Difficulty – Hard

Hard brain teasers and puzzles

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Brain Teasers and Puzzles

Prison Keys Strategy

A prison warden was feeling capricious and played a game with the prison keys:

  1. Each prisoner is handed a key to another prisoner’s cell.
  2. Each prisoner will know which other prisoner was initially given the key to their cell (but does not know whose key they were handed).
  3. Each day, when all prisoners are out of their cells and no one is watching, each prisoner is allowed to place keys in another prisoner’s cell.
  4. Each night, each prisoner can collect any keys placed in their cell.
  5. The prisoners can summon the warden when they’re sure everyone has their own key – but if they are wrong, they’re immediately executed.
  6. The prisoners can discuss a strategy beforehand but cannot communicate in any way after keys are handed out.

What is the fewest number of days it would take for the prisoners to be sure everyone has their key? What was the strategy to achieve this?

More info and solution “Prison Keys Strategy”
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Brain Teasers and Puzzles

Sharing Game Theory Puzzle

In this sharing game theory puzzle, 3 friends take turns taking from a jar of 1000 candies to share. For example, the 1st friend could take 500 candies, then the 2nd friend could take 400, and the 3rd friend would take the remaining 100.

No one wants to be seen as greedy, but no one wants to end up with the fewest candies either. As such, their goals are (in order of preference):

  1. Not end up with the most candies, nor the fewest candies
  2. End up with as many candies as possible

All of them are logical, rational, know each other’s goals, but cannot communicate before or during sharing. How many candies should each friend end up with?

More info and solution “Sharing Game Theory Puzzle”
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Brain Teasers and Puzzles

Converge to an Integer

Given the iterative formula:

xn+1 = sqrt(xn) + a

There are some values of a for which a positive starting value for x results in this formula converging to an integer.

For example, if a = 2, you would observe that xn+1 = sqrt(xn) + 2 eventually converges to 4 for any positive starting value.

What form must a take in order for this formula to converge to an integer?

More info and solution “Converge to an Integer”
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Brain Teasers and Puzzles

Mate in 4 Chess Puzzle 1

Mate in 4 Chess Puzzle 1

White to play and mate in 4.

More info and solution “Mate in 4 Chess Puzzle 1”
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Brain Teasers and Puzzles

Consecutive Numbers Grid Puzzle

13
10
416
7

The cells in this 4×4 grid puzzle contain the numbers 1-16 each once. There are two rules to the arrangement of the numbers:

  1. Any two consecutive numbers must share a row/column (1 and 16 should be considered consecutive).
  2. No row/column can contain three consecutive numbers.

Fill in the remaining numbers.

More info and solution “Consecutive Numbers Grid Puzzle”
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Brain Teasers and Puzzles

The Impossible Puzzle

This puzzle was coined the “Impossible Puzzle” by Martin Gardner, a famous math and science writer that liked to create and write about math games and puzzles. The puzzle is named as such because it appears to provide insufficient information to solve, but it is in fact solvable! Read on for the puzzle and the (very difficult) solution.

There are two distinct whole numbers greater than 1, we can call them x and y (where y > x). We know the sum of x and y is no more than 100.

Sam and Prada are perfect logicians. Sam (“sum”) is told x + y and Prada (“product”) is told x * y, and both of them know all the information provided so far.

Sam and Prada have this conversation in which they truthfully deduce the numbers:

  1. Sam: I know Prada does not know x and y.
  2. Prada: Well now I know x and y.
  3. Sam: Ah, now I also know x and y.

Can you figure out x and y using this information?

More info and solution “The Impossible Puzzle”
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Brain Teasers and Puzzles

Four Lines of Four Coins

Form four lines of four coins by moving exactly 3 of the coins below:

Four Rows of Four Coins

Rules:

  • Coins cannot be stacked on top of one another.
  • A line of five or more coins is still only considered one line.
  • The centers of the coins must be precisely on the same straight line, but can be anywhere on the line.
  • Lines can be any length, and can be horizontal, vertical, or diagonal.
More info and solution “Four Lines of Four Coins”
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Brain Teasers and Puzzles

Truel

A truel is a three-way duel. The three participants take turns firing one shot at whichever opponent they choose, until only one is remaining.

Allison, Ben, and Chase are in a truel, and have varying degrees of accuracy: Allison has a 50% chance of hitting her intended target, Ben has a 80% chance, and Chase has a 100% chance. Allison gets to shoot first, then Ben, then Chase, and repeating in that order until only one person is remaining.

Assume that the accuracy of all participants are publicly known, and everyone is trying to maximize their chances of winning. What is Allison’s optimal strategy, and what is her likelihood of winning under that strategy?

More info and solution “Truel”
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Brain Teasers and Puzzles

Mate in 2 Chess Puzzle 1

Mate in 2 Chess Puzzle 1

White to play and mate in 2.

More info and solution “Mate in 2 Chess Puzzle 1”
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Brain Teasers and Puzzles

Battery Test Puzzle

You have four good/new batteries and four bad/used batteries but don’t know which are which. You have a flashlight that uses two batteries, which will only work if both batteries are good – if it doesn’t work, you won’t know if both batteries are bad or just one.

How many pairs of batteries do you need to test to guarantee you find a good pair?

