Categories

## The Monty Hall Problem

You are on a game show in which there are three identical doors, one with a car behind it and two with goats behind them. You must pick one door, and you win if that door has the car behind it.

After you pick a door, the host of the game show always opens a door you didn’t choose that has a goat behind it. This leaves the door you chose and one other remaining door, and you are given the option to switch your choice to the other remaining door.

Should you switch or should you stick to your original choice? What chance of winning would that give you?

## The History

The Monty Hall Problem is a classic probability puzzle, named for its similarity to the game show “Let’s Make a Deal”, which was hosted by Monty Hall. The problem was made famous when Marilyn vos Savant answered it correctly in her column in a popular magazine, and thousands of readers wrote letters to the magazine arguing her solution was wrong!

The solution can be counter-intuitive, so give it some thought and then scroll down to see the Monty Hall Problem explained.

Categories

## What are brain teasers and why do interviewers ask them?

### What Are Brain Teasers

Brain teasers can be thought of broadly as types of puzzles that test problem-solving and critical thinking, and potentially other related skills such as logic, math, and creativity.

### Why Do Interviewers Ask Brain Teasers

An interviewer may ask a brain teaser to see how you would approach a problem or challenge, and to assess your critical thinking skills and how you think under pressure. Some brain teasers also test your ability to be flexible, creative, and adaptable.

### What Kinds of Interviews Might Have Brain Teasers

You will most commonly see brain teaser interview questions in these industries and roles:

• Quantitative finance – including institutional and prop trading, hedge funds, quant trading and modeling
• Consulting – including management and strategy consulting
• Engineering interviews – including software engineering and data science & analytics

Back in the day, Google and Microsoft were notorious for asking brain teasers, and every trader interview featured a brain teaser.

In recent years, brain teasers have largely fallen out of favor with interviewers, partly because they typically do a poor job of assessing on-the-job performance. But you may still encounter them for certain roles and firms, so read on and learn how to prepare!

## How to prepare for brain teaser interview questions

### Types of brain teasers

Familiarize yourself with the types of brain teasers you are likely to encounter in an interview.

#### Logic puzzles

As the name suggests, logic puzzles test logical thinking and deduction, and usually feature multiple logical steps in which you use clues to rule out certain possibilities to arrive at a clever and tidy solution.

Often encountered in: finance interviews, some consulting interviews, and software engineering interviews.

Example: There are three boxes: one with two red widgets, one with two blue widgets, and one with one red and one blue widget. All three boxes are labeled incorrectly. What is the fewest number of widgets you have to take out in order to correctly re-label all the boxes? See the solution here

#### Math puzzles

There are many types of math puzzles, but interview brain teasers most commonly test probability because it can be relevant to a lot of technical jobs.

Often encountered in: technical finance interviews, software engineering, and data analytics interviews.

Example: In a best of 3 tennis match, the player that first wins 2 sets wins the match. For a 3-set tennis match, would you bet on it finishing in 2 or 3 sets? See the solution here

#### Estimation problems

Estimation problems test your ability to estimate some unknown value using limited information – typically solved by using some common knowledge and piecing together reasonable assumptions to arrive at a good estimate. These don’t usually have a single correct solution, the interviewer just wants to see if you can figure out what information you can bring in and apply to make a difficult problem a more tractable one.

Often encountered in: strategy/management consulting interviews (which often frame them in terms of “market-sizing questions”).

Example: How many people in Germany have iPhones?

While there is technically a correct answer, there isn’t just one correct solution – the idea is to identify relevant pieces and put them together logically. One potential approach is to break down the problem into a couple of steps like this: (1) rough population of Germany, (2) smartphone ownership rates, (3) iPhone market share (you may have a general sense and a guess as to whether that share might be higher or lower in Germany).

#### Riddles

Riddles use a cryptic clue to describe an item or concept that you must guess. These often require some lateral and creative thinking.

Our take: riddles do not belong in interviews, but that doesn’t stop interviewers from asking them! These are uncommon precisely because they are not good at testing critical thinking, but in some cases they can be suitable for testing more creative types of problem-solving, like forensic investigation. We would recommend not using these to prepare, unless you have reason to believe a particular interviewer or company asks these.

Example: What is black when you buy it, red when you use it, and gray when you throw it away? See the solution here

### Prepare by practicing

Prepare with lots of good practice problems. The key to becoming proficient at pretty much anything is to practice. For brain teaser interviews, make sure to practice with suitably difficult questions that may get asked in an interview.

Don’t waste your time with super easy brain teasers (you’ll get them anyway), or the wrong kind of brain teaser.

If you just search up “brain teasers”, you might come across unsuitable brain teasers like this:

How far can a squirrel run into the woods?

The answer is: halfway – after that, the squirrel is running back out of the woods.

While this is a clever and cheeky brain teaser, it isn’t a brain teaser that you are likely to encounter in an interview.

