\(N\) young men and \(N\) young ladies gathered on the dance floor. How many ways can they split into pairs in order to participate in the next dance?
The Russian Chess Championship is made up of one round. How many games are played if 18 chess players participate?
Prove that the product of any three consecutive natural numbers is divisible by 6.
Between the nine planets of the solar system, a cosmic messaging system is introduced. Rockets fly along the following routes: Earth – Mercury, Pluto – Venus, Earth – Pluto, Pluto – Mercury, Mercury – Venus, Uranus – Neptune, Neptune – Saturn, Saturn – Jupiter, Jupiter – Mars and Mars – Uranus. Is it possible to get from Earth to Mars?
There are 9 cities in the country Number with the names 1, 2, 3, 4, 5, 6, 7, 8, 9. The traveller discovered that two cities are connected by an airline if and only if a two-digit number made up of the digit-names of these cities, is divisible by 3. Is it possible to get from city 1 to city 9?
In a city, there are 15 telephones. Can I connect them with wires so that each phone is connected exactly with five others?
In a state there are 100 cities, and from each of them there are 4 roads. How many roads are there in the state?
There are 30 people in the class. Can it be that 9 of them have 3 friends (in this class), 11 have 4 friends, and 10 have 5 friends?
In the city Smallville there are 15 telephones. Can they be connected by wires so that there are four phones, each of which is connected to three others, eight phones, each of which is connected to six, and three phones, each of which is connected to five others?
A king divided his kingdom into 19 counties who are governed by 19 lords. Could it be that each lord’s county has one, five or nine neighbouring counties?