Questions tagged [checkerboard]

A puzzle involving checkerboards: grids of squares alternating black and white in color, most commonly an 8x8 board.

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16
votes
3answers
720 views

Placing seven point-sized pawns

Can you place seven (point-sized) pawns on a $7\times7$ checkerboard, so that every pawn is placed precisely in the middle of one of the little checkerboard squares, and all distances between pairs ...
6
votes
2answers
154 views

Swapping knights

On a $3\times3$ grid we have: with $8$ moves needed to swap the red and blue knights. What is the minimum numbers of moves to swap the knights on a $4\times4$ grid?
11
votes
2answers
1k views

Professor Halfbrain and the 99x99 chessboard (Part 2)

This puzzle is the continuation of "Professor Halfbrain and the 99x99 chessboard (Part 1)". The difference is that "four corners of a rectangle" has now become "$2\times2$ subsquare". Professor ...
12
votes
3answers
2k views

Professor Halfbrain and the 99x99 chessboard (Part 1)

Professor Halfbrain has spent the last weekend with filling the squares of a $99\times99$ chessboard with real numbers from the interval $[-1,+1]$. Whenever four squares form the corners of a ...
41
votes
1answer
3k views

Crippled King Crossing a Canyon

A chess king has been injured in battle against an evil wizard, and can no longer move northeast or southwest. This king is on the North rim of a canyon, and must flee to safety on the South rim. ...
6
votes
3answers
487 views

Crosses on a 10x11 chessboard

A cross consists of six unit squares and looks as follows: O OOOO O (this may also be rotated by several right turns, or horizontally ...
9
votes
5answers
1k views

Professor Halfbrain and the 9x9 chessboard (Part 2)

This puzzle is the continuation of "Professor Halfbrain and the 9x9 chessboard (Part 1)". The difference is that "distance at least $2$" has now become "distance more than $2$". Professor Halfbrain ...
10
votes
3answers
961 views

Professor Halfbrain and the 9x9 chessboard (Part 1)

Professor Halfbrain has spent has spent the last few days with placing pawns on a $9\times9$ chessboard; each of the $81$ squares on the chessboard had side length $1$. Halfbrain always started with ...
25
votes
2answers
2k views

Rooks on a 15x15 chessboard

On a 15x15 chessboard there are 15 rooks that do not attack each other (via ordinary rook moves). Then each of the rooks makes one move like that of a knight. Is it possible that after all this is ...
19
votes
1answer
1k views

Dominos on a checkerboard

What's the maximal number of dominos (2x1 tiles) that can be placed on a checkerboard (8x8 square) so that every domino covers exactly 2 squares of the checkerboard and no two dominos form a 2x2 ...
20
votes
3answers
680 views

Concentrating tokens on an infinite board

One token is placed on each square of an infinite checkerboard. One square is marked with an X. You want to get as many tokens on the marked square as possible. To do this, you may make any finite ...
7
votes
4answers
288 views

Tiling an Odd Polygon with Dominoes

There is a polygon whose edge lengths are all odd integers. Prove that this polygon's interior cannot be tiled by dominoes whose dimensions are $1\times 2$. An example of such a polygon is a "...
20
votes
2answers
1k views

Desegregate the Knights

You are given a 3 by 3 chessboard with a knight on each corner, where the knights in the top row are black and in the bottom row are white. On each turn, you may move a knight of either color (the ...
29
votes
5answers
3k views

Switch The Knights

On a small $ 4 \times 3$ chessboard, the top row is filled with black knights and the bottom row with white knights. On each move, you may move one knight (as it moves in chess) to an unoccupied ...
9
votes
2answers
499 views

Trapping pieces on an infinite chessboard

Alice and Bob are playing an unusual game of chess. We begin with a piece on a square of an infinite chessboard. On each of her turns, Alice moves the piece. On each of his turns, Bob destroys a ...
10
votes
1answer
629 views

Let's revisit this later (a two player knight's game)

Alice and Bob play a game on the following irregular chessboard. (Note the blacked out squares are not legal moves.) Alice starts the game by placing a knight on any square she chooses. They then ...
3
votes
1answer
122 views

