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You have a glass with perfectly cylindrical shape which has some water in it. How can you figure out if the glass is exactly half full without using any measurement tools (like a ruler)?
Use the geometry of the cylinder. Start tilting the glass until the water surface gets either to the top or the bottom edge. If the glass is exactly half full, then the surface should touch both edges simultaneously.
Get from START to FINISH, moving only over smooth lines, without reversing and making sharp turns.
The solution is shown below.
Cross out nine letters so that the remaining letters spell a single word.
N A I S N I E N L G E L T E T W E O R R S D
Cross out every other letter starting with the first one – N I N E L E T T E R S.
The remaining letters are – A S I N G L E W O R D.
Look at the picture below and answer the following questions:
2. People are sowing the crops, so it is early spring.
8. Since it is spring, birds are flying North.
1. Since birds are flying North, the shadows are pointing East, and therefore it is morning.
4. Judging by the water around the buoy, the river is flowing South.
3. Since there is a buoy, the river must be navigable.
5. The fishing line is long, so the river must be deep.
6. There would not be a ferry if there was a bridge nearby.
7. The guy on the left looks like a railroad worker, so probably the railroad is nearby.
As a birthday present last year, I received some fridge magnets. They didn’t come as a puzzle, so I don’t know if they have a solution, but I made a puzzle out of them anyway. The magnets are tetrominoes. There are 7 of each shape. Is it possible to arrange them into a 7×28 rectangle so that they are all used and all inside the rectangle? The closest I have managed is this:
No, it is impossible. Imagine you are placing the tetrominoes on a 7×28 chess board. All of them, except for the T-shaped ones cover exactly 2 black and 2 white cells. Each of the T-shaped tetrominos covers either 2 more black cells than white cells or 2 more white cells than black cells. Since there are 7 of them, combined they will cover either more black cells than white cells or more white cells than black cells. Therefore all pieces on the picture can not cover perfectly a rectangle, which contains an equal number of black and white cells.
One man is trying to cross the desert to reach the neighboring village. It takes 4 days to get there, but his camel can carry bananas which will feed him for 3 days only. How can the man reach the neighboring village without starving?
The man travels one day, leaves one portion of bananas in the desert and returns back to his village. Then he leaves again with 3 new portions of bananas, picks the portion left in the desert on his way and ends up in the neighboring village on the sixth day.
Which is the next symbol in the following sequence?
This is the sequence 1, 2, 3, 4, 5… with all numbers stuck to their mirrored images.
Each night one of three prisoners has steak for dinner, while the other two have fish tacos. Also every night, each of the three prisoners votes for one of the following two options:
If a majority go with option 2, nothing happens that night. If a majority go with option 1, then they are set free if they are right and executed if they are wrong. The distribution of votes is kept secret, so it is unknown what each of the others voted. Also, it is known that every prisoner eventually will get a steak.
The three prisoners can have a brief strategy meeting and after that, they are not allowed to communicate. What should the prisoners’ strategy be?
The prisoner who gets a steak the first night should always vote 2, whereas the other two prisoners should vote 2 until the night they get a steak, and 1 every night after.
You have a rectangular grid and arbitrary real numbers in its cells. You are allowed repeatedly to multiply the elements in any row or any column by -1. Prove that you can make all row sums and all column sums non-negative simultaneously.
If there is any row or column in the grid with a negative sum, multiply it by -1. Since on every step the total sum of the numbers in the grid increases, we will be able to do this procedure only finitely many times. In the end, all row sums and column sums will be non-negative.
Why are manholes round?
Because it is impossible for the covers to drop inside the holes, no matter how you position them on top. Also, when they have
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