Brain Teasers

Few people enjoy brain teasers, especially in a high-pressured interview setting. The good news is that brain teasers are not common in the consulting recruiting process. The bad news, however, is that they are still fair game.

Just like case interviews, it’s important to showcase logic, thoughtfulness, and structure when working through a brain teaser during an interview. What most people don’t know is that brain teasers can be practiced just like math. The more you familiarize yourself with brain teasers, the easier they become due to pattern recognition. With that said, let’s go into some of the common types of brain teasers that you might come across in your consulting interviews.

Pro tip: try to solve these brain teasers before you read the answers! Build a structure, talk it through out-loud, then calculate an answer before reading to our result.

Market Sizing Brain Teasers

Question Example: How many neckties are sold in the United States every year?

Advice: Market sizing questions are the most common type of brain teasers in consulting. Take solace in the fact that your interviewer usually does not know the right answer. In fact, the interviewer doesn’t really care about you getting the right answer!

Instead, focus on structuring your assumptions in an organized manner. Think of whether you should take a top-down or bottom-up approach. The former means starting from a large number – like total population – and making assumptions to reduce irrelevant groups of data and arrive at your answer. The latter means starting with a smaller figure like the number of neckties bought by one person each year and multiplying “upwards” to arrive to your larger answer.

Additionally, you’re going to make a lot of assumptions based on facts you know or your intuition. Most times, you won’t know if your assumptions are on target, but as long as you base them on reasonable logic, your interviewer will give you a pass. Remember – they don’t know the right answers either, so as long as you have structure and have a reasonable answer, you will be in good shape.

Sample Answer: Let’s take a top-down approach. Assuming the U.S. population is 320 million and the average life expectancy is 80 years, there are an average of 4 million people in each age year (i.e. 4 million people aged 1, 4 million people aged 2, and etc.)

Let’s assume for simplification purposes that women aren’t buying neckties. This leaves 160 million males. Next, break down these 160 million males into 4 age groups: 0-20, 20-40, 40-60, and 60-80. There are 40 million people in each of these age brackets (based on our previous assumption). Let’s assume the following:

Ages 0-20: 0.1 neckties a year; males don’t require neckties until they are employed
Ages 20-40: 1 necktie a year; some males may buy more per year but stop once they have a collection
Ages 40-60: 0.5 neckties a year; by this time, most males have many neckties and are still working but don’t need many more
Ages 60-80: 0.25 neckties a year; same reason as above but necktie collections are even larger, reducing the need for more

Multiplying our assumptions together, we get:

0.1 x 40mm + 1 x 40mm + 0.5 x 40mm + 0.25 x 40mm = 4mm + 40mm + 20mm + 10mm = 74mm

74 million neckties are sold in the United States every year.

Math Brain Teasers

Question Example: An owner of a zoo has tigers and cages. If he puts one tiger into each cage, the owner has one tiger too many. If he puts in two tigers per cage, the owner has one cage too many. How many tigers and cages does the owner have?

Advice: Use variables and algebra to organize your thoughts.

T = number of tigers
C = number of cages

T – C = 1 à C = T – 1
C – (T ÷ 2) = 1

Using the substation method:
(T – 1) – (T ÷ 2) = 1
2T – 2 – T = 2

T = 4
C = 4 – 1 = 3

Answer: The owner has four tigers and three cages.

Logic Brain Teasers

Question Example: You have a 3 gallon jug, a 5 gallon jug, and an unlimited supply of water. How do you measure out exactly 4 gallons?

Advice: You know that you need to use the two jugs together in order to get to the 4 gallons. If you have a pen and paper out in your interview, draw out the jugs in order to visualize the problem at hand. Use trial and error and play around with the different scenarios until you start getting close to an answer.

Answer: First, fill the 3 gallon jug with water and dump it into the 5 gallon jug. Then, fill the 3 gallon jug with water again and carefully fill up the 5 gallon jug until it’s filled to the top. Since you’ve just added 2 gallons into the 5 gallon jug (which previously had 3 gallons of water inside), you’re left with 1 gallon of water in the 3 gallon jug.

Dump the water completely out of the 5 gallon jug and pour the 1 gallon of water in the 3 gallon jug into the 5 gallon jug. Lastly, fill in the 3 gallon jug with water and dump it into the 5 gallon jug. Now you have 4 gallons!

Out-of-the-Box Brain Teasers

Question Example: What is unique about the following sequence of numbers?

{8, 11, 5, 4, 9, 1, 7, 6, 10, 3, 12, 2, 0}

Advice: At first glance, this brain teaser seems like a math problem. However, many brain teasers will often have answers that are completely counter intuitive. Instinctually, you may try to find the differences from one number to the next or see if there are any relationships between the odd and even numbers.

When you run into issues that are leading you nowhere, remember that you are solving an interview brain teaser. Open up your mind and think outside the box. In this case, the question actually wants you to think of letters, not numbers.

Answer: The numbers are in alphabetical order. (Eight, eleven, five, four, nine, one, seven, six, ten, three, twelve, two, zero)

Concluding Thoughts

Brain teasers are often tricky because the answers require both structured and creative thinking. When presented with one during an interview, try to keep an open mind to all possibilities and use your intuition and logic to guide you towards patterns from previous brain teasers you’ve seen before. If you do that, no brain teaser should get in the way between you and a successful interview!

More Content:

• Mental Math for Consulting
• Fast Math: Quant Skills
• Case Interview: Complete Prep Guide