Problems

Can the equality \(K \times O \times T = U \times W \times E \times N \times H \times Y\) be true if the numbers from 1 to 9 are substituted for the letters? Different letters correspond to different numbers.

Deep in a forest there is a small town of talking animals. Elephant, Crocodile, Rabbit, Monkey, Bear, Heron and Fox are friends. They each have a landline telephone and each two telephones are connected by a wire. How many wires were required?

In a room, there are 85 balloons – red and blue ones. It is known that: 1) at least one of the balls is red, 2) from each arbitrarily chosen pair of balls at least one is blue. How many red balls are there in the room?

Liz is 8 years older than Natasha. Two years ago Liz’s age was 3 times greater than Natasha’s. How old is Liz?

When Gulliver came to Lilliput, he found that there all things were exactly 12 times shorter than in his homeland. Can you say how many Lilliputian matchboxes fit into one of Gulliver’s matchboxes?

There are 6 locked suitcases and 6 keys to them. At the same time, it is not known to which suitcase each key fits. What is the smallest number of attempts you need to make in order to open all the suitcases for sure? And how many attempts will it take there are not 6 but 10 keys and suitcases?

Decipher the following rebus (see the figure). Despite the fact that only two figures are known here, and all others are replaced by asterisks, the example can be restored.

The parliament of a certain country has two houses with an equal number of members. In order to make a decision on an important issue all the members voted and there were no abstentions. When the chairman announced that the decision had been taken with a 23-vote advantage, the opposition leader declared that the results had been rigged. How did he know it?

Among some number of mathematicians, every seventh is a philosopher, and among some number of philosophers every ninth is a mathematician. Who are there more of: philosophers or mathematicians?

Find numbers equal to twice the sum of their digits.