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## 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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## 7 Matchstick Equation Puzzles

In matchstick puzzles, you are presented with an incorrect equation made using matchsticks, and you must move 1 or more matchsticks to turn it into a valid equation.

You must use all of the matches, and you are not allowed to make an inequality symbol such as ≠, ≥, >, <, or ≤.

Try these 7 fun matchstick equation puzzles!

7 + 2 = 9

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## Color of the Last Ball

There is a bag with 20 blue balls and 13 red balls. Randomly remove 2 balls from the bag:

• If they are the same color, replace them with a blue ball
• If they are different colors, replace them with a red ball

Repeat this process until there is just 1 ball remaining. What is the color of the last ball?

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## Multiple of 24

Prove that for all prime numbers p > 3, (p2 – 1) is a multiple of 24.

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## Same Birthday in Line

There is a long line of people waiting to see a new movie. They announce that the first person to have the same birthday as someone standing before them in the line gets to meet one of the actors in the movie.

What place in line would maximize your chances of winning? Assume birthdays are uniformly distributed through the year.

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## Digital Sum of the Sum

The digital sum of a number is just the sum of the individual digits. E.g., the digital sum of 123 is 1 + 2 + 3 = 6.

• The digital sum of x is 42
• The digital sum of y is 67
• When x and y are added, you have to “carry the 1” exactly five times (e.g., if you add 8 and 9, you “carry the 1” once to get a “1” in the tens digit, thereby getting 17)

What is the digital sum of z = x + y?

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## I Am Normally Below You Riddle

I am a 5-letter word. I am normally below you.

If you remove my 1st letter, I am normally above you.

If you remove my 1st and 2nd letters, I am all around you.

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## How Many Friday the 13th in a Year

Some consider Friday the 13th (any Friday that falls on the 13th day of a month) unlucky.

In a calendar year, how many Friday the 13th can occur? Find both the minimum and maximum.

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## Ferry Boat Problem

Two ferry boats serve the same route on a river, but travel at different speeds. They depart from opposite ends of the river at the same time, meeting at a point 720 yards from the nearest shore.

When each boat reaches the other side, it takes 10 minutes to unload and load passengers, then begins the return trip. This time, the boats meet at a point 400 yards from the other shore.

How wide is the river?

The ferry boat problem is created by well-known puzzle author Sam Loyd.

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## Sum of Powers

Find a combination of three different positive integers x, y, and z such that:

x3 + y3 = z4

Hint: there’s a better way than brute force / trial & error.

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## Finish My Wordle

These Wordle puzzles are nearly complete, you just need to finish them by figuring out the correct word. In each of these 5 puzzles, there should be just one or two valid answers remaining – find them!

Haven’t played Wordle before? You get 6 tries to guess the 5-letter English word, and for each guess you are shown whether each letter is in the right place in the word (green), in the word but in the wrong place (yellow), or not in the word at all (gray).

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## Russian Roulette Riddle

In the morbid game of Russian Roulette, a partially loaded revolver with a six-chamber cylinder is randomly spun, pointed at one of the players, and fired. If the revolver landed on an empty chamber, the lucky player is safe, and the process is repeated with the next player. The obvious objective of the game is to not get shot.

You find yourself stuck in a game of Russian Roulette. A freshly loaded revolver is aimed at the first player, and it turns out to be an empty chamber. Your turn is next, and you are given the choice to either:

• Spin the cylinder before pulling the trigger (i.e., you get a random new chamber)
• Or just pull the trigger (i.e., let the revolver fire whatever is in the next chamber)

Which choice should you pick if the revolver was originally:

1. Loaded with one bullet?
2. Loaded with bullets in two random chambers?
3. Loaded with bullets in two consecutive chambers?

Assume the revolver cannot misfire, and that spinning the cylinder lands on all chambers with equal probability.

Some variation of this Russian Roulette riddle was once asked in interviews at Jane Street, Susquehanna International Group (SIG), Facebook (now Meta), UBS, Capital One, and more.

