Difficulty: Advanced

 

Fold Into a Cube

Section: Brain Teasers Category: Practical
Difficulty:

Cut out the black shape and then fold it perfectly into a cube, without overlapping.

If the size of each of the small edges of the shape is equal to 1cm, then its area is equal to 30cm². Thus, the cube that is formed by the shape must have an edge of length √5cm. If we set the center of the black shape to be the center of one of the cube’s faces, then the four closest vertices must lie on the boundary of the shape, √2.5cm away. This determines uniquely the folding of the shape which is shown on the simulation below.

created by Wossname
Source:

Puzzling Stackexchange

puzzle
plasticinsect March 13, 2024
0 Comments

Average Salary

Section: Brain Teasers Category: Insight
Difficulty:

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?

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.

puzzle
Unknown Author March 10, 2024
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Protect the Treasure

Section: Brain Teasers Category: Mathematics
Difficulty:

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?

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.

puzzle
Unknown Author February 19, 2024
2 Comments

Fish Eat Fish

Section: Brain Teasers Category: Mathematics
Difficulty: Tags:

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?

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?”

puzzle
Unknown Author February 7, 2024
0 Comments

Mountain Hike

Section: Brain Teasers Category: Insight
Difficulty:

A man decides to climb a mountain. He starts at sunrise from the bottom of the mountain and arrives at the top at sunset. He sleeps there and on the next day he goes back the same way, descending at higher speed. Prove that there is some point of his path, on which the man will be at the same time on both days.

Imagine a second man who starts climbing from the bottom of the mountain on the second day and following the first hiker’s first day movements. At some point the first and the second hiker will meet each other, and this will be the point you are looking for.

puzzle
Unknown Author January 7, 2024
0 Comments

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