Peter marks several cells on a $5\times 5$ board. Basil wins if he can cover all marked cells with three-cell corners. The corners must be inside the board and not overlap. What is the least number of cells Peter should mark to prevent Basil from winning? (Cells of the corners must coincide with the cells of the board).

In a $3n \times 3n$ grid, each square is either black or red. Each red square not on the edge of the grid has exactly five black squares among its eight neighbor squares.. On every black square that not at the edge of the grid, there are exactly four reds in the adjacent squares box. How many black and how many red squares are in the grid?

A table consists of $17 \times 17$ squares. In each square one positive integer from $1$ to $17$ is written, every such number is written in exactly $17$ squares. Prove that there is a row or a column of the table that contains at least $5$ different numbers.

Consider a $10 \times 10$ grid. On every move, we colour $4$ unit squares that lie in the intersection of some two rows and two columns. A move is allowed if at least one of the $4$ squares is previously uncoloured. What is the largest possible number of moves that can be taken to colour the whole grid?

Find the greatest number of depicted pieces composed of $4$ unit squares that can be placed without overlapping on an $n \times n$ grid (where n is a positive integer) in such a way that it is possible to move from some corner to the opposite corner via uncovered squares (moving between squares requires a common edge). The shapes can be rotated and reflected. [img]https://cdn.artofproblemsolving.com/attachments/b/d/f2978a24fdd737edfafa5927a8d2129eb586ee.png[/img]

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 $6$?

Each of the $16$ squares in a $4 \times 4$ table contains a number. For any square, the sum of the numbers in the squares sharing a common side with the chosen square is equal to $1$. Determine the sum of all $16$ numbers in the table. (R Zhenodarov)

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 $6$?

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$.

The square table $n\times n$ is filled by integers. If the fields have common side, the difference of numbers in them doesn't exceed $1$. Prove that some number is encountered not less than a) not less than $[n/2]$ times ($[ ...]$ mean the whole part), b) not less than $n$ times.

From a squared sheet of paper of size $29 \times 29, 99$ pieces, each a $2\times 2$ square, are cut off (all cutting is along the lines bounding the squares). Prove that at least one more piece of size $2\times 2$ may be cut from the remaining part of the sheet. (S Fomin, Leningrad)

(a) On each square of a squared sheet of paper of size $20 \times 20$ there is a soldier. Vanya chooses a number $d$ and Petya moves the soldiers to new squares in such a way that each soldier is moved through a distance of at least $d$ (the distance being measured between the centres of the initial and the new squares) and each square is occupied by exactly one soldier. For which $d$ is this possible? (Give the maximum possible $d$, prove that it is possible to move the soldiers through distances not less than $d$ and prove that there is no greater $d$ for which this procedure may be carried out.) (b) Answer the same question as (a), but with a sheet of size $21 \times 21$. (SS Krotov, Moscow)

Every cell of a $4\times 4$ square grid net, has $1\times 1$ size. Is it possible to represent this net as a union of the following sets: a) Eight broken lines of length five each? b) Five broken lines of length eight each?

Initially, all the squares of an $8\times 8$ grid are white. You start by choosing one of the squares and coloring it gray. After that, you may color additional squares gray one at a time, but you may only color a square gray if it has exactly $1$ or $3$ gray neighbors at that moment (where a neighbor is a square sharing an edge). For example, the configuration below (of a smaller $3\times 4$ grid) shows a situation where six squares have been colored gray so far. The squares that can be colored at the next step are marked with a dot. Is it possible to color all the squares gray? Justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/1/c/d50ab269f481e4e516dace06a991e6b37f2a85.png[/img]

Each field of the $1987\times 1987$ board is filled with numbers, which absolute value is not greater than one. The sum of all the numbers in every $2\times 2$ square equals $0$. Prove that the sum of all the numbers is not greater than $1987$.

