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?
John, coming back from Disneyland, told me that there are seven islands on the enchanted lake, each of which is lead to by one, three or five bridges. Is it true that at least one of these bridges necessarily leads to the shore of the lake?
Prove that the number of people who have ever lived on Earth and who shook hands an odd number of times is even.
Is it possible to draw 9 segments on a plane so that each intersects exactly three others?
In the country Seven there are 15 cities, each of which is connected by roads with no less than seven other cities. Prove that from every city you can get to any other city (possibly passing through other cities).
There is a group of islands connected by bridges so that from each island one can get to any of the other islands. The tourist has bypassed all the islands, walking on each bridge exactly once. He visited the island of Three-isle three times. How many bridges are there on Three-isle if the tourist
a) did not start on it and did not finish on it?
b) started on it, but did not finish on it?
c) started on it and finished on it?
There are three groups of stones: in the first – 10, in the second – 15, in the third – 20. During one turn, you are allowed to split any pile into two smaller ones; the one who cannot make a move loses.
Numbers from 1 to 20 are written in a row. Players take turns placing pluses and minuses between these numbers. After all of the gaps are filled, the result is calculated. If it is even, then the first player wins, if it is odd, then the second player wins. Who won?