Two weighings. There are 7 coins which are identical on the surface, including 5 real ones (all of the same weight) and 2 counterfeit coins (both of the same weight, but lighter than the real ones). How can you find the 3 real coins with the help of two weighings on scales without weights?
a) Two in turn put bishops in the cells of a chessboard. The next move must beat at least one empty cell. The bishop also beats the cell in which it is located. The player who loses is the one who cannot make a move.
b) Repeat the same, but with rooks.
There are two piles of sweets: one with 20 sweets and the other with 21 sweets. In one go, one of the piles needs to be eaten, and the second pile is divided into two not necessarily equal piles. The player that cannot make a move loses. Which player wins and which one loses?
The game begins with the number 0. In one go, it is allowed to add to the actual number any natural number from 1 to 9. The winner is the one who gets the number 100.
In a certain realm there are magicians, sorcerers and wizards. The following is known about them: firstly, not all magicians are sorcerers, and secondly, if the wizard is not a sorcerer, then he is not a magician. Is it true that not all magicians are wizards?
A traveller on the planet of liars and truth tellers met four people and asked them: “Who are you?”. They received the following answers:
1st: “We are all liars.”
2nd: “Among us is exactly one liar.”
3rd: “Among us there are two liars.”
4th: “I have never lied and I’m not lying”.
The traveller quickly realised who the fourth resident was. How did they do it?
In the lower left corner of an 8 by 8 chessboard is a chip. Two in turn move it one cell up, right or right-up diagonally. The one who puts the chip in the upper right corner wins. Who will win in a regular game?
a) There are 10 coins. It is known that one of them is fake (by weight, it is heavier than the real ones). How can you determine the counterfeit coin with three weighings on scales without weights?
b) How can you determine the counterfeit coin with three weighings, if there are 27 coins?
A family went to the bridge at night. The dad can cross it in 1 minute, the mum in 2 minutes, the child in 5 minutes, and the grandmother in 10 minutes. They have one flashlight. The bridge only withstands two people. How can they cross the bridge in 17 minutes? (If two people cross, then they pass with the lower of the two speeds. They cannot pass along the bridge without a flashlight. They cannot shine the light from afar. They cannot carry anyone in their arms. They cannot throw the flashlight.)
Several stones weigh 10 tons together, each weighing not more than 1 ton.
a) Prove that this load can be taken away in one go on five three-ton trucks.
b) Give an example of a set of stones satisfying the condition for which four three-ton trucks may not be enough to take the load away in one go.