Fill the free cells of the “hexagon” (see the figure) with integers from 1 to 19 so that in all vertical and diagonal rows the sum of the numbers, in the same row, is the same.
Six chess players participated in a tournament. Each two participants of the tournament played one game against each other. How many games were played? How many games did each participant play? How many points did the chess players collect all together?
Is it possible to fill a \(5 \times 5\) table with numbers so that the sum of the numbers in each row is positive and the sum of the numbers in each column is negative?
Find out the principles by which the numbers are depicted in the tables (shown in the figures below) and insert the missing number into the first table, and remove the extra number from the second table.
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?
Two play a game on a chessboard \(8 \times 8\). The player who makes the first move puts a knight on the board. Then they take turns moving it (according to the usual rules), whilst you can not put the knight on a cell which he already visited. The loser is one who has nowhere to go. Who wins with the right strategy – the first player or his partner?
What is the minimum number of squares that need to be marked on a chessboard, so that:
1) There are no horizontally, vertically, or diagonally adjacent marked squares.
2) Adding any single new marked square breaks rule 1.
Initially, on each cell of a \(1 \times n\) board a checker is placed. The first move allows you to move any checker onto an adjacent cell (one of the two, if the checker is not on the edge), so that a column of two pieces is formed. Then one can move each column in any direction by as many cells as there are checkers in it (within the board); if the column is on a non-empty cell, it is placed on a column standing there and unites with it. Prove that in \(n - 1\) moves you can collect all of the checkers on one square.