There are six natural numbers, all different, which sum up to 22. Can you find those numbers? Are they unique, or is there another bunch of such numbers?
In how many ways can you rearrange the numbers 1, 2, ..., 100 so the neighbouring numbers differ by not more than 1?
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?
Alice finally decided to do some arithmetic. She took four different integer numbers, calculated their pairwise sums and products, and the results ( the pairwise sums and products) wrote down in her wonderful book. What could be the smallest number of different numbers Alice wrote in her book?
Alice wants to arrange the numbers from \(1\) to \(16\) so that any two neighbors add up to a square number. The Hatter claims this can be done in a line but not in a circle. Show that he is right.
Show that \(\frac{x}{y} + {\frac{y}{z}} + {\frac{z}{x}} = 1\) is not solvable in natural numbers.
There are six cities in Wonderland. Her Majesty’s principal secretary of state for transport has a plan of building six new railways. The only condition for these railways is that each of them joins some pair of cities having other four cities equally distributed on both sides of a line containing the segment of the railway. Is it possible to implement such a plan for some configuration of cities?
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.
Draw \(6\) points on a plane and join some of them with edges so that every point is joined with exactly \(4\) other points.