Best Answer bonanova, 11 April 2014 - 11:05 AM

I interpret the question to ask the number * m* of additional and unvisited squares that

are within reach of a **knight** after *n* moves. Summing these numbers gives the number

of squares a knight can reach with *n* moves or fewer. I do not double-count any squares.

Note that interpreting * m* as an incremental quantity will make it linear in

*n.*

The partial sums will be quadratic in *n*.

Here are the squares that are first reached for *n*=1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 moves, color coded.

It's easier to count the moves by looking at even and odd *n*, respectively:

The numbers of dots in the colored regions are 8, 32, 68, 96, 120, 148, 176, 204, 232, and 260.

This discloses the *general* result that the number of additional squares increases by 28 with each move.

The *anomalous* values of ** m** are

** m**(2) = 32 instead of 36, dark green

** m**(3) = 68 instead of 64, dark blue, and

* m*(4) = 96 instead of 92, orange.

These are the moves numbers for which the knight has unvisited squares back toward the original square.

For larger numbers, concentric regions for even and odd moves do not intermingle.

For these values of *n*,

* m*(

*n*) = 4(7

*n*-5).

Again, this is the number of squares that move *n* adds to the total.

Repeating the solution for a **bishop**:

A bishop on an infinite board can reach every square of its color in two moves.

The cardinality of that set of squares is aleph null.

That is, the black (or white) squares on an infinite chessboard are countable.