What is the minimum number of knights you need to place on a 10x10 chess board, such that every empty cell is attacked by at least one knight?
Good luck!
Answer
A possible way to cover the board with indicating the positions of the knights with "X" is:
The conclusion is that it is possible to cover the board with 16 knights.
Next I will prove that with 15 knights it is not possible.
This is a 3x3 area/board inside of a 7x7 board. If a knight is placed on any of the squares of the middle part, the number how many squares are covered by it inside the middle part is written on its square. Any single knight can cover up to 3 squares in this 3x3 area. However, 3 knights cannot cover all 9 of the central squares because a knight in any of the 8 positions that cover 3 squares cannot cover the center square. The conclusion is that at least 4 knights are needed to cover any 3x3 area/board.
Lastly, consider this arrangement.
This is already a 10x10 board. A minimum of 4 knights are needed to cover each of the four 3x3 lettered areas. Since a knight cannot cover squares in two different lettered areas a total of at least 16 knights are required. In case of a 9x9 board there would be at least one knight that could cover one or more squares in two different lettered areas.
The conclusion:
It has been shown that it is possible to cover the 10x10 board with 16 knights. It also has been shown that at least 16 knights are required to cover the board, therefore it is the minimum number.
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