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Two monks are standing on the two sides of a 2-dimensional mountain, at altitude 0. The mountain can have any number of ups and downs, but never drops under altitude 0. Prove that the monks can climb or descend the mountain at the same time on both sides, always staying at the same altitude, until they meet at the same point.
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In a parliament
For each split of the people into 2 groups, compute the animosity level of each person by subtracting the number of enemies in the opposite group from the number of enemies in his own group. Then, split the people into 2 groups so that the total animosity level of all of them is as little as possible. If there is a person, who has more enemies in his own group than the opposite one, then by transferring him to the other group, we will reduce the total animosity level of the people and will get a contradiction.
White to play and mate in 17 moves.
An evil warden holds you as a prisoner but offers you a chance to escape. There are 3 doors A, B, and C. Two of the doors lead to freedom and the third door leads to lifetime imprisonment, but you do not which door is what type. You are allowed to point to a door and ask the warden a single yes-no question. If you point to a door that leads to freedom, the warden does answer your question truthfully. But if you point to the door that leads to imprisonment, the warden answers your question randomly, saying either “YES” or “NO” by chance. Can you figure out a way to escape the prison?
You can point towards door A and ask whether door B leads to freedom. If the warden says “YES”, then you open door B. It can not lead to imprisonment because this would mean that door A leads to freedom and the warden must have told you the truth. If the warden says “NO”, then you open door C. This is because either the warden lied, and then the imprisonment door is A, or he told you the truth, and then the imprisonment door is B.
Somehow, you end up in a room that has three doors. Behind the first door, there is a deadly poisonous gas. Behind the second door, there are trained assassins with knives. Behind the third door, there are lions that have not eaten in years. Which door would you choose to open?
You should open the door with the lions. If they have not eaten in years, they will be dead already.
You place two stones on a chessboard and start moving them one by one, every time shifting a stone to an adjacent cell – left, right, up, or down, without letting the stones at any time occupy the same cell. Is it possible to find such a sequence of moves, that all possible combinations for the position of the stones
No, such sequence does not exist. Call a position of the stones “matching” if the stones occupy cells of the same color, and “opposite” if the stones occupy cells of opposite colors. If such sequence exists, then the number of matching and opposite positions must differ by at most 1. However, the number of matching positions is 64×31, whereas the number of opposite positions is 64×32, so this is not true.
There is an island on a planet and infinitely many planes on it. You need to make one of these planes fly all around the world and land back to the island. However, each of the planes can carry fuel which is enough to travel just half of the way, and fuel
3 planes are enough, label them A, B, C. They leave the island simultaneously in a clockwise direction, and after 1/8 of the way, A refuels B and C completely, then turns back towards the island. B and C continue to fly until they get to 1/4 of the way, where B refuels C completely and turns back towards the island. When C gets mid-way, A and B leave the island counter-clockwise, and after 1/8 of the way, A refuels B completely and turns back towards the island. B continues towards C, and when the two planes meet, they share their fuel, then fly together towards the island. In the meantime A arrives on the island, refuels completely, and starts flying again counter-clockwise towards B and C, so that it can meet them and give them enough fuel, so that all of them arrive safely on the island.
It is easy to see that 2 planes are not enough.
You and eight of your team members are trying to escape the Temple of Doom. You are running through a tunnel away from a deadly smoke and end up in a large hall. There are four paths ahead, and exactly one of them leads to the exit. It takes 20 minutes to explore any of the four paths one way, and your group has only 60 minutes until the deadly smoke suffocates you. The problem is that two of your friends are known to be delirious and it is possible that they do not tell the truth, but nobody knows which ones they are. How should you split the group and explore the tunnels, so that you have enough time to figure out which is the correct path and escape the temple?
You explore the first path. You send two of your teammates to explore the second path. You send the remaining six teammates in groups of three to explore each of the two remaining paths. If your path leads to the exit, then everything is good. Otherwise, you ask the two groups of three whether their paths lead to the exit. If in both groups everyone answers consistently, then nobody is lying, and you will escape. If in both groups there is a person whose answer is different from the others in the group, then the majority in both groups says the truth. Once again, you will know which path leads to the exit. Finally, if in exactly one of the groups everyone answers consistently, you ask the group of two. If the team members there answer consistently with each other, then they say the truth. You will have two groups which tell the truth and will know which path leads to the exit. If the answers of the teammates in the group of two differ, then in the inconsistent group of three the majority will be saying the truth. Again, you will be able to deduce which path leads to the exit.
A mother has 4 children. On Halloween, she decides to give them 200 candies one by one, starting with the youngest child, then the second youngest, third youngest, etc. At some point, they have to interrupt the procedure and eat dinner, but when they continue, nobody remembers which was the last child to get a candy. How can the mother distribute the remaining candies to the kids without starting over again or counting how many candies have been given so far?
The mother can start giving the remaining candies one by one in the reverse order, starting with the oldest child, then the second oldest, etc.
When a car is making a turn, are its front wheels parallel to each other?
The front wheels are not parallel to each other. The reason is that when steering, all wheels are turning along arks that have a common center. Otherwise
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