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Answer :
To determine which board sizes allow for a Knight's Tour, we consider some known results and facts about the problem:
1. 8x8 Board:
- A Knight's Tour is possible on an 8x8 chessboard.
- The Knight's Tour is a classic problem that works on the standard chessboard size of 8x8.
2. 5x5 Board:
- A Knight's Tour is also possible on a 5x5 chessboard.
- This is because a knight can move sufficiently on a 5x5 board to cover all squares without getting trapped.
3. 4x4 Board:
- A Knight's Tour is generally NOT possible on a 4x4 chessboard.
- The 4x4 board is too small for the knight to move around freely without getting stuck.
4. 3x6 Board:
- A Knight's Tour is NOT possible on a 3x6 chessboard.
- The long and narrow shape limits the knight's movement, making it impossible to cover all squares.
Thus, among the given board sizes, only the 5x5 and 8x8 boards allow for a Knight's Tour.
In summary, the board sizes that allow for a Knight's Tour are:
- 8x8
- 5x5
In contrast, the board sizes that do not allow for a Knight's Tour are:
- 4x4
- 3x6
So, for the given options:
- 4x4 - Not possible
- 5x5 - Possible
- 3x6 - Not possible
- 8x8 - Possible
1. 8x8 Board:
- A Knight's Tour is possible on an 8x8 chessboard.
- The Knight's Tour is a classic problem that works on the standard chessboard size of 8x8.
2. 5x5 Board:
- A Knight's Tour is also possible on a 5x5 chessboard.
- This is because a knight can move sufficiently on a 5x5 board to cover all squares without getting trapped.
3. 4x4 Board:
- A Knight's Tour is generally NOT possible on a 4x4 chessboard.
- The 4x4 board is too small for the knight to move around freely without getting stuck.
4. 3x6 Board:
- A Knight's Tour is NOT possible on a 3x6 chessboard.
- The long and narrow shape limits the knight's movement, making it impossible to cover all squares.
Thus, among the given board sizes, only the 5x5 and 8x8 boards allow for a Knight's Tour.
In summary, the board sizes that allow for a Knight's Tour are:
- 8x8
- 5x5
In contrast, the board sizes that do not allow for a Knight's Tour are:
- 4x4
- 3x6
So, for the given options:
- 4x4 - Not possible
- 5x5 - Possible
- 3x6 - Not possible
- 8x8 - Possible
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