Problems

Can the equality \(K \times O \times T\) = \(U \times W \times E \times H \times S \times L\) be true if instead of the letters in it we substitute integers from 1 to 9 (different letters correspond to different numbers)?

Rebus. Solve the numerical rebus \(AAAA-BBB + SS-K = 1234\) (different letters correspond to different numbers, but the same letters each time correspond to the same numbers)

Multiplication of numbers. Restore the following example of the multiplication of natural numbers if it is known that the sum of the digits of both factors is the same.

Restore the example of the multiplication.

What is the maximum number of kings, that cannot capture each other, which can be placed on a chessboard of size \(8 \times 8\) cells?

Prove that the number of all arrangements of the largest possible amount of peaceful bishops (figures that move on diagonals and don’t threaten each other) on the \(8\times 8\) chessboard is an exact square.

Petya and Misha play such a game. Petya takes in each hand a coin: one – 10 pence, and the other – 15. After that, the contents of the left hand are multiplied by 4, 10, 12 or 26, and the contents of the right hand – by 7, 13, 21 or 35. Then Petya adds the two results and tells Misha the result. Can Misha, knowing this result, determine which hand – the right or left – contains the 10 pence coin?

The evil stepmother, leaving for the ball, gave Cinderella a bag in which rice and cous-cous were mixed, and ordered for them to be sorted. When Cinderella was leaving for the ball, she left three bags: one was rice, the other – cous-cous, and in the third – not yet disassembled mixture. In order not to confuse the bags, Cinderella attached to each of them a sign saying: “Rice”, “Cous-cous” and “Mixture”.

The stepmother returned from the ball first and deliberately swapped all the signs in such a way that on every sack there was an incorrect sign. The Fairy Godmother managed to warn Cinderella that now no sign on bags is true. Then Cinderella took out only one single grain from one sack and, looking at it, immediately guessed what that mixture was. How did she do it?

There is a 12 litre barrel filled with water, and two empty barrels (one is 5 litres in volume and the other is 8 litres). Using these barrels, try to

a) divide the water into 2 parts with volumes: 3 litres and 9 litres;

b) divide the water into 2 parts with equal volume.

In a bookcase, there are four volumes of the collected works of Astrid Lindgren, with each volume containing 200 pages. A worm who lives on this bookshelf has gnawed its way from the first page of the first volume to the last page of the fourth volume. Through how many pages has the worm gnawed its way through?