Three children wanted to make a table-game. For that purpose they wished to enumerate the $mn$ squares of an $m \times n$ game-board by the numbers $1, ... ,mn$ in such way that the numbers $1$ and $mn$ lie in the corners of the board and the squares with successive numbers have a common edge. The children agreed to place the initial square (with number $1$) in one of the corners but each child wanted to have the final square (with number $mn$ ) in different corner. For which numbers $m$ and $n$ is it possible to satisfy the wish of any of the children?

Initially we have a $2 \times 2$ table with at least one grain of wheat on each cell. In each step we may perform one of the following two kinds of moves: $i.$ If there is at least one grain on every cell of a row, we can take away one grain from each cell in that row. $ii.$ We can double the number of grains on each cell of an arbitrary column. a) Show that it is possible to reach the empty table using the above moves, starting from the position down below. b) Show that it is possible to reach the empty table from any starting position. c) Prove that the same is true for the $8 \times 8$ tables as well.

Given some $m\times n$ table, and some numbers in its fields. You are allowed to change the sign in one row or one column simultaneously. Prove that you can obtain a table, with nonnegative sums over each row and over each column.

Given a natural number $k>1$. Find the smallest number $\alpha$ satisfying the following condition. Suppose that the table $(2k + 1) \times (2k + 1)$ is filled with real numbers not exceeding $1$ in absolute value, and the sums of the numbers in all lines are equal to zero. Then you can rearrange the numbers so that each number remains in its row and all the sums over the columns will be at most $\alpha$.

A grid measuring $10 \times 11$ is given. How many "crosses" covering five unit squares can be placed on the grid? (pictured right) so that no two of them cover the same square? [img]https://cdn.artofproblemsolving.com/attachments/a/7/8a5944233785d960f6670e34ca7c90080f0bd6.png[/img]

In the cells of a square table $n$ x $n$ the numbers $1,2,...,n^2$ are written in an arbitrary way. Prove that there exist two adjacent cells, for which the difference between the numbers written in them is no lesser than $n$.

Consider a triangular table of positive integers \[ \begin{matrix} & & & a_{11} & a_{12} & a_{13} & & & \\ & & a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & & \\ & a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} & a_{37} & \\ \iddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{matrix} \] The first row consists of odd numbers only. For $i>1,j\ge1$ we have \[a_{ij}=a_{i-1,j-2}+a_{i-1,j-1}+a_{i-1,j}.\] If we get out of range with the second index, we consider such $a$ to be zero (eg. $a_{22}=0+a_{11}+a_{12}$ and $a_{37}=a_{25}+0+0$). Show that for every $i>1$ there is $j\in\{1,\ldots,2i+1\}$ such that $a_{ij}$ is even.

We are given a rectangular table with a positive integer written in each of its cells. For each cell of the table, the number in it is equal to the total number of different values in the cells that are in the same row or column (including itself). Find all tables with this property.

Take a $n \times n$ chess page.Determine the $n$ such that we can put the numbers $1,2,3, \ldots ,n$ in the squares of the page such that we know the following two conditions are true: a) for each row we know all the numbers $1,2,3, \ldots ,n$ have appeared on it and the numbers that are in the black squares of that row have the same sum as the sum of the numbers in the white squares of that row. b) for each column we know all the numbers $1,2,3, \ldots ,n$ have appeared on it and the numbers that are in the black squares in that column have the same sum as the sum of the numbers in the white squares of that column.

a) We are playing the following game on this table: In each move we select a row or a column of the table, reduce two neighboring numbers in that row or column by $1$ and increase the third one by $1$. After some of these moves can we get to a table with all the same entries? b) This time we have the choice to arrange the integers from $1$ to $9$ in the$ 3 \times3$ table. Still using the same moves now our aim is to create a table with all the same entries, maximising the value of the entries. What is the highest possible number we can achieve?

We will call one of the cells of a rectangle 11 x 13 “[i]peculiar[/i]” , if after removing it the remaining figure can be cut into squares 2 x 2 and 3 x 3. How many of the 143 cells are “[i]peculiar[/i]”?

Let $x$ and $y$ be odd positive integers. A table $x$ x $y$ is given in which the squares with coordinates $(2,1)$, $(x - 2, y)$, and $(x, y)$ are cut. The remaining part of the table is covered in dominoes and squares [b]2 x 2[/b]. Prove that the dominoes in a valid covering of the table are at least $\frac{3}{2}(x+y)-6$

For each natural $n \ge 4$, find the smallest natural number $k$ that satisfies following condition: For an arbitrary arrangement of $k$ chips of two colors on $n\times n$ board, there exists a non-empty set such that all columns and rows contain even number ($0$ is also possible) of chips each color.

