Puzzled Knight
A knight is standing at one corner of an
chessboard and it has to reach the opposite corner.The minimum number of steps required by him to do so is 6.
Find the number of paths by which it can reach the other end in 6 moves.
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1 solution
Note: this solution works from top left to bottom right. The number of squares the knight needs to go through to reach the opposite square is 14 (7 horizontal, 7 vertical). 6 moves can go a maximum of 18 squares (2 squares one direction, 1 square in another), so the moves go backwards a total of 2 squares. There are two different ways to do that. One way is for one knight move to go up 2 squares, right 1 square, or left 2 squares, down 1 square. We can just work on that case first and worry about the other case later.
The one knight move going 2 steps back is symmetrical along the diagonal, meaning that every way we find with one of the knight moves going backwards 2 steps can be done with the other knight move going backwards 2 steps, and every other move is also flipped along the diagonal. We can just work with the knight move going 2 squares up, one square right, and the number of ways with that set of knight moves can be multiplied by 2.
If one knight move is 2U1R (I'll abbreviate stuff from now on), then there must be four knight moves with 2D1R and one with 1D2R to get to the opposite corner. The total number of ways to rearrange those moves is 6*5=30, but some of them go outside the board and don't work. The knight move 2U1R cannot be first or last, or else the knight would've went outside the board, or started outside the board to reach the opposite corner. There are 10 ways (in total for both cases) to rearrange the other moves so 30-10 = 20 cases work from the remaining. If the first or last move is 1D2R, the second or second to last move (respectively) can't be 2U1R for the same reasons. The other 4 moves for both of those cases are all the same, so there is an additional 2 set of moves that don't work. 20-2=18. You can verify that there aren't any other cases that don't work.
So far, we have counted 18 ways, so multiplying that by 2 gives 36. Going on the the alternate case, there must be 2 distinct knight moves that go backwards one step. Testing this case out, the knight moves must have two 1D2R, two 2D1R, one 1U2R, and one 2D1L. You can show that this case has at least one set of moves that work, and it's unnecessary to show what it equals (and I'm lazy) since there's only one choice with an answer greater than 36 so the answer is 48.