Start with Brain Easer’s curated collection of interview brain teasers, which feature puzzles that are suitable for interviews, many of which are sourced from actual interviews. The logic and math puzzles with more than one insight or logical leap are particularly suitable for interviews, because an interviewee can make progress and demonstrate their critical thinking skills even if they don’t arrive at the final answer.

### How to approach the interview

The last step is learning and becoming comfortable with how you should be approaching the brain teaser at the interview itself.

#### 1. Clarify the question

You need to fully understand the question in order to come up with a good solution. Ask clarifying questions.

#### 2. Talk through your reasoning

• You don’t necessarily need to solve the brain teaser to do well, you just need to demonstrate how you would tackle a difficult problem. This is like getting partial credit in school for showing your work.
• Talking through your reasoning can buy you some time to think, and for some people it also helps organize your thoughts.
• This gives the interviewer an opportunity to steer you off the wrong track. You may misinterpret a piece of information or make an invalid assumption, and if the interviewer is able to jump in a correct your assumptions, you will waste less time.

Remember that this is still an interview. While the brain teaser tests your critical thinking, you also should demonstrate other relevant skills to the job, such as organization, communication, and presentation.

So after you piece together the answer, package it together and clearly present the steps of your solution.

#### 4. Impress the interviewer (optional)

If you’ve already solved the brain teaser, there is room to go above and beyond! If you have time left over, you can talk about variations and extensions on the brain teaser. Proactively solving a more difficult problem is likely to impress any interviewer.

Example: You and a friend play “first to 100”, a game in which you start with 0, and you each take turns adding an integer between 1 and 10 to the sum. Whoever makes the sum reach 100 is the winner. What is the winning strategy? See the solution here

But after getting the right answer, you might add a restriction – neither player is allowed to add 11 minus what the other player just added – and show that there is still a (different) guaranteed winning strategy.

Naturally, this is difficult to do, so don’t worry if you are not able to.

Another way to engage meaningfully with the brain teaser after you’ve solved it is to discuss why it was interesting to solve and how it might relate to the type of thinking you would do on the job.

At the end of the day, brain teaser interview questions are like any other interview question – try to take it as an opportunity to demonstrate why you are a good candidate for the job. Hopefully this guide helped take some of the pressure off solving the brain teaser itself and will allow you to put your best foot forward at the interview.

Categories

## Rebus Puzzles

A rebus is a visual word puzzle that uses the positioning of words, letters, and/or symbols to represent a common phrase, sometimes in an indirect or tricky manner. They are sometimes referred to as “hidden meaning” puzzles.

Here are some common things to look for that can help you solve rebus puzzles. Happy decoding!

## Positioning

The positioning of words and letters relative to each other is often used to replace a word or part of a word.

##### Positioning Example 1

NEAFRIENDED

The words “A FRIEND” appears in the word “NEED”, which means this represents “a friend in need”.

##### Positioning Example 2

PANTS
FIRE

The word “PANTS” appears on the word “FIRE”, which means this represents “pants on fire”.

##### Try it Out

Use what you just learned about positioning of words in rebuses and try this fun rebus.

## Highlighting

There may be an arrow, circle, or square highlighting one part of the rebus, which is often a clue pointing to an adjective to describe the word shown.

##### Highlighting Example 1

AID ←
AID
AID

There are multiple “AID” words, but the first one is highlighted, which means this represents “first aid”.

##### Highlighting Example 2

SECRET ←
SECRET
SECRET

There are multiple “SECRET” words, but the top one is highlighted, which means this represents “top secret”.

## Font Properties

Displaying the word in a different color, size, direction, or style is likely a clue to an adjective or verb to pair with that word.

##### Font Properties Example 1

E
K
A
W

“Wake” is displayed with the letters reading upward, which means this represents “wake up“.

##### Font Properties Example 2

DEAL

“Deal” is displayed in really big font, which means this represents “big deal”.

##### Try it Out

Use what you just learned about fonts in rebuses and try this fun rebus.

## Homophones

To be clever or tricky, rebuses sometimes lead you to a word that sounds like (but is not spelled like) another word or part of another word. Tougher and more creative rebuses use this quite often.

##### Homophones Example 1

11111
A TIME

A bunch of 1’s are on top of the word “TIME”. “Ones” sounds like “once”, which means this represents “once upon a time”.

## Repetition

Some rebuses contain multiples of words, and the number of times the word appears can usually be interpreted as a word or part of a word in the phrase. The number is sometimes replaced with like-sounding words (see homophones above) in the phrase.

##### Repetition Example 1

TIRED TIRED WALK WALK

“Tired” and “walk” show up two times each. Sometimes this just represents the word “two”, but sometimes it could represent the like-sounding words “too” or “to” instead. In this case, this represents “too tired to walk”.

##### Repetition Example 2

INSTANT
INSTANT
INSTANT
INSTANT

“Instant” shows up four times. Sometimes this just represents the word “four”, but sometimes it could represent the like-sounding word “for”. In this case, this represents “for instance”.