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## Vocabulary Puzzle

Quick vocabulary puzzle – given the middle four letters of each word, figure out these 8-letter words:

1. _ _ TABA _ _
2. _ _ CUME _ _
3. _ _ NERO _ _
4. _ _ IDEN _ _
5. _ _ NETA _ _
6. _ _ EASA _ _
7. _ _ ENAR _ _
8. _ _ WERF _ _
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## Matchstick Math 3

Rearrange exactly 2 matches to make this a valid equation.

In matchstick math puzzles, you are not allowed to make an inequality symbol such as ≠, ≥, >, <, or ≤. That would make it too easy!

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## Mate in 3 Chess Puzzle 1

White to play and mate in 3.

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## I Like the Numbers 4 and 5

I like the numbers 4 and 5. I like to add 4’s and 5’s together to make other numbers, such as 4 + 4 + 5 + 5 + 5 = 23. How many numbers from 1 to 1000 can be written as the sum of 4’s and 5’s?

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## Lions and Sheep

On a magical island, there are 100 lions and 1 sheep, all of which can live by eating the plentiful grass on the island. Any lion that eats the sheep will magically turn into a sheep afterward, such that there will always be a sheep on the island.

Every lion would like to eat a sheep, but would much rather prefer to not be eaten (they wouldn’t mind turning into a sheep if they wouldn’t be eaten).

If all the lions act rationally and know all the other lions act rationally, how many lions will remain on the island in the end?

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## Throwing Rocks Off a Boat

A man is throwing rocks off a boat floating in the middle of a lake. The rocks sink quickly to the bottom of the lake.

Does the water level in the lake rise, fall, or stay the same after the rocks are thrown off the boat and sink to the bottom of the lake?

This question was asked in an actual mechanical engineering interview.

Preparing for a brain teaser interview? Check out our ultimate guide to brain teaser interviews.

#### Hint

Consider Archimedes’ principle and how the rocks affect the water level while they are on the boat vs. how the rocks affect the water level when they are in the lake.

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## Same Number of Handshakes

In some cultures, it is typical for guests at an event to shake hands when they meet other guests (when there isn’t a pandemic). Some might be more social and shake hands with a lot of other guests, while others may be less social and shake hands with few or no other guests.

Can you prove that, regardless of the number of guests at the event, there must be at least two guests with the same number of handshakes at the event? In other words, can you prove that it is impossible for every guest to have shaken a different number of hands?

Assume guests can’t shake hands with themselves.

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## Mate in 2 Chess Puzzle 2

White to play and mate in 2.

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## Middle Four Letters

The middle four letters of these 8-letter words are shown. Can you figure out the full 8-letter words? There may be multiple suitable answers for some of the words.

1. _ _ EQUA _ _
2. _ _ DUST _ _
3. _ _ CUBA _ _
4. _ _ COLA _ _
5. _ _ DIRE _ _
6. _ _ MESA _ _
7. _ _ TACO _ _
8. _ _ OPIC _ _
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## Inscribed Squares

Above are two identical isosceles right triangles containing two inscribed squares.

In one, a perfect square has been inscribed such that two sides line up with the two legs of the right triangle. In the other, a perfect square has been inscribed such that one side lines up with the hypotenuse of the right triangle.

Is the inscribed square on the left larger or the one on the right?

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## Matchstick Minimization 1

What is the smallest number you can make by moving exactly 2 matches above?

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## Card Flipping Endgame

There are n playing cards lined up face-down in a row. Every turn, a pair of adjacent cards with the left card face-down is randomly selected (i.e., a pair of cards has no chance of being selected if the left card is face-up, otherwise all pairs are equally likely to be selected). Both cards are then flipped over (face-down to face-up or face-up to face-down).

Prove that after enough turns, it will eventually be impossible to select a pair of cards with the left card face-down.

When you reach this card flipping endgame, will the rightmost card be face-up or face-down?

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## Calculator Error Riddle

A kid is adding consecutive integers on a calculator, one at a time, starting with 1 + 2 + 3 + … and so on. At one point you notice the sum is now 100, but that shouldn’t be possible if the kid was adding this way. The kid tells you that he made an error and subtracted exactly one of the numbers he was supposed to add.

What is the number he subtracted?