Learning

Learning materials and explanations of nifty concepts to expand your knowledge and skills

The Monty Hall Problem Explained

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.

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Ultimate Guide to Brain Teaser Interview Questions

Puzzle Pieces

Brain Easer’s guide to brain teaser interview questions

1. Why Brain Teasers

To understand how to tackle brain teasers at an interview, first understand what brain teasers are and why they are used.

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.

The Goal of 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.

Types of Interviews with Brain Teasers

You will most commonly encounter brain teaser interview questions in these industries or 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
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How to Solve Rebus Puzzles

What are 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.

So if you see words or letters placed in unusual arrangements, think about what words that appear in common phrases could describe that positioning. This could include “on,” “above,” “below,” “down,” “between,” and many more.

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.

So if you see any emphasis on certain words or letters, think about how to describe that emphasis in the context of a common phrase.


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.

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.

There’s no trick to solving these, but now that you know it’s possible, you’ll know that sometimes you have to think outside of the box and maybe try saying things out loud.


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.

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.

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:

Arrange the Digits (easy)
Polydivisible Number (hard)

Berkson’s Paradox

Berkson’s Paradox is a counterintuitive or unexpected trend observed in a sample due to a particular type of selection bias. This bias arises when the sample is selected based on the combination of two characteristics.

Also known as Berkson’s bias or collider bias, Berkson’s Paradox pertains to situations where a group is selected based on the combination of two characteristics and results in some false observation of correlation between the two characteristics – the correlation might be observed in the sample only because those without those two characteristics were not selected to be in the group in the first place.

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Base Rate Fallacy

What is the Base Rate Fallacy?

In simple terms, it’s a common error we make in assessing likelihoods due to (a) over-emphasizing the rate of something within a group and (b) under-emphasizing how common that group is in the first place (i.e., the base rate).

For example, let’s say you see a chess set in a building with 1 avid chess player and 1000 other people. You might assume it belongs to the chess player, even though it’s more likely to belong to one of the others because there are so many of them – if only 1% of regular people own chess sets, there would likely be ~10 of them in a group of 1000, outnumbering the 1 chess player.

Sometimes also referred to as Base Rate Bias or Base Rate Neglect, this is a cognitive bias arising from the tendency to place too much emphasis on event-specific information, at the expense of relevant base rate information. Often this results in a sense of probabilities or rates that are very far from reality!

To understand what this means, let’s look at a few more examples:

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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!

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