Arrange in a row the numbers from 1 to 100 so that any two neighbouring ones differ by at least 50.
The numbers from 1 to 9999 are written out in a row. How can I remove 100 digits from this row so that the remaining number is a) maximal b) minimal?
Four aliens — Dopey, Sleepy, Happy, and Moody — from the planet of liars and truth-tellers are having a conversation.
Dopey says to Sleepy: “You are a liar.”
Happy says to Sleepy: “You are a liar.”
Moody says to Happy: “They are both liars.” Then, after thinking for a moment, Moody adds a final statement: “You are also a liar”
Which of the four aliens are telling the truth?
One of five brothers baked a cake for their Mum. Alex said: “This was Vernon or Tom.” Vernon said: “It was not I and not Will who did it.” Tom said: “You’re both lying.” David said: “No, one of them told the truth, and the other was lying.” Will said: “No David, you’re wrong.” Mum knows that three of her sons always tell the truth. Who made the cake?
The order of books on a shelf is called wrong if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from \(n\) books of different heights, if: a) \(n = 4\); b) \(n = 5\)?
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 travels to the planet of Liars and Truth-Tellers with an official guide and meets a local. “Are you a truth-teller?” the adventurer asks. The alien replies, “Yrrg”, which means either “yes” or “no.” The adventurer turns to the guide for help. The guide says, “‘Yrrg’ means ‘yes’. Also, the local is actually a liar.” Is the local alien a liar or a 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 residents 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 alien 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?