In an ordinary set of dominoes, there are 28 tiles. How many tiles would a set of dominoes contain if the values indicated on the tiles did not range from 0 to 6, but from 0 to 12?
A square piece of paper is cut into 6 pieces, each of which is a convex polygon. 5 of the pieces are lost, leaving only one piece in the form of a regular octagon (see the drawing). Is it possible to reconstruct the original square using just this information?
Given an endless piece of chequered paper with a cell side equal to one. The distance between two cells is the length of the shortest path parallel to cell lines from one cell to the other (it is considered the path of the center of a rook). What is the smallest number of colors to paint the board (each cell is painted with one color), so that two cells, located at a distance of 6, are always painted with different colors?
We are given 101 rectangles with integer-length sides that do not exceed 100.
Prove that amongst them there will be three rectangles \(A, B, C\), which will fit completely inside one another so that \(A \subset B \subset C\).
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.
On the grid paper, Theresa drew a rectangle \(199 \times 991\) with all sides on the grid lines and vertices on intersection of grid lines. How many cells of the grid paper are crossed by a diagonal of this rectangle?
The school cafeteria offers three varieties of pancakes and five different toppings. How many different pancakes with toppings can Emmanuel order? He has to have exactly one topping on each pancake.
How many six-digit numbers are there whose digits all have the same parity? That is, either all six digits are even, or all six digits are odd.
Donald’s sister Maggie goes to a nursery. One day the teacher at the nursery asked Maggie and the other children to stand a circle. When Maggie came home she told Donald that it was very funny that in the circle every child held hands with either two girls or two boys. Given that there were five boys standing in the circle, how many girls were standing in the circle?