[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?

Source text last modified Thu Apr 23 12:11:12 1998.

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