a) Find the biggest 6-digit integer number such that each digit, except for the two on the left, is equal to the sum of its two left neighbours.
b) Find the biggest integer number such that each digit, except for the rst two, is equal to the sum of its two left neighbours. (Compared to part (a), we removed the 6-digit number restriction.)
Some inhabitants of the Island of Multi-coloured Frogs speak only the truth, and the rest always lie. Three islanders said:
Bree: There are no blue frogs on our island.
Kevin: Bree is a liar. She herself is a blue frog!
Clara: Of course, Bree is a liar. But she’s a red frog.
Are there any blue frogs on this island?
In the family of happy gnomes there is a father, a mother and a child. The names of the family members: Alex, Charlie and Jo. At the dinner table two gnomes made two statements.
Charlie said: “Alex and Jo are of different genders. Alex and Charlie are my parents”.
Alex said: “I am Jo’s father. I am the daughter of Charlie”.
Who is who? That is, what is the name of the father, the mother and the child, if it is known that each gnome lied once, and each told the truth once.
On the first day of school, in all three of the first year classes (1A, 1B, 1C), there were three lessons: Maths, French and Biology. Two classes cannot have the same lesson at the same time. 1B’s first lesson was Maths. The Biology teacher praised the students in 1B: “You have even better marks than 1A”. 1A’s second lesson was not French. Which class’s last lesson was Biology?
Will thought of a number: 1, 2 or 3. You can ask him only one question, to which he can answer “yes”, “no” or “I do not know”. Can you guess the number by asking just one question?
On an island there are 1,234 residents, each of whom is either a knight (who always tells the truth) or a liar (who always lies). One day, all of the inhabitants of the island were broken up into pairs, and each one said: “He is a knight!" or “He is a liar!" about his partner. Could it eventually turn out to be that the number of “He is a knight!" and “He is a liar!" phrases is the same?
Solving the problem: “What is the solution of the expression \(x^{2000} + x^{1999} + x^{1998} + 1000x^{1000} + 1000x^{999} + 1000x^{998} + 2000x^3 + 2000x^2 + 2000x + 3000\) (\(x\) is a real number) if \(x^2 + x + 1 = 0\)?”, Vasya got the answer of 3000. Is Vasya right?
There are five chain links with 3 rings in each. What is the smallest number of rings that need to be unhooked and hooked together to connect these links into one chain?
In Wonderland, an investigation was conducted into the case of a stolen soup. At the trial, the White Rabbit said that the soup was stolen by the Mad Hatter. The Cheshire Cat and the Mad Hatter also testified, but what they said, no one remembered, and the record was washed away by Alice’s tears. During the court session, it became clear that only one of the defendants had stolen the soup and that only he had given a truthful testimony. So, who stole the soup?
The stepmother, leaving for the ball, gave Cinderella a sack which contained a mixture of poppy and millet, and ordered them to be sorted. When Cinderella was leaving for the ball, she left three sacks: one contained millet, the other contained poppy, and in the third – a mixture that had not yet been sorted. In order not to confuse the sacks, Cinderella attached a label to each of them that said: “Poppy seed”, “Millet” and “Mixture”. The stepmother returned from the ball first and deliberately swapped all of the labels in such a way that on each sack there was an incorrect inscription. The fairy godmother managed to warn Cinderella that now none of the labels on the sacks were correct. Then Cinderella took out only one single grain from one sack and, looking at it, immediately guessed what was in each sack. How did she do this?