Pawn-free squares

The chessboard below contains two pawns. How many squares (of any size) does not contain a pawn?
28
votes
3answers
4k views

A lonely pawn on the chessboard

Alice and Bob and the referee Conny play the following game with a pawn on a standard $8\times8$ chessboard: In the beginning, Conny places the pawn into the center of a randomly chosen square. All $...
5
votes
2answers
719 views

A prime number game

Alice and Bob play the following game on a $9\times11$ chessboard. First Alice divides the chessboard into $33$ smaller rectangles of dimensions $1\times3$ and $3\times1$. Then Bob labels each of ...
12
votes
3answers
2k views

The knight's game

Alice and Bob play the following game on a standard $8\times8$ chessboard. In the very beginning, Alice picks a square on the chessboard and places a knight on this square. Then Bob and Alice ...
2
votes
1answer
272 views

Professor Halfbrain's chessboard dissection theorem

Professor Halfbrain has spent another weekend on analyzing and dissecting chessboards. His efforts resulted in the following surpring theorem. Professor Halfbrain's chessboard dissection theorem: ...
3
votes
1answer
120 views

Keep the grid differences below $x$

Inspired by Keep the chessboard differences below eight Given an $n\times m$ grid of squares, what is the smallest possible integer bound $x$ for which it is possible to fill the squares with the ...
7
votes
2answers
261 views

Keep the chessboard differences below eight

Can one place the numbers $1-64$ in the squares of a chessboard so that every two squares that share an edge have a difference of at most seven?
8
votes
5answers
645 views

Professor Halfbrain's infinite chessboard theorems

After his first set of theorems involving queen-accessibility, Professor Halfbrain started wondering about infinite chessboards rather than just ones with arbitrary large dimensions. He came up with ...
8
votes
3answers
1k views

The Erasmus chessboard theorem

Professor Erasmus informed me today that he has proved another fascinating theorem about chessboards. Consider a standard $8\times8$ chessboard with 32 black and 32 white squares. Pick an ...
13
votes
4answers
1k views

Professor Halfbrain's chessboard theorems

Professor Halfbrain has spent his last weekend with analyzing $n\times n$ chessboards. Halfbrain says that a subset $S$ of squares on such a chessboard is queen-connected, if a chess queen can move ...
6
votes
1answer
654 views

3D Chess Cube - Queen Puzzle

The basis of this puzzle, is based on a common 2D chess puzzle, where you try to fit 8 queens on a chessboard without the queens being able to attack each other. Extend this concept into a 3D cube (...
5
votes
2answers
272 views

Checkerboard City versus the ACDWNPP

The Association for the Construction, Development and Wellbeing of Nuclear Power Plants (or simply ACDWNPP for short) plans to build a new nuclear power plant in Checkerboard City. As you all surely ...
3
votes
3answers
531 views

Checkerboard tour

An caterpillar hatches on one corner of a checkerboard. At every second, it moves from one square to another adjacent (sharing a side) one. However, it can not return to any cell it has visited before ...
23
votes
7answers
5k views

Explore the square with 100 hops

This is a puzzle that was a fad when I was back in school. (It's not sooo long ago, but way before Smartphones with AngryBirds or DoodleJump came up...). For quite a while, everybody was scribbling ...
10
votes
2answers
488 views

Maze Solving Robot, More Difficult Variant

You are trying to design a maze solving robot, capable of solving "valid" mazes. A valid maze is an 8 by 8 checkerboard, surrounded by walls, where Between each pair of orthogonally adjacent squares, ...
21
votes
4answers
1k views

Checkerboard Infection: The Aftermath

As in this previous puzzle, we are again playing with our favorite bacteria, C. Coli, whose natural habitat is an 8x8 checkers board. These are very nasty bacteria - once they are in a cell of a ...
19
votes
4answers
3k views

Checkerboard Infection

There is a dangerous bacteria, Checkerichia Coli, (or C. Coli for short) which lives in the squares of checkerboards. Squares with this bacteria living in their digestive systems are said to have "C-...
25
votes
4answers
2k views

Black wants to go first!

