After you have aced the Maths and Sequence quizzes, you will be invited for an interview or assessment centre. You will be asked brain teasers to assess your:

- Analytical skills
- Ability to problem solve
- Calm and measured approach
- Logical solution
- Review of new information quickly
- Clear communication of the answer

**The final answer is rarely as important as your approach to the brain teaser itself.** Some useful tips when answering brain teasers:

- Take a minute to gather your thoughts
- Ask counter questions to clarify if you are unsure
- State your assumptions

We have 88 brain teasers for you to practice before your trading interview. Each brain teaser has a **hint** if you get stuck and answer to check your thought process. A huge thank you to Nigel Caldwell, who has spent many hours answering each of these brain teasers for you!

Different types of brain teasers:

**Problem Simplification**

Reduced, reduce, reduce. If the original brain teaser seems too difficult, try a simplified version of the problem. Usually, you can start with a subset of the original population and work your way upwards. For example, the classic Tiger and Sheep problem:

*One hundred tigers and one sheep are on a magic island that only has grass. Tigers can live on grass, but they want to eat sheep. If a tiger bites a sheep then it will become a sheep itself. If 2 tigers attack a sheep, only the first tiger to bite converts into a sheep. Tigers do not mind being a sheep, but they have a risk of getting eaten by another tiger. All tigers are intelligent and want to survive. Will the sheep survive?*

This seems a little complicated. Let us **simplify** the subset of tigers:

1 tiger and 1 sheep – The tiger will eat the sheep as it will not have to worry about being eaten afterwards. Hence, the sheep will not survive.

2 tigers and 1 sheep – Each tiger knows that if they eat the sheep, they will turn into a sheep themselves and then the other tiger will eat them without the risk of another tiger eating them. Hence, the sheep survives.

3 tigers and 1 sheep – Each tiger knows that if they eat the sheep, then they will be in the same situation as above, 2 tigers and 1 sheep. In this situation, the sheep survives. Hence, all the 3 tigers will try to eat the original sheep and it does not survive.

Now you will start to notice a pattern that if there are an odd number of tigers, the sheep does not survive. If there is an even number, the sheep survives. Solved through simplification of the original population.

**Pigeon Hole Principle**

The pigeon hole principle states that if m items are put into n containers, with m > n then at least one container must contain more than one item. Simply put, if you have more “objects” than you have “holes,” at least one hole must have multiple objects in it. This somewhat obvious principle can be applied to a lot of brain teasers.

*A bag of marbles contains 4 which are red, 4 which are blue and 4 which are green. How many marbles must be chosen from the bag to guarantee that two are the same colour?*

Here we have 3 pigeon holes – the colours, red, blue and green. Therefore, we need to remove 4 individual marbles to be sure that we have at least two marbles of matching colour.

**Applying Symmetry**

*Two players are playing a game. The game board is a circular table. The players have access to an ample supply of equal-sized circular coins. The players alternate turns, with each turn adding a single coin to the table. The coins are not allowed to overlap. Once a coin is placed on the table, it is not allowed to be moved. The player who has no place to put their next coin loses. Develop a winning strategy for the player who starts.*

Here, you want to go first and place the first coin in the very centre of the table. Now your opponent will play a coin. You must then copy the placement of their coin, on the opposing side – using symmetry. Proceed to copy each placement of their coin on the opposing side of the table. It follows that they must be the first player to not lay a coin flat on the table as they are always now the “first “to act.

**Summation of Numbers**

Other brain teasers will require you to use maths, such as summing numbers. It is worth knowing the following formula: to sum the numbers from 1 to n. For example summing 1 to 6, . Easy.

*A clock fell off the wall and broke into three pieces that the sums of the numbers on each piece are equal. What are the numbers on each piece?*

We start by summing the numbers from 1 to 12. Using the above formula, .

We have three equal piles of numbers that sum to 78, therefore, each pile must = 26.

Now you should realise that you must pair up the bigger numbers with smaller numbers. Summing in order the largest with smallest, 12 + 1 = 13, and 11 + 2 = 13. It quickly follows that summing the biggest and smallest in order gives you two pairs of 13s and a piece that sums to 26.

You subsets:

{12, 1, 11, 2}

{10, 3, 9, 4}

{8, 5, 7, 6}

**Modular Arithmetic**

The modulo or mod is the remainder when dividing. For example, 8 mod 3 = 2 which means 2 is the remainder when you divide 8 by 3. Seemingly simply, it will help you to answer a lot of brain teasers.

*What day of the week it will be 100 days after Monday?*

We know there are 7 days in a week, 100 mod 7 = 2, as 100/7= 14 remainder 2

Adding two days to Monday answers Wednesday.

**Fermi Brain Teaser**

A Fermi question is asked to calculate a rough approximation to a question that is difficult to measure directly. With this sort of brain teaser, the interviewer rarely will care if your final answer is way off. They are looking that you can remain calm, communicate clearly and answer logically.

Remember:

- Take a minute to think
- State your assumptions
- Ask questions for clarity
- Break the problem down
- Make the maths
**easy**

*How many piano tuners are there in London?*

*“*Please may I have a moment to think”

Whilst I am thinking… Do you mean the city of London or Greater London? Greater London the interviewer responds.

*It is good to give a road map of how you will answer the question.* “Firstly, I will estimate the number of people in London that own a piano. Then calculate the time spent tuning these pianos every year and finally divide to calculate the number of piano tuners”

- “I will assume that there are 10 million people in Greater London.”
*Although this is over, this is close and working with a number like 10 is EASY.* - “I will also assume that on average there are 3 people per household.”
- “Out of my close connections which I take as a normal distribution, I would say 1 in 20 owns a piano in their house.”
*– It is good to back up with real-life evidence if possible.* - “This gives 10m / 3 = 3.3m households in total, with 3.3m/20 = 165,000 pianos. I will say 150,000 for simplicity.”
- “Now I am going to assume that the market dictates the number of piano tuners.”
- “I will assume a piano is tuned on average once a year and each tuning take roughly 2 hours, including travel time.”
- “If each piano tuner works 8 hours a day, 5 days a week for 50 weeks a year, that’s 2000 hours. Each piano takes 2 hours, meaning each piano tuner tunes 1000 pianos a year.”
- “Dividing that into the 150,000 pianos, gives 150 piano tuners in London.”

**Logical Reasoning**

These type of brain teasers often require you to work through the possible solutions step-by-step with little maths. You may find that using a trial and error approach will help you to answer the questions. Logically working through each scenario. It may be better to think through these in your head rather than using a trial and error approach aloud.

*You have two ropes coated in oil to help them burn. Each rope will take exactly 1 hour to burn all the way through. However, the ropes do not burn at constant rates—there are spots where they burn a little faster and spots where they burn a little slower, but it always takes 1 hour to finish the job.With a lighter to ignite the ropes, how can you measure exactly 45 minutes?*

First, label the ropes A and B.

If I set either rope A or B alight it will take 1 hour – not helpful

If I cut either rope A or B in half and set alight, however, they do not burn equally – not helpful

If I light rope A or B at both ends it will take 30 minutes – slightly more helpful

If I light rope A from both ends and rope B from one end at the same time, as rope A burns out in 30 minutes, I then light the other end of rope B, that will burn out in another 15 minutes, 45 minutes – bingo!

**2584 STUDENTS ENROLLED**