Problems

Solve the rebus \(AC \times CC \times K = 2002\) (different letters correspond to different integers and vice versa).

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?

Giuseppe has a sheet of plywood, measuring \(22 \times 15\). Giuseppe wants to cut out as many rectangular blocks of size \(3 \times 5\) as possible. How should he do it?

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 – an unsorted 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 none of the signs on the bags are true. Then Cinderella took out only one single grain from one sack and, looking at it, immediately worked out what was in each bag. How did she do it?