Billy and Matthew decide to play a game of Chess. They live far away, so they decide to do it online. Matthew wants to play first, but the game randomly gave him the color black (who goes second.) ...
25
votes
4answers
2k views

Lions and Zebras on a Chess Board

The black knight is our lion chasing 8 white bishop zebras which can't capture. Can the zebras evade the lion forever, if team zebra positions all the pieces and has the first turn? Rules: The game ...
11
votes
2answers
347 views

Combinatorial Agriculture

You have just acquired a $64$ acre farm, in the shape of a square, and divided into an eight by eight array of one acre subplots. You have $21$ crops to plant. Each crop requires its own $3$ acre plot ...
5
votes
2answers
223 views

Patriotic Solitaire

You are given a $3\times n$ checkerboard, covered with $n$ red, $n$ white, and $n$ blue checkers. Call a board patriotic if every column has a red, white and blue checker. You want to make the given ...
6
votes
1answer
396 views

Checkers Against The King

The king has invented a new game called King's Checkers. The rules are simple, it's exactly like regular Checkers, the only difference? The king plays first and all his pieces are automatically "king",...
12
votes
2answers
2k views

Checkers with the devil

After however much inordinate time passed for the tree-pruning game to finish, you ended up winning, and Satan was infuriated. He was sure he'd come up with a game keep you in hell. But once again, ...
6
votes
1answer
717 views

2 bishops versus a lone king

There are 4 pieces on the board, the white and black kings, as well as 2 white bishops (on differently colored squares). The only way a stalemate occurs is if the black king is not in check, but ...
9
votes
3answers
5k views

Tiling a Chessboard with tetrominos

Is it possible to tile a $10\times10$ chessboard with (non-overlapping) T-tetrominos? If so, how? If not, prove it's impossible. Bonus: Which Tetris pieces can used to tile a 10$\times$10 ...
25
votes
4answers
1k views

Arrows on a Chessboard

I've taken an $n$ by $n$ chessboard and drawn an arrow on each square, pointing in one of the eight compass directions. I've done this in such a way that arrows in (orthogonally) adjacent squares ...
5
votes
2answers
579 views

Pedro's pawn game

Yesterday, my good friend Pedro told me about Pedro's pawn game which is played with $7$ white and $7$ black pawns on a $7\times7$ checkerboard. In the starting situation, there is a white pawn on ...
8
votes
2answers
671 views

Dominoes on a chessboard

Mary has a box with special $2\times1$ dominoes. Each dominoe has two red corners and two blue corners, and these dominoes come in two different types: The first type has the lower left and the upper ...
4
votes
5answers
1k views

Coin flipping game

An $8\times8$ checkerboard is filled with two-sided coins (that are blue on one side and red on the other side). The following picture shows three examples of a cross (multiplication sign): the five ...
37
votes
3answers
5k views

The coolest checkerboard magic trick

In the small town of Terni (Italy), there's a couple of young friends named Marco and Leonardo, who like to perform magic tricks to a restricted audience of common friends and relatives. They like to ...
3
votes
3answers
950 views

Knights on a 5x5 chess board

What is the maximum number of knights that can be positioned on a $5\times5$ chess board, so that each knight attacks exactly two other knights?
6
votes
2answers
716 views

Block the snake

On a $9\times10$ checkerboard with $90$ squares, a game proceeds as follows: First, you place $x$ rocks on the board. Each rock occupies a single square. No rock may touch the edge of the ...
11
votes
1answer
1k views

Two rooks for Bobby Fischer

Bobby Fischer liked to play the following game on a standard $8\times8$ chessboard: In his first step, Bobby placed a white rook and a black rook somewhere on the chessboard (on two different ...
12
votes
2answers
567 views

Polyominoes on a checkerboard

Professor Halfbrain has spent his entire weekend by cutting lots of wooden $50\times50$ checkerboards into lots of polyominoes. He looked at various pattern polyominoes with area $49$, and always ...