A number is written on each edge of a cube. The sum of the 4 numbers on the adjacent edges is written on each face. Place the numbers \(1\) and \(-1\) on the edges so that the numbers written on the faces are all different.
It is known that among the members of the government of the Planet of Liars and truth tellers, consisting of 20 members, there is at least one honest one, and also that from any two at least one is a bribe taker. How many bribe takers are there in the government?
An adventurer is travelling to the planet of liars and truth tellers with an official guide and is introduced to a local. “Are you a truth teller?” asked the adventurer. The alien answers “Yrrg,” which means either “yes” or “no”. The adventurer asks the guide for a translation. The guide says “"yrrg" means "yes". I will add that the local is actully a liar.” Is the local alien liar or truth teller?
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?
Prove that in a game of noughts and crosses on a \(3\times 3\) grid, if the first player uses the right strategy then the second player cannot win.
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?
Vincent makes small weights. He made 4 weights which should have masses (in grams) of 1, 3, 4 and 7, respectively. However, he made a mistake and one of these weights has the wrong mass. By weighing them twice using balance scales (without the use of weights other than those mentioned) can he find which weight has the wrong mass?
There are some coins on a table. One of these coins is fake (has a different weight than a real coin). By weighing them twice using balance scales, determine whether the fake coin is lighter or heavier than a real coin (you don’t need to find the fake coin) if the number of coins is: a) 100; b) 99; c) 98?