##### Try it Out

Use what you just learned about repetition in rebuses and try this tough rebus.

## Context and Clues

There are many, many more ways a puzzle designer can cleverly represent a a hidden meaning. Some more challenging rebus puzzles may include words that are there just to provide context, or you may have to replace a word with a synonym. You will usually know when you get the right answer to a well-designed rebus, so think creatively and keep trying – deciphering the clues is why rebuses are fun!

Still having trouble? Try some easy rebus puzzles to warm up and start thinking in the right direction.

Categories

## Divisibility Rules

A divisibility rule is a shortcut you can use to see if an integer is evenly divisible by another integer, without doing the actual division. Most divisibility rules involve looking at the digits of the number.

Here are some easy divisibility rules for 1 through 11:

## 1

All integers are divisible by 1.

## 2

Last digit is even.

Examples:

• Divisible: 2556, because the last digit (6) is an even number
• Not Divisible: 2655, because the last digit (5) is not an even number

## 3

Sum of digits is divisible by 3.

Examples:

• Divisible: 2334, because 2 + 3 + 3 + 4 = 12 is divisible by 3
• Not Divisible: 2443, because 2 + 4 + 4 + 3 = 13 is not divisible by 3

## 4

Last two digits form a number divisible by 4.

Examples:

• Divisible: 2512, because the number formed by the last two digits (12) is divisible by 4
• Not Divisible: 2242, because the number formed by the last two digits (42) is not divisible by 4

## 5

Last digit is 0 or 5.

Examples:

• Divisible: 2185, because the last digit is a 5
• Not Divisible: 2953, because the last digit (3) is not a 0 or 5

## 6

Divisible by both 2 and 3.

Examples:

• Divisible: 5322, because the last digit (2) is even and the sum of the digits (5 + 3 + 2 + 2 = 12) is divisible by 3
• Not Divisible: 4994, because although the last digit (4) is even, the sum of the digits (4 + 9 + 9 + 4 = 26) is not divisible by 3

## 7

Subtracting double the last digit from the number formed by the remaining digits gives a result that is divisible by 7.

Examples:

• Divisible: 532, because 53 – (2 x 2) = 49 is divisible by 7
• Not Divisible: 270, because 27 – (0 x 2) = 27 is not divisible by 7

## 8

Last three digits form a number divisible by 8.

Examples:

• Divisible: 36136, because the number formed by the last 3 digits (136) is divisible by 8
• Not Divisible: 20238, because the number formed by the last 3 digits (238) is not divisible by 8

## 9

Sum of digits is divisible by 9.

Examples:

• Divisible: 1431720, because 1 + 4 + 3 + 1 + 7 + 2 + 0 = 18 is divisible by 9
• Not Divisible: 2299, because 2 + 2 + 9 + 9 = 22 is not divisible by 9

## 10

Last digit is 0.

Examples:

• Divisible: 35480, because the last digit is 0
• Not Divisible: 30005, because the last digit is not 0

## 11

Alternating sum of digits is divisible by 11. To get the alternating sum, add every other digit starting from the left, and subtract all the other digits.

Examples:

• Divisible: 3729, because 3 – 7 + 2 – 9 = -11, which is divisible by 11
• Not Divisible: 4311, because 4 – 3 + 1 – 1 = 1, which is not divisible by 11

## Larger Divisors

Some larger composite numbers also have simple divisibility rules. For example, a number is divisible by 99 if it is both divisible by 9 and divisible by 11.

## Relevant Brainteasers and Puzzles

Some seemingly difficult brainteasers can be solved by using divisibility rule shortcuts:

##### Polydivisible Number (hard)
Categories

Berkson’s Paradox (also known as Berkson’s bias or collider bias) is the observation of a counterintuitive and usually incorrect statistical result due to selection bias or sampling bias.

Specifically, Berkson’s Paradox pertains to situations where a group is selected based on the combination of two characteristics. One might observe within the group that those characteristics are negatively correlated – but those characteristics may in fact be positively correlated or uncorrelated in the population, and the observed negative correlation could be because those without those two characteristics were not selected to be in the group.

Categories

Simpson’s Paradox (also sometimes known as the reversal paradox or Simpson’s reversal) refers to the phenomenon in which a trend or result that appears in multiple groups of data no longer appears—or in fact reverses—when the groups are combined.

Categories

## Base Rate Fallacy

The Base Rate Fallacy (also known as Base Rate Bias or Base Rate Neglect) is the misguided tendency to place too much emphasis on event-specific information, at the expense of relevant base rate information – which will sometimes result in very poor estimates of probabilities or incidences!

Categories

## Pigeonhole Principle

The Pigeonhole Principle is a simple and elegant concept:

If you place more than n pigeons into n pigeonholes, at least one of the pigeonholes must contain more than one pigeon.

More generally, if you need to make more than n selections and only have n options, at least one of the options must be selected more than once.

This is also known as Dirichlet’s box principle, named after a German mathematician who applied this concept in a mathematical proof back in 1834!