Chessboard Patterns

[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

• move one square South-East and then one square North-East
• move one square North-East and then one square South-East
• move two squares South-East, one square North-East, and then one square North-West.
• ...

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

• move two squares right and then one square down
• move one square down and then two squares right
• move five squares right, one square down, then three squares left
• move four squares down, three squares right, two squares left, three squares up, and then one square right
• ...

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?