Prove that \(n^2 + 1\) is not divisible by \(3\) for any natural \(n\).
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?
The board has the form of a cross, which is obtained if corner boxes of a square board of \(4 \times 4\) are erased. Is it possible to go around it with the help of the knight chess piece and return to the original cell, having visited all the cells exactly once?
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?
Can there be exactly 100 roads in a state in which three roads leave each city?