How many ways can a rook get from one corner of a chess board to the opposite corner?

A rook can move 1 or 8 spaces. Please explain your reasoning. (using a chess board with 64 squares)|||Additional details needed:


1. Are you asking for the number of paths, or are you asking for the number of possible move-lists? (The rook can get to the opposite corner in 2 moves by moving to the intermediate corner and then to the destination corner -- but it could take exactly the same path by moving one square at a time, which would take 14 moves.)


2. Do you really mean "1 or 8 spaces"? I play chess, and the rook can move, from one corner, any number from 1 to 7 spaces. (Note that it moves 7 spaces maximum). In fact, at any point in time, if there are no other pieces on the board, the rook always has 14 legal moves.





I'm going to guess that for #1 you mean how many paths, and for #2 you really mean "1 or 8 spaces" -- so that the rook would never move 3 squares at a time.





It turns out that measuring the paths simplifies things, because you can just assume that the rook moves one square at a time. (A move of N squares is equivalent to N single-square moves in the same direction, from the point of view of the path taken.) The problem then resembles Pascal's Triangle, I believe.





By hand I got the answer, which is 3432. See the attached image. They don't go exactly to the depth you want, but the value 3432 is calculated by adding 1716 + 1716.





Hope that helps!|||A chessboard is an 8 by 8 grid. Because the rook begins on one of the squares, it can't actually move 8 spaces without falling off the board -- this is why I'm saying 7 spaces below.





I think the answer you're looking for it 8. My reasoning is If it chose to move 7 then it couldn't move 1 along that side -- and if it choose to move 1 then it couldn't move 7 along that side. With 4 sides and two different ways to do each the answer is 8 different ways.





For example


it would move all 7 then all 7 again


all 7 then 1, 1, 1, 1, 1, 1, 1


1, 1, 1, 1, 1, 1, 1 then all 7


and finally 1, 1, 1, 1, 1, 1, 1, then 1, 1, 1, 1, 1, 1, 1





And the same for the path through the other corner would make 8 different ways.





Hope that answers your question :)