More info and solution “Battery Test Puzzle”
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Brain Teasers and Puzzles

Chess Tactic Puzzle

Chess Tactic Puzzle

Black is down significant material, but has a strong series of moves to turn things around. Can you find it?

More info and solution “Chess Tactic Puzzle”
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Brain Teasers and Puzzles

Food with 5 Letters

I am a food with 5 letters.

Take away my first letter, and I become a form of energy.

Take away my first two letters, and I become something you need to live.

Rearrange my last three letters, and I become something you drink.

More info and solution “Food with 5 Letters”
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Brain Teasers and Puzzles

Make Five into Four

Four Squares Match Puzzle

Move exactly 3 matches to make these five squares into four squares of equal size, without any matches left over.

More info and solution “Make Five into Four”
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Brain Teasers and Puzzles

Fake Coins Proof

You are a rare coins expert and have determined there are 7 fake coins out of 14 gold coins. Now you need to prove to the judge which ones are fake.

It is known that that real coins all weigh the same, fake coins all weigh the same, and fake coins weigh less than real ones (but are otherwise identical).

Using a traditional double-pan balance scale just 3 times, can you prove exactly which of the 14 coins are fake?

More info and solution “Fake Coins Proof”
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Brain Teasers and Puzzles

Rebus Puzzle 13

What common word or phrase is this rebus referring to?

GOT GOT GOT GOT HERO HERO HERO HERO HERO HERO HERO HERO HERO HERO

Solution

Forgotten heroes (“four got ten heroes”)

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Brain Teasers and Puzzles

Four Squares Match Puzzle

Four Squares Match Puzzle

Move exactly 2 matches to make these five squares into four squares of equal size, without any matches left over.

More info and solution “Four Squares Match Puzzle”
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Brain Teasers and Puzzles

Polydivisible Number

A polydivisible number is a number for which the first n digits form a number evenly divisible by n for all n between 1 and the number of digits of that number.

In other words, the first 2 digits form a number divisible by 2, first 3 digits form a number divisible by 3, first 4 digits form a number divisible by 4, etc. for all of the digits of the number.

Can you arrange the digits 1-9 to form a 9-digit polydivisible number? Each digit 1-9 must be used exactly once.

More info and solution “Polydivisible Number”
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Brain Teasers and Puzzles

Last Chocolate Game

Two friends are playing a game with two boxes of chocolates. They take turns taking out some number of chocolates from the boxes, and whoever takes the last chocolate wins. Each turn they can take chocolates one of two ways:

  • Take any number of chocolates from a single box
  • Or take an equal number of chocolates from each box

If there are 25 chocolates in one box and 35 in the other, what is the winning strategy?

More info and solution “Last Chocolate Game”
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Brain Teasers and Puzzles

Vacant Room Probability

Your workplace has a phone room for employees to quietly make personal calls. The room has no windows, just a sign that can be switched from “Vacant” to “Occupied”. However, employees differ in how consistently they use the sign:

  • 1/2 of them always switch to “Occupied” when they enter and “Vacant” when they exit.
  • 1/4 of them ignore the sign altogether – the sign will always read the same before, during, and after their visit.
  • 1/4 of them always switching to “Occupied” when they enter, but always forget to switch back to “Vacant” when they exit.

If the room is actually occupied exactly 1/2 of the time, what is the probability the room is actually vacant when the sign reads “Vacant”?

More info and solution “Vacant Room Probability”
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Brain Teasers and Puzzles

Break the Cipher

The CIA is tracking a criminal organization and have reason to believe they are about to launch some sort of attack on New York City. One morning, they intercept this message, can you break the cipher and figure out what it means?

8 HIJP
11 IATC
2 BEIO
7 NTYH
4 AAWC
15 LLAT
12 NOML
6 TONF

More info and solution “Break the Cipher”
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Rebus Puzzle 7

What common word or phrase is this rebus referring to?

Knee

FULL OF

Solution

Full of baloney (“below knee”)

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Brain Teasers and Puzzles

Four Triangles with Matches

Four Triangles with Matches Puzzle

Move exactly 3 matches to make these three equilateral triangles into four equilateral triangles of any size, without any matches crossing or overlapping.

More info and solution “Four Triangles with Matches”
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Pirate Survivor Puzzle

The democratic pirates are at it again! Since last time, they have become very successful and have now expanded to 100 pirates. They decide this is too large of a group for plundering, so true to their democratic roots, they want to settle it with a vote:

  1. They vote on whether to kick out the newest (least senior) pirate.
  2. If a majority votes “aye”, then the newest pirate is kicked out, and the process repeats with the remaining pirates.
  3. If half or more of the remaining pirates vote “nay”, then the vote is over and everyone remaining is safe.

Each pirate wants to stay in the group, but if that is assured, they would prefer to kick out as many other pirates as possible (fewer ways to split the treasure).

How many pirates will remain at the end of this process?

More info and solution “Pirate Survivor Puzzle”
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Brain Teasers and Puzzles

Sum of Squares

If x is an even integer that can be written as the sum of two perfect squares, prove that x / 2 can also be written as the sum of two perfect squares.

More info and solution “Sum of Squares”
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Brain Teasers and Puzzles

Rebus Puzzle 4

What common word or phrase is this rebus referring to?

♣ OFFICE
♠ SCHOOL
♥ HOME
♦ STORE

Solution

Home is where the heart is