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?
Can you draw 9 line segments in such a way that each segment crosses exactly 3 other segments?
There are one hundred natural numbers, they are all different, and sum up to 5050. Can you find those numbers? Are they unique, or is there another bunch of such numbers?
Can you pay 20 p in seven coins?
Can you pay 20 p in seven coins using 1p and 5p coins only?
Can you pay 25 p in eight coins using 1p and 5p coins only?
(a) Show that it is impossible to find five odd numbers which all add to 100.
(b) Alice wrote several odd numbers on a piece of paper. The Hatter did not see the numbers, but says that if he knew how many numbers Alice wrote down, than he would say with certainty if the sum of the numbers is even or odd. How can he do it?