My mum once told me the following story: she was walking home late at night after sitting in the pub with her friends. She was then surrounded by a group of unfriendly looking people. They demanded: “money or your life?!” She was forced to give them her purse. She valued her life more, since she was pregnant with me at that time. According to her story she gave them two purses and two coins. Moreover, she claimed that one purse contained twice as many coins as the other purse. Immediately, I thought that the mum must have made a mistake or could not recall the details because of the shock and the amount of time that passed after that moment. But then I figured out how this could be possible. Can you?
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?
Is it possible to find a way of arranging numbers from \(0\) to \(9\) in a row so that each pair of consecutive numbers adds up to a multiple of \(5\), \(7\), or \(13\)?
Each pair of cities in Wonderland is connected by a flight operated by "Wonderland Airlines". How many cities are there in the country if there are \(105\) different flights? We count a flight from city \(A\) to city \(B\) as the same as city \(B\) to city \(A\) - i.e. the pair \(A\) to \(B\) and \(B\) to \(A\) counts as one flight.
(a) In a regular 10-gon we draw all possible diagonals. How many line segments are drawn? How many diagonals?
(b) Same questions for a regular 100-gon.
(c) Same questions for an arbitrary convex 100-gon.
A hedge fund is intending to buy 50 computers and connect each of them with eight other computers with a cable. Please do not ask why they need to do that, that is a top secret never to be made public! A friend of mine said that it’s related to some cryptocurrency research, but you should immediately forget all I just told you; it would be unwise to spread rumours! Let’s go back to the mathematical part of this story and stop the unrelated talk. The question is, how many cables do they need?
At a party there are people dressed in either blue or green. Every person dressed in blue had a chance to dance with exactly \(7\) people in green, only once with each one. On the other hand, every person in green danced exactly with \(9\) people in blue, also only once with each. Were there more people dressed in blue or in green at the party?