Unknown Author - Puzzle Prime

Split the Cube

Section: Brain Teasers Category: Mathematics

Difficulty: Hard

Can a cube be split into finely many smaller cubes, all with different sizes?

SOLUTION

It is impossible. Assume the opposite and consider the smallest cube on the bottom, C1. Since all cubes surrounding it are larger, they must “wall in” its top face. We repeat the same argument for the smallest cube which lies on the top face of C1, call it C2. If all surrounding cubes are larger, they must wall in its top face. Thus, we can create an infinite sequence of cubes with decreasing sizes lying on top of each other: C1, C2, C3, etc. Since the initial cube is split into a finite number of smaller cubes, we get a contradiction.

Unknown Author March 25, 2024

8 Comments

Airplane on a Conveyor Belt

Section: Brain Teasers Category: Science

Difficulty: Advanced

A Boeing 747 is sitting on a conveyor belt, as wide and long as a runway. The conveyor belt is designed to exactly match the speed of the wheels, moving in the opposite direction. Will the plane take off?

SOLUTION

Solution coming soon.

Unknown Author March 19, 2024

5 Comments

Had Had Had Had Had

Section: Brain Teasers Category: Deduction

Difficulty: Advanced Tags: Words

A teacher in English had asked James and John to describe a man who had suffered from a cold in the past. James while John had had had had had had had had had had had a better effect on the teacher.

Add punctuation to the sentence in bold, so that it makes sense.

SOLUTION

A teacher in English had asked James and John to describe a man who had suffered from a cold in the past. James, while John had had “had”, had had “had had”; “had had” had had a better effect on the teacher.

The meaning is that John had used the phrase “had a cold”, whereas James had used the phrase “had had a cold”. The latter, being more grammatical, had resulted in a better impression on the teacher.

Unknown Author March 16, 2024

0 Comments

Average Salary

Section: Brain Teasers Category: Insight

Difficulty: Advanced

Three friends, A, B and C, want to find out what their average salary is without disclosing their own salaries to the others. How can they do it using only verbal communication?

SOLUTION

A tells B some number, then B adds his salary to it and tells the result to C, then C adds his salary and tells the result to A. Now A subtracts the number he told B in the beginning, adds his own salary and divides by 3. Repeat the same procedure with B and C starting first.

Unknown Author March 10, 2024

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Eminent King

Section: Casual Puzzles Category: Riddle

Difficulty: Medium

Five hundred begins it,
five hundred ends it,
five in the middle is seen.

The first of all letters,
the first of all numbers
take up their stations between.

Put them all together
and bring before you
the name of an eminent king.

SOLUTION

The answer is DAVID. In Roman numerals D=500, V=5, I=1.

Unknown Author March 4, 2024

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Six Glasses

Section: Brain Teasers Category: Insight

Difficulty: Medium

Six identical glasses are placed in a row on the table – first three filled with water, and then three empty ones. Can you move just one glass, so that empty and full glasses alternate?

SOLUTION

Take the second full glass, pour all the water into the second empty glass, and then put it back in its place.

Unknown Author March 1, 2024

2 Comments

Protect the Treasure

Section: Brain Teasers Category: Mathematics

Difficulty: Advanced

Nine pirates have captured a treasure chest. In order to protect it, they decide to lock it using multiple locks and distribute several keys for each of these locks among them, so that the chest can be opened only by a majority of the pirates. What is the minimum number of keys each of the pirates should get?

SOLUTION

First, we show that for every four pirates, there exists a lock which cannot be opened only by them and can be opened by everyone else. We choose an arbitrary group of four pirates. If they can open every lock, then they can access the treasure without the need of a majority. If any of the remaining pirates cannot open that lock, then he, together with the initial group of four still cannot access the treasure. Thus, the claim is proved and to each group of four pirates we can assign a unique lock. These are $\binom{9}{4}=126$ locks in total. Finally, every pirate should get keys for $\binom{8}{4}=70$ of these locks, one for each group of four additional pirates he can be a group of.

Unknown Author February 19, 2024

2 Comments

Buried Up to Neck

Section: Brain Teasers Category: Deduction

Difficulty: Medium

Three friends, Adam, Bob, and Charlie are buried in the sand up to their necks, all facing West. Charlie can see both Adam and Bom, Bom can see only Adam, and Adam cannot see anyone. Black and white hats are placed on their heads. The three friends are told that there is at least one hat from each color, and then they are asked whether anyone can guess the color of their own hat.

After a few minutes, one of them answers. Who is that?

SOLUTION

If Adam and Bob had hats with identical colors, then Charlie would immediately be able to deduce that his hat has the opposite color. Charlie doesn’t do that, so Adam and Bob are able to figure out that their hats have opposite colors. Since Bob is the one who can see the color of Adam’s hat, he is the one that answers the question.

Unknown Author February 13, 2024

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The Letter E

Section: Brain Teasers Category: Practical

Difficulty: Expert

Click the image to enlarge it and download it. Then print it, cut out the three pieces, and rearrange them so that you get the letter “E”.

SOLUTION

Rearrange the pieces, so that they form the shadow of the letter “E”, as shown below.

Unknown Author February 10, 2024

1 Comment

Fish Eat Fish

Section: Brain Teasers Category: Mathematics

Difficulty: Advanced Tags: Probability

A hundred fish are swimming along a stream at different velocities. If one fish catches up to another fish, it eats it and continues swimming. What is the expected number of fish that will survive?

SOLUTION

Notice that the N-th fish in the stream survives if and only if it is the fastest among the first N fish. The probability of this event happening is equal to 1/N. Since the expected number of fish that survive is equal to the sum of the survival probabilities for each of them, the answer is 1+1/2+1/3+…+1/N.

For more details on the last claim, consider reading our blog post “How Many Times on Average?”

Unknown Author February 7, 2024

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