Problems
A new airline "Capitals Direct" has direct flights operating on the following routes (in both directions): Paris - London, Paris - Lisbon, Rome - London, Rome - Madrid, Berlin - Helsinki, Berlin - Amsterdam, Amsterdam - Prague. These are the only flights that the company offers. I would like to travel from Paris to Amsterdam. I cannot buy a direct ticket for sure, but can "Capitals Direct" offer me a connecting flight?
Can one arrange numbers from \(1\) to \(9\) in a row so that each pair of consecutive numbers forms a two-digit multiple of \(7\) or a multiple of \(13\)?
In the Royal Grammar School all Year \(9\) students were gathered in the Queen’s Hall for an important announcement. They have been waiting for it for a while and everyone had enough time to greet every other student with a handshake. Assuming there are \(100\) Year \(9\) students at the school, how many handshakes were made before the announcement?
In \(2149\) a regular transport connection between nine planets of the Solar System was introduced. Space capsules are flying between the following pairs of planets: 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 travel from Earth to Mars by using this type of transport with possible changes at other planets?
A graph is called Bipartite if it is possible to split all its vertices into two groups in such a way that there are no edges connecting vertices from the same group. Find out whic of the following graphs are bipartite and which are not:
Thirteen boys and girls met to play a football match. Eleven of them shook hands with everybody else in the group. The last two shook hands with everybody else but not each other, because they were siblings and arrived together. How many handshakes took place?
The city of Konigsberg has seven bridges as depicted on the layout below. \[\includegraphics[scale=0.5]{https://problems-static.s3.amazonaws.com/production/task_images/2829/WSP-000146.png}\] Is it possible for the great mathematician Leonard Euler to have an excursion in Konigsberg visiting all islands and land banks, but crossing each bridge exactly once?
There are \(6\) people at a party. Each two people either know each other or not, and the knowledge goes both ways: if \(A\) knows \(B\), then \(B\) knows \(A\). Show that there either is a trio of people who all know each other or a trio of people who all don’t know each other.
An elven village in the woods has \(11\) treehouses. We want to link the houses by ropes so that each house is connected to exactly \(4\) others. How many ropes do we need?
Can you draw a house like the one below without lifting your pen from the paper, nor going over the same edge twice? \[\includegraphics[scale=0.5]{https://problems-static.s3.amazonaws.com/production/task_images/2833/WSP-000149.png}\]