Seven coins are arranged in a circle. It is known that some four of them, lying in succession, are fake and that every counterfeit coin is lighter than a real one. Explain how to find two counterfeit coins from one weighing on scales without any weights. (All counterfeit coins weigh the same.)
Four people discussed the answer to a task.
Harry said: “This is the number 9”.
Ben: “This is a prime number.”
Katie: “This is an even number.”
And Natasha said that this number is divisible by 15.
One boy and one girl answered correctly, and the other two made a mistake. What is the actual answer to the question?
Peter recorded an example of an addition on a board, after which he replaced some digits with letters, with the same figures being replaced with the same letters, and different figures with different letters. He did it such that he was left with the sum: \(CROSS + 2011 = START\). Prove that Peter made a mistake.
Can 100 weights of masses 1, 2, 3, ..., 99, 100 be arranged into 10 piles of different masses so that the following condition is fulfilled: the heavier the pile, the fewer weights in it?
Matt, Conrad and Louie ate some sweets. Their surnames are Smith, Jones and Cooper. Smith ate 2 sweets fewer than Matt, Jones – 2 sweets fewer than Conrad, and Conrad ate more than anyone. Which of them has which last name?
When cleaning her children’s room, a mother found \(9\) socks. In a group of any \(4\) of the socks at least two belonged to the same child. In a group of any \(5\) of the socks no more than \(3\) had the same owner. How many children are there in the room and how many socks belong to each child?
A bag contains balls of two different colours – black and white. What is the minimum number of balls you need to remove, without looking, to guarantee that within the removed balls at least two are the same colour.
Imogen’s cat always sneezes before it rains. Today the cat sneezed. “So, it will rain” thinks Imogen. Is she right?
Three tortoises crawl along the road in a line. “Two tortoises are crawling behind me,” says the first. “One tortoise is crawling behind me, and one tortoise is crawling in front of me,” says the second. “Two tortoises are crawling in front of me, and one tortoise is crawling behind me,” says the third. How can this be?
Is it possible to fill a \(5 \times 5\) board with \(1 \times 2\) dominoes?