Is it possible to place 12 identical coins along the edges of a square box so that touching each edge there were exactly: a) 2 coins, b) 3 coins, c) 4 coins, d) 5 coins, e) 6 coins, f) 7 coins.
You are allowed to place coins on top of one another. In the cases where it is possible, draw how this could be done. In the other cases, prove that doing so is impossible.
At the end of the term, Billy wrote out his current singing marks in a row and put a multiplication sign between some of them. The product of the resulting numbers turned out to be equal to 2007. What is Billy’s term mark for singing? (The marks that he can get are between 2 and 5, where 5 is the highest mark).
Sarah believes that two watermelons are heavier than three melons, Anna believes that three watermelons are heavier than four melons. It is known that one of the girls is right, and the other is mistaken. Is it true that 12 watermelons are heavier than 18 melons? (It is believed that all watermelons weigh the same and all melons weigh the same.)
Peter has some coins in his pocket. If Peter pulls \(3\) coins from his pocket, without looking, there will always be a £1 coin among them. If Peter pulls \(4\) coins from his pocket, without looking, there will always be a £2 coin among them. Peter pulls \(5\) coins from his pocket. Identify these coins.
Let’s denote any two digits with the letters \(A\) and \(X\). Prove that the six-digit number \(XAXAXA\) is divisible by 7 without a remainder.
In Neverland, only elves and gnomes live. Gnomes lie about their gold, but in any other instances they tell the truth. Elves lie when talking about gnomes, but in other instances they tell the truth. One day two neverlandians said:
\(A\): All my gold I stole from the Dragon.
\(B\): You’re lying.
Determine whether each of them is an elf or a gnome.
Hannah has a calculator that allows you to multiply a number by 3, add 3 to the number or (4 if the number is divisible by 3 to make a whole number) divide by 3. How can the number 11 be made on this calculator from the number 1?
In the \(4 \times 4\) square, the cells in the left half are painted black, and the rest – in white. In one go, it is allowed to repaint all cells inside any rectangle in the opposite colour. How, in three goes, can one repaint the cells to get the board to look like a chessboard?
A class has 25 pupils. It is known that for any two girls in the class, the number of male friends they have in the class is different. What is the maximum number of girls that it is possible for there to be in the class?
In the king’s prison, there are five cells numbered from 1 to 5. In each cell, there is one prisoner. Kristen persuaded the king to conduct an experiment: on the wall of each cell she writes at one point a number and at midnight, each prisoner will go to the cell with the indicated number (if the number on the wall coincides with the cell number, the prisoner does not go anywhere). On the following night at midnight, the prisoners again must move from their cell to another cell according to the instructions on the wall, and they do this for five nights. If the location of prisoners in the cells for all six days (including the first) is never repeated, then Kristen will be given the title of Wisdom, and the prisoners will be released. Help Kristen write numbers in the cells.