Found problems: 189
A chessboard is covered with $32$ dominos. Each domino covers two adjacent squares. Show that the number of horizontal dominos with a white square on the left equals the number with a white square on the right.
On a chessboard (with $64$ squares) there are situated $32$ white and $32$ black pools. We say that two pools form a mixed pair when they are with different colors and they lie on the same row or column. Find the maximum and the minimum of the mixed pairs for all possible situations of the pools.
[i]K. Dochev[/i]
(a) Decide whether the fields of the $8 \times 8$ chessboard can be numbered by the numbers $1, 2, \dots , 64$ in such a way that the sum of the four numbers in each of its parts of one of the forms
[list][img]http://www.artofproblemsolving.com/Forum/download/file.php?id=28446[/img][/list]
is divisible by four.
(b) Solve the analogous problem for
[list][img]http://www.artofproblemsolving.com/Forum/download/file.php?id=28447[/img][/list]
What is the smallest number of squares of a chess board that can be marked in such a manner that
(a) no two marked squares may have a common side or a common vertex, and
(b) any unmarked square has a common side or a common vertex with at least one marked square?
Indicate a specific configuration of marked squares satisfying (a) and (b) and show that a lesser number of marked squares will not suffice.
(A. Andjans, Riga)
Let $n$ be a natural number. Given a chessboard sized $m \times n$. The sides of the small squares of chessboard are not on the perimeter of the chessboard will be colored so that each small square has exactly two sides colored. Prove that a coloring like that is possible if and only if $m \cdot n$ is even.
A number is written in each square of a chessboard, so that each number not on the border is the mean of the $4$ neighboring numbers. Show that if the largest number is $N$, then there is a number equal to $N$ in the border squares.
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.
Sixteen points are placed in the centers of a $4 \times 4$ chess table in the following way:
• • • •
• • • •
• • • •
• • • •
(a) Prove that one may choose $6$ points such that no isoceles triangle can be drawn with the vertices at these points.
(b) Prove that one cannot choose $7$ points with the above property.
$16$ pieces of the shape $1\times 3$ are placed on a $7\times 7$ chessboard, each of which is exactly three fields. One field remains free. Find all possible positions of this field.
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]
Chess piece called [i]skew knight[/i], if placed on the black square, attacks all the gray squares.
[img]https://i.ibb.co/HdTDNjN/Kyiv-MO-2021-Round-1-11-2.png[/img]
What is the largest number of such knights that can be placed on the $8\times 8$ chessboard without them attacking each other?
[i]Proposed by Arsenii Nikolaiev[/i]
The numbers from $1$ to $64$ must be written on the small squares of a chessboard, with a different number in each small square. Consider the $112$ numbers you can make by adding the numbers in two small squares which have a common edge. Is it possible to write the numbers in the squares so that these $112$ sums are all different?
Sixteen squares of an $8\times 8$ chessboard are chosen so that there are exactly lwo in each row and two in each column. Prove that eight white pawns and eight black pawns can be placed on these sixteen squares so that there is one white pawn and one black pawn in each row and in cach colunm.
Let $n>1$ be a positive integer. Each unit square in an $n\times n$ grid of squares is colored either black or white,
such that the following conditions hold:
$\bullet$ Any two black squares can be connected by a sequence of black squares where every two consecutive squares in the sequence share an edge;
$\bullet$ Any two white squares can be connected by a sequence of white squares where every two consecutive squares in the sequence share an edge;
$\bullet$ Any $2\times 2$ subgrid contains at least one square of each color.
Determine, with proof, the maximum possible difference between the number of black squares and white squares in this grid (in terms of $n$).
For which positive integers $n$ is it possible to cover a $(2n+1) \times (2n+1)$ chessboard which has one of its corner squares cut out with tiles shown in the figure (each tile covers exactly $4$ squares, tiles can be rotated and turned around)?
[img]https://cdn.artofproblemsolving.com/attachments/6/5/8fddeefc226ee0c02353a1fc11e48ce42d8436.png[/img]
On a $300 \times 300$ board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook. At what least $k$, it is possible to state that there is at least one rook in each $k\times k$ square ?
Consider arrangements of the numbers $1$ through $64$ on the squares of an $8\times 8$ chess board, where each square contains exactly one number and each number appears exactly once.
A number in such an arrangement is called super-plus-good, if it is the largest number in its row and at the same time the smallest number in its column. Prove or disprove each of the following statements:
(a) Each such arrangement contains at least one super-plus-good number.
(b) Each such arrangement contains at most one super-plus-good number.
Proposed by Gerhard J. Woeginger
On a chessboard with dimensions 1000 by 1000 and squares colored in the usual way in white and black, there is a set A consisting of 1000 squares. Any two fields of set A can be connected by a sequence of fields of set A so that subsequent fields have a common side. Prove that there are at least 250 white fields in set A.
We visit all squares exactly once on a $n\times n$ chessboard (colored in the usual way) with a king. Find the smallest number of times we had to switch colors during our walk.
[i]Proposed by Dömötör Pálvölgyi, Budapest[/i]
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]
Two players play a game on an $8$ by $8$ chessboard according to the following rules. The first player places a knight on the board. Then each player in turn moves the knight , but cannot place it on a square where it has been before. The player who has no move loses. Who wins in an errorless game , the first player or the second one? (The knight moves are the normal ones. )
(V . Zudilin , year 12 student , Beltsy (Moldova))
Eight pawns are placed on a chessboard, so that there is one in each row and column. Show that an even number of the pawns are on black squares.
Consider a chessboard $6 \times 6$, made up of $36$ single squares. We want to place $6$ chess rooks on this board, one rook on each square, so that there are no two rooks on the same row, nor two rooks on the same column. Note that, once the rooks have been placed in this way, we have that, for every square where a rook has not been placed, there is a rook in the same row as it and a rook in the same column as it. We will say that such rooks are in line with this square.
For each of those $30$ houses without rooks, color it green if the two rooks aligned with that same house are the same distance from it, and color it yellow otherwise. For example, when we place the $6$ rooks ($T$) as below, we have:
(a) Is it possible to place the rooks so that there are $30$ green squares?
(b) Is it possible to place the rooks so that there are $30$ yellow squares?
(c) Is it possible to place the rooks so that there are $15$ green and $15$ yellow squares?
The $64$ cells of an $8 \times 8$ chessboard have $64$ different colours. A Knight stays in one cell. In each move, the Knight jumps from one cell to another cell (the $2$ cells on the diagonal of an $2 \times 3$ board) also the colours of the $2$ cells interchange. In the end, the Knight goes to a cell having common side with the cell it stays at first. Can it happen that: there are exactly $3$ cells having the colours different from the original colours?
Some bishops and knights are placed on an infinite chessboard, where each square has side length $1$ unit. Suppose that the following conditions hold:
[list]
[*] For each bishop, there exists a knight on the same diagonal as that bishop (there may be another piece between the bishop and the knight).
[*] For each knight, there exists a bishop that is exactly $\sqrt{5}$ units away from it.
[*] If any piece is removed from the board, then at least one of the above conditions is no longer satisfied.
[/list]
If $n$ is the total number of pieces on the board, find all possible values of $n$.
[i]Sheldon Kieren Tan[/i]