[A problem by Stephanie Wilcox and Sam Rebelsky, inspired by a problem on p. 10 of Cofman.]
A chessboard is unbounded on all sides. A rook is placed on a square A in the board, and can move horizontally or vertically.
. . . . . . . . . . . . . | . | . | . | . | . --+---+---+---+---+-- . . . | | | | | . . . --+---+---+---+---+-- . . . | | A | | B | . . . --+---+---+---+---+-- . . . | | | | C | . . . --+---+---+---+---+-- . . . | | | | | . . . --+---+---+---+---+-- . | . | . | . | . | . . . . . . . . . . . . .
For each square on the board, there are a number of paths the bishop can take to the square. For example, to get from A to B, the rook might
Obviously, some paths cover fewer squares than others. A shortest path is a path such that covers the fewest number of squares when moving from one square to another. In the example above, the shortest paths are all of length four (including A and C).
For each square, find the number of shortest paths the rook can take from A to that square, and write this number in the square.
How do your answers change if the piece is a bishop? A knight?
Note that the paths are signficantly different. For example, to get from A to C, a rook might
The following part of the question makes no sense to me, given that there is no right-hand corner -- SamR
Investigate the following number pattern. Can you predict what number will be in the right-hand corner ten layers away from the upper left-hand corner? How about 100?