribute the cookies, they have agreed to a rule: according to the order of their initial letters in the alphabet, Anton, Betty, Carl, Danielle and Emily each make a suggestion on how to split the cookies. So Anton starts first and Emily is last. A person is kicked out (and consequently doesn't receive any cookies), if they don't secure at least 50% of the votes with their suggestion. In that case, the next one makes a suggestion until an agreement is reached. What is the maximum Anton can demand without being kicked out?

**Solution:** Backwards induction will provide you with the answer. It is difficult to start right away with Anton’s decision, as we don’t know what the others would decide. So we have to figure out the others' decisions first:

If the game continues long enough for Emily to make a suggestion, she will be the only one left. So she will give herself everything.

If Danielle decides, she and Emily are left. Whatever Danielle says will be the final decision, because she makes up 50% of the votes. So Danielle will take everything herself.

If Carl decides, he has to convince at least one of the other two girls left. If he gives 1 cookie to Emily and keeps 9, Emily will agree because if Emily doesn’t agree, Danielle will decide. The case above then occurs, where Emily gets nothing.

If Betty decides, she also has to convince one of the other three left to receive 50% of the votes. If she gives 1 cookie to Danielle and keeps 9, Danielle will agree. If Danielle doesn’t agree, Carl decides. The case above then occurs where Danielle gets nothing.

If Anton decides, he needs two votes. If he gives 1 cookie to Carl, 1 to Emily and keeps 8, both Carl and Emily will agree because they are better off this way. If they don’t agree, Betty will decide. The case above then occurs, where Carl and Emily receive nothing.

Therefore, Anton can demand at max 8 cookies.

**Solve with Brute Force**

Similar to mathematical questions, if you get a brainteaser with a restricted amount of possible solutions, try a backwards approach. First, assume one of the solutions is correct and see if works with the information you were given.

**Example**: I have a horse. Do you know what color it is? Allan said, "I guess it is not black". Brian said, "It is either brown or gray". Charlie said, "I know it is brown". I said, "At least one of you is right and at least one of you is wrong." What is the color of my horse if the color is one of the above?

**Solution:** The horse is brown, black or gray. So three options you have to check:

If the horse is brown, then everyone is right. This is not the answer. If the horse is black, then everyone is wrong. This isn't the answer either. Therefore, the horse is gray. To verify the answer, Allan was right, Brian was right, but Charlie was wrong.

## Takeaways:

- Brain teasers are not very common in consulting case interviews
- Your creative and out-of-the-box thinking is usually tested in brain teasers
- Brain teasers can usually be solved with brute force, logic or equations

I don't think brute force is feasible in your example with xx • x = xxx, where each x represents one of 1, 2, 3, 4, 5, or 6. Keep in mind that there are 6! = 720 possible solutions. Even if you deduce, that 1 cannot be in the 3rd position, 5! • 5 = 600 candidates remain.