Robinson Crusoe’s friend Friday was looking at \(3\)-digit numbers with the same first and third digits. He soon noticed that such number is divisible by \(7\) if the sum of the second and the third digits is divisible by \(7\). Prove that he was right.
2016 digits are written in a circle. It is known, that if you make a number reading the digits clockwise, starting from some particular place, then the resulting 2016-digit number is divisible by 27. Show that if you start from some other place, and moving clockwise make up another 2016-digit number, then this new number is also divisible by 27.
Louise is confident that all her classmates have different number of friends. Is she right?
There are 100 cities all connected by roads. Each city has 6 roads coming in (or going out). How many roads do connect those cities?
There are 15 cities in a country named The Country of Fifteen Cities. The king ordered his main architect to build roads in such a way that each city was connected with other cities by exactly 5 roads, otherwise he would hang the architect. Do you think that the architect can accomplish the task or should he flee that country immediately?
The architect decided to flee The Country of 15 Cities and began to travel around the world. He arrived to a country, where every city had exactly 3 roads going to and from it. Can there be all together 100 roads in that country?
There are 9 cities named City 1, City 2, City 3, …, and City 9 in a country named The Country of the Nine Cities. Two cities are connected by a road only if the sum of the numbers made up by their names is divisible by 3. Can our travelling architect reach City 9 by starting his journey from City 1 and travelling along those roads?
Show that among any 6 people there are always either 3 people who all know each other or 3 total strangers.
Show that the number of people who ever lived and made an odd number of handshakes is even.
Is it possible to trace the lines in the figures below in such a way that you trace each line only once?