Found problems: 189
The nonnegative integer $n$ and $ (2n + 1) \times (2n + 1)$ chessboard with squares colored alternatively black and white are given. For every natural number $m$ with $1 < m < 2n+1$, an $m \times m$ square of the given chessboard that has more than half of its area colored in black, is called a $B$-square. If the given chessboard is a $B$-square, fi nd in terms of $n$ the total number of $B$-squares of this chessboard.
The squares of an $8 \times 8$ board are coloured alternatingly black and white. A rectangle consisting of some of the squares of the board is called [i]important[/i] if its sides are parallel to the sides of the board and all its corner squares are coloured black. The side lengths can be anything from $1$ to $8$ squares. On each of the $64$ squares of the board, we write the number of important rectangles in which it is contained. The sum of the numbers on the black squares is $B$, and the sum of the numbers on the white squares is $W$. Determine the difference $B - W$.
We consider the division of a chess board $8 \times 8$ in p disjoint rectangles which satisfy the conditions:
[b]a)[/b] every rectangle is formed from a number of full squares (not partial) from the 64 and the number of white squares is equal to the number of black squares.
[b]b)[/b] the numbers $\ a_{1}, \ldots, a_{p}$ of white squares from $p$ rectangles satisfy $a_1, , \ldots, a_p.$ Find the greatest value of $p$ for which there exists such a division and then for that value of $p,$ all the sequences $a_{1}, \ldots, a_{p}$ for which we can have such a division.
[color=#008000]Moderator says: see [url]https://artofproblemsolving.com/community/c6h58591[/url][/color]
Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?
2500 chess kings have to be placed on a $100 \times 100$ chessboard so that
[b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);
[b](ii)[/b] each row and each column contains exactly 25 kings.
Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.)
[i]Proposed by Sergei Berlov, Russia[/i]
Determine the smallest positive integer $k{}$ satisfying the following condition: For any configuration of chess queens on a $100 \times 100$ chequered board, the queens can be coloured one of $k$ colours so that no two queens of the same colour attack each other.
[i]Russia, Sergei Avgustinovich and Dmitry Khramtsov[/i]
Place eight rooks on a standard $8 \times 8$ chessboard so that no two are in the same row or column. With the standard rules of chess, this means that no two rooks are attacking each other. Now paint $27$ of the remaining squares (not currently occupied by rooks) red. Prove that no matter how the rooks are arranged and which set of $27$ squares are painted, it is always possible to move some or all of the rooks so that:
• All the rooks are still on unpainted squares.
• The rooks are still not attacking each other (no two are in the same row or same column).
• At least one formerly empty square now has a rook on it; that is, the rooks are not on the same $8$ squares as before.
Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?
Consider an infinite grid $\mathcal G$ of unit square cells. A [i]chessboard polygon[/i] is a simple polygon (i.e. not self-intersecting) whose sides lie along the gridlines of $\mathcal G$.
Nikolai chooses a chessboard polygon $F$ and challenges you to paint some cells of $\mathcal G$ green, such that any chessboard polygon congruent to $F$ has at least $1$ green cell but at most $2020$ green cells. Can Nikolai choose $F$ to make your job impossible?
[i]Nikolai Beluhov[/i]
Is it possible to cover an $n \times n$ chessboard which has its center square cut out with tiles shown in the picture (each tile covers exactly $4$ squares, tiles can be rotated and turned around) if
a) $n = 5$,
b) $n = 2003$?
[img]https://cdn.artofproblemsolving.com/attachments/6/5/8fddeefc226ee0c02353a1fc11e48ce42d8436.png[/img]
A "labyrinth" is an $8 \times 8$ chessboard with walls between some neighboring squares. If a rook can traverse the entire board without jumping over the walls, the labyrinth is "good" ; otherwise it is "bad" . Are there more good labyrinths or bad labyrinths?
(A Shapovalov)
In a $4\times 4$ chessboard, in how many ways can you place $3$ rooks and one bishop such that none of these pieces threaten another piece?
For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition.
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
A natural number is written in each square of an $ m \times n$ chess board. The allowed move is to add an integer $ k$ to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.