We may permute the rows and the columns of the table below. How may different tables can we generate? 1 2 3 4 5 6 7 7 1 2 3 4 5 6 6 7 1 2 3 4 5 5 6 7 1 2 3 4 4 5 6 7 1 2 3 3 4 5 6 7 1 2 2 3 4 5 6 7 1

In each square of the $100\times 100$ square table, type $1, 2$, or $3$. Consider all subtables $m \times n$, where $m = 2$ and $n = 2$. A subtable will be called [i]balanced [/i] if it has in its corner boxes of four identical numbers boxes . For as large a number $k$ prove, that we can always find $k$ balanced subtables, of which no two overlap, i.e. do not have a common box.

Each of the cells of an $m \times n$ table is coloured either black or white. For each cell, the total number of the cells which are in the same row or in the same column and of the same colour as this cell is strictly less than the total number of the cells which are in the same row or in the same column and of the other colour as this cell. Prove that in each row and in each column the number of white cells is the same as the number of black ones. (A Shapovalov)

Consider the main diagonal and the cells above it in an \( n \times n \) grid. These cells form what we call a tabular triangle of length \( n \). We want to place a real number in each cell of a tabular triangle of length \( n \) such that for each cell, the sum of the numbers in the cells in the same row and the same column (including itself) is zero. For example, the sum of the cells marked with a circle is zero. It is known that the number in the topmost and leftmost cell is $1.$ Find all possible ways to fill the remaining cells.

Alice and Brian are playing a game on a $1\times (N + 2)$ board. To start the game, Alice places a checker on any of the $N$ interior squares. In each move, Brian chooses a positive integer $n$. Alice must move the checker to the $n$-th square on the left or the right of its current position. If the checker moves off the board, Alice wins. If it lands on either of the end squares, Brian wins. If it lands on another interior square, the game proceeds to the next move. For which values of $N$ does Brian have a strategy which allows him to win the game in a finite number of moves?

Let $n$ be a positive integer and let $t$ be an integer. $n$ distinct integers are written on a table. Bob, sitting in a room nearby, wants to know whether there exist some of these numbers such that their sum is equal to $t$. Alice is standing in front of the table and she wants to help him. At the beginning, she tells him only the initial sum of all numbers on the table. After that, in every move he says one of the $4$ sentences: $i.$ Is there a number on the table equal to $k$? $ii.$ If a number $k$ exists on the table, erase him. $iii.$ If a number $k$ does not exist on the table, add him. $iv.$ Do the numbers written on the table can be arranged in two sets with equal sum of elements? On these questions Alice answers yes or no, and the operations he says to her she does (if it is possible) and does not tell him did she do it. Prove that in less than $3n$ moves, Bob can find out whether there exist numbers initially written on the board such that their sum is equal to $t$

Given is a table 4x4 and in every square there is 0 or 1. In a move we choose row or column and we change the numbers there. Call the square "zero" if we cannot decrease the number of zeroes in it. Call "degree of the square" the number zeroes in a "zero" square. Find all possible values of the degree.

Prove that it is impossible to fill the cells of an $8 \times 8$ table with the numbers from $ 1$ to $64$ (each number must be used once) so that for each $2\times 2$ square, the difference between products of the numbers on it’s diagonals will be equal to $ 1$.

Let $n$ be a positive integer. In how many ways can an $n \times n$ table be filled with integers from $0$ to $5$ such that a) the sum of each row is divisible by $2$ and the sum of each column is divisible by $3$ b) the sum of each row is divisible by $2$, the sum of each column is divisible by $3$ and the sum of each of the two diagonals is divisible by $6$?

Find all positive integers $n$ for which a square[b][i] n x n[/i][/b] can be covered with rectangles [b][i]k x 1[/i][/b] and one square [b][i]1 x 1[/i][/b] when: a) $k = 4$ b) $k = 8$

A square is divided into 169 identical small squares and in every small square is written 0 or 1. It isn’t allowed in one row or column to have the following arrangements of adjacent digits in this order: 101, 111 or 1001. What is the the biggest possible number of 1’s in the table?

A magician and his assistant perform in front of an audience of many people. On the stage there is an $8$×$8$ board, the magician blindfolds himself, and then the assistant goes inviting people from the public to write down the numbers $1, 2, 3, 4, . . . , 64$ in the boxes they want (one number per box) until completing the $64$ numbers. After the assistant covers two adyacent boxes, at her choice. Finally, the magician removes his blindfold and has to $“guess”$ what number is in each square that the assistant. Explain how they put together this trick. $Clarification:$ Two squares are adjacent if they have a common side