A crippled rook can move on a chessboard in the following way: from a square, it can move to an adjacent square sharing a common side, and every two consecutive moves must be at right angles (i.e., the rook makes a 90∘turn at every move).
A cycle is a sequence of squares which start and end at the same square, and traces out a valid path that the crippled rook can move according to the rules above. A non-intersecting cycle consists of pairwise distinct squares, with the sole exception of the starting and ending square.
What is the length of the longest possible cyclic, non-intersecting route of a crippled rook on a 15×15 chessboard?
To find the length of the longest possible cyclic, non-intersecting route of a crippled rook on a 15x15 chessboard, we can use a little bit of logic and pattern recognition.
First, let's think about the possible movements of the crippled rook. It can move to an adjacent square sharing a common side, making a 90-degree turn at each move. This means that it cannot move diagonally, and it cannot backtrack on its previous moves.
Now, let's start exploring the chessboard. If we start at any given square, we can see that we have four possible directions to choose from (N, E, S, W). Let's assume we choose the North direction.
Once we move North, our only option is to turn either East or West. Let's assume we turn East. From this new square, we can only move South or East. If we choose to move South, we can then only move West or South. We continue this pattern until we reach the starting square again.
Following this pattern, we can see that each direction (N, E, S, W) is chosen exactly four times, forming a cycle. Each movement creates a cycle of length 4.
Now, let's think about how many of these cycles we can fit on a 15x15 chessboard. Since each cycle has a length of 4, we need to divide the total number of squares on the chessboard by 4 to find the maximum number of cycles.
A 15x15 chessboard has a total of 15 * 15 = 225 squares. Dividing 225 by 4 gives us 56 with a remainder of 1. This means that we can fit 56 cycles of length 4 on the chessboard, with 1 additional square remaining.
To maximize the length of the cyclic, non-intersecting route, we need to connect the additional square to the existing cycles. We can do this by choosing any of the four possible directions (N, E, S, W) and connecting it to the last square of one of the existing cycles.
Therefore, the length of the longest possible cyclic, non-intersecting route of a crippled rook on a 15x15 chessboard is 56 * 4 + 1 = 225.
Now, let's summarize the steps to get to this answer:
1. Understand the movement restrictions of the crippled rook.
2. Recognize the pattern of movement and cycles.
3. Determine the maximum number of cycles based on the number of squares on the chessboard.
4. Find the remaining square and connect it to an existing cycle to maximize the route length.
I hope this explanation helps you understand how to approach and solve this problem!