A minus sign is placed on one square of a $5 \times 5$ board and plus signs are placed on the remaining squares. A move is to select a $2 \times 2, 3 \times 3, 4 \times 4$ or $5 \times 5$ square and change all the signs in it. Which initial positions allow a series of moves to change all the signs to plus?

a) A square table with $49$ small squares is filled with numbers $1$ to $7$ so that in each row and in each column all numbers from $1$ to $7$ are present. Let the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers $1, 2, 3, . . . , 7$ are present. b) A square table with $n^2$ small squares is filled with numbers $1$ to $n$ so that in each row and in each column all numbers from $1$ to $n$ are present. Let $n$ be odd and the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers $1, 2, 3, . . . , n$ are present.

A $5\times 5$ board is covered by eight hooks (a three unit square figure, shown in the picture) so that one unit square remains free. Determine all squares of the board that can remain free after such covering. [img]https://cdn.artofproblemsolving.com/attachments/6/8/a8c4e47ba137b904bd28c01c1d2cb765824e6a.png[/img]

a) The fields of the square table $4\times 4$ are filled with the "+" or "-" signs. You are allowed to change the signs simultaneously in the whole row, column, or diagonal to the opposite sign. That means, for example, that You can change the sign in the corner square, because it makes a diagonal itself. Prove that you will never manage to obtain a table containing pluses only. b) The fields of the square table $8\times 8$ are filled with the "+" or signs except one non-corner field with "-". You are allowed to change the signs simultaneously in the whole row, column, or diagonal to the opposite sign. That means, for example, that You can change the sign in the corner field, because it makes a diagonal itself. Prove that you will never manage to obtain a table containing pluses only.

The "Sea battle" game. a) You are trying to find the $4$-field ship -- a rectangle $1x4$, situated on the $7x7$ playing board. You are allowed to ask a question, whether it occupies the particular field or not. How many questions is it necessary to ask to find that ship surely? b) The same question, but the ship is a connected (i.e. its fields have common sides) set of $4$ fields.

Every unit square of a $n \times n$ board is colored either red or blue so that among all 2 $\times 2$ squares on this board all possible colorings of $2 \times 2$ squares with these two colors are represented (colorings obtained from each other by rotation and reflection are considered different). a) Find the least possible value of $n$. b) For the least possible value of $n$ find the least possible number of red unit squares

In an $8 \times 8$ grid, $n$ disks, numbered $1$ to $n$ are stacked, with random order, in a pile in the bottom left comer. The disks can be moved one at a time to a neighbouring cell either to the right or top. The aim to move all the disks to the cell at the top right comer and stack them in the order $1,2,...,n$ from the bottom. Each cell, except the bottom left and top right cell, can have at most one disk at any given time. Find the largest value of $n$ so that the aim can be achieved.

Given a table $4\times 4$. a) Find, how $7$ stars can be put in its fields in such a way, that erasing of two arbitrary lines and two columns will always leave at list one of the stars. b) Prove that if there are less than $7$ stars, You can always find two columns and two rows, such, that if you erase them, no star will remain in the table.

The figure shows a figure of $5$ unit squares, a Greek cross. What is the largest number of Greek crosses that can be placed on a grid of dimensions $8 \times 8$ without any overlaps, with each unit square covering just one square in a grid?

a) The numbers $1, 2, . . . , 49$ are arranged in a square table as follows: [img]https://cdn.artofproblemsolving.com/attachments/5/0/c2e350a6ad0ebb8c728affe0ebb70783baf913.png[/img] Among these numbers we select an arbitrary number and delete from the table the row and the column which contain this number. We do the same with the remaining table of $36$ numbers, etc., $7$ times. Find the sum of the numbers selected. b) The numbers $1, 2, . . . , k^2$ are arranged in a square table as follows: [img]https://cdn.artofproblemsolving.com/attachments/2/d/28d60518952c3acddc303e427483211c42cd4a.png[/img] Among these numbers we select an arbitrary number and delete from the table the row and the column which contain this number. We do the same with the remaining table of $(k - 1)^2$ numbers, etc., $k$ times. Find the sum of the numbers selected.