There is a \(5\times 9\) rectangle drawn on squared paper. In the lower left corner of the rectangle is a button. Kevin and Sophie take turns moving the button any number of squares either to the right or up. Kevin goes first. The winner is the one who places the button in upper right corner. Who would win, Kevin or Sophie, by using the right strategy?
Tickets cost 50 cents, and \(2n\) buyers stand in line at a cash register. Half of them have one dollar, the rest – 50 cents. The cashier starts selling tickets without having any money. How many different orders of people can there be in the queue, such that the cashier can always give change?
Prove that the Catalan numbers satisfy the recurrence relationship \(C_n = C_0C_{n-1} + C_1C_{n-2} + \dots + C_{n-1}C_0\). The definition of the Catalan numbers \(C_n\) is given in the handbook.
A \(1 \times 10\) strip is divided into unit squares. The numbers \(1, 2, \dots , 10\) are written into squares. First, the number 1 is written in one square, then the number 2 is written into one of the neighboring squares, then the number 3 is written into one of the neighboring squares of those already occupied, and so on (the choice of the first square is made arbitrarily and the choice of the neighbor at each step). In how many ways can this be done?
A game takes place on a squared \(9 \times 9\) piece of checkered paper. Two players play in turns. The first player puts crosses in empty cells, its partner puts noughts. When all the cells are filled, the number of rows and columns in which there are more crosses than zeros is counted, and is denoted by the number \(K\), and the number of rows and columns in which there are more zeros than crosses is denoted by the number \(H\) (18 rows in total). The difference \(B = K - H\) is considered the winnings of the player who goes first. Find a value of B such that
1) the first player can secure a win of no less than \(B\), no matter how the second player played;
2) the second player can always make it so that the first player will receive no more than \(B\), no matter how he plays.
Tile a \(5\times6\) rectangle in an irreducible way by laying \(1\times2\) rectangles.
Does there exist an irreducible tiling with \(1\times2\) rectangles of
(a) \(4\times 6\) rectangle;
(b) \(6\times 6\) rectangle?
Irreducibly tile a floor with \(1\times2\) tiles in a room that is
(a) \(5\times8\); (b) \(6\times8\).
Having mastered tiling small rooms, Robinson wondered if he could tile big spaces, and possibly very big spaces. He wondered if he could tile the whole plane. He started to study the tiling, which can be continued infinitely in any direction. Can you help him with it?
Tile the whole plane with the following shapes:
Robinson Crusoe was taking seriously the education of Friday, his friend. Friday was very good at maths, and one day he cut 12 nets out of hardened goat skins. He claimed that it was possible to make a cube out of each net. Robinson looked at the patterns, and after some considerable thought decided that he was able to make cubes from all the nets except one. Can you figure out which net cannot make a cube?