A young mathematician had quite an odd dream last night. In his dream he was a knight on a \(4\times4\) board. Moreover, he was moving like a knight moves on the usual chessboard. In the morning he could not remember what was actually happening in his dream, though the young mathematician is pretty sure that either
(a) he has passed exactly once through all the cells of the board except for the one at the bottom leftmost corner, or
(b) he has passed exactly once through all the cells of the board.
For each possibility examine if it could happen or not.
A \(7 \times 7\) square was tiled using \(1 \times 3\) rectangular blocks. One of the squares has not been covered. Which one can it be?
Ten pairwise distinct non-zero numbers are such that for each two of them either the sum of these numbers or their product is a rational number.
Prove that the squares of all numbers are rational.
On every cell of a \(9 \times 9\) board there is a beetle. At the sound of a whistle, every beetle crawls onto one of the diagonally neighbouring cells. Note that, in some cells, there may be more than one beetle, and some cells will be unoccupied.
Prove that there will be at least 9 unoccupied cells.
A city in the shape of a triangle is divided into 16 triangular blocks, at the intersection of any two streets is a square (there are 15 squares in the city). A tourist began to walk around the city from a certain square and travelled along some route to some other square, whilst visiting every square exactly once. Prove that in the process of travelling the tourist at least 4 times turned by \(120^{\circ}\).
What is the minimum number of \(1\times 1\) squares that need to be drawn in order to get an image of a \(25\times 25\) square divided into 625 smaller 1x1 squares?
A grasshopper can make jumps of 8, 9 and 10 cells in any direction on a strip of \(n\) cells. We will call the natural number \(n\) jumpable if the grasshopper can, starting from some cell, bypass the entire strip, having visited each cell exactly once. Find at least one \(n > 50\) that is not jumpable.
Picasso got a new set of crayons. He started colouring various things. First, Picasso coloured a line under the following condition: each point on a line is coloured either red or blue. Show that there are three different points \(A,B,C\) on the line of the same colour such that \(AB = BC\).
Monet has coloured a plane. He wants to colour the entire plane in such a way that each straight line can only have points of three or fewer different colours. Show that he can use however many different colours he wants and still be able to achieve this goal.
Frida colours the plane. She decided to use \(9\) different colours, but she wants to colour the entire infinite plane in such a way that if two points are distance \(1\) apart, then they must be different colours. Show that this is always possible.