65 votes
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Cover 63 squares of a chess board

These should do it: Just to show another example:
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56 votes
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Crippled King Crossing a Canyon

Explanation:
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50 votes

How can 3 queens control the white squares?

I think this arrangement works for the bonus question:
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46 votes
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How can 3 queens control the white squares?

I think this will do it
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41 votes
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How many chess pieces does it take to "cover" all spaces on a chessboard?

Yes. The minimum number of pieces required is 5. 5 queens can be places such that they cover every space on the board, as in the following example: There are 12 such arrangements, along with ...
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38 votes
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A lonely pawn on the chessboard

Strategy: How this works:
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38 votes
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Rooks on a 15x15 chessboard

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37 votes
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Black wants to go first!

Proof: Initially, there are an even number of knights on white squares (namely, there are two of them, at b1 and g8). Every time a knight moves, the number of knights on white squares either ...
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36 votes

Chessboard Rook Problem

There are more ones. Proof:
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34 votes
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How many queens can be on a chessboard without attacking each other?

Because: According to Wolfram-Alpha, there are One possible solution is: A list (and images!) of all
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34 votes
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Dominos on a checkerboard

Looks like: Thanks to @Gamow's comment, this number's maximality can be proved by self-contradiction of the assumption that it is not maximal. Any more dominos would cover all 64 squares. Assumption ...
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34 votes

Two Knights, two Bishops, two Rooks and two Kings on a 4x4 chessboard

I'm not trying to solve the puzzle, I'm just interested in how many solutions there are, since the OP claims he doesn't know. I brute forced it with a program. There are First of all, there are ...
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34 votes
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A chess board with numbers

The smallest sum that can be achieved is Because Reasoning
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32 votes
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16 pawns on a chess board with no three collinear: how do I go about solving this?

The Algoritm is : Then Here is the complete solution from Achim Flammenkamp Ph.D. There are totally 57 solutions
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30 votes
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Mutilated chessboard

I believe this works as a short proof.
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30 votes
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Could you solve a chessboard math puzzle at gunpoint?

Suppose that we have a chessboard with the desired properties. Find the greatest number in each row. Out of these numbers, let the smallest be $m_i$ in row $i$. Find the smallest number in each row....
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29 votes
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Desegregate the Knights

Give these names to all the squares: 163 4 8 725 Each number can only be accessed by way of the numbers before and after it (where 8 wraps around to 1). That ...
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29 votes
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Chessboard Rook Problem

There are Proof: Now I bet the ratio of good to bad is I wrote a little Python program (possibly buggy, so apply some skepticism, but I've tested the bit most likely to have bugs and it seems OK) ...
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29 votes

Chess solitaire: The King's longest walk

First solution - 50 moves Second solution - 59 moves Current solution - 66 moves (beaten by Retudin - 70 moves and Rewan Demontay - 139 moves) Moves:
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27 votes

Switch The Knights

I found a solution that uses 16 moves. After exhaustively checking that there is no solution in 14 moves, I conclude that 16 moves is optimal, because after any odd number of moves the number of ...
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27 votes
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Two Knights, two Bishops, two Rooks and two Kings on a 4x4 chessboard

I hope I didn't make any mistakes: Edit : Replace queens by king
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27 votes
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Playing with one Queen on a chessboard

We can work backwards to figure this out: Conclusion:
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26 votes

The coolest checkerboard magic trick

This indeed is an old puzzle. One possible source (but certainly not the first one) is: Andy Liu: Two Applications of a Hamming Code The College Mathematics Journal 40, (Jan 2009), pp. 2-5 The ...
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25 votes
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The Popular Letter Chessboard

The rules of the question state that: On your final grid, a letter (actually several is also mathematically possible) will be more frequent than all the other letters. Your aim is that this letter ...
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24 votes

How many chess pieces does it take to "cover" all spaces on a chessboard?

This type of chess puzzle is known as a domination problem, and as @Xynariz points out, only five queens are needed for the 8x8 board. It's also interesting to note that five queens are also ...
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24 votes

The Knight's Romp

I have a computer program for solving packing problems, and found a way to use it to solve this problem. One of the solutions it found is below: Note that this is very close to the attempted solution ...
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23 votes
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Two rooks for Bobby Fischer

Answer:
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23 votes

Two Knights, two Bishops, two Rooks and two Kings on a 4x4 chessboard

Here is mine with passive kings, some minutes late :
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22 votes
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Moving a pawn around on a chessboard

No. Every move the pawn makes it switches from a white to a black square or vice versa. Therefore it must either touch an equal number of white and black squares with an even number of moves, or one ...
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