Q1. Dividing the 8 x 8 chessboard into identical individual squares that do not intersect and continue for as long as possible, how many lines do you need? Can a generic solution be found for an n x n board?
Example: a 2 x 2 board requires three lines. First one divides A1 and B1 from A2 and B2 (or A1 and A2 from B1 and B2), the second A1 from B1, the third A2 from B2.
Q2. Dividing the 8x8 board into rectangles of different shapes or sizes, keeping the individual squares intact, what is the largest number of rectangles possible? Can a generic solution be found for an n x n board?
Example: a 3 x 3 board can be split into three rectangles with different dimentions, either 2x3, 1x2 and 1x1, or 2x2, 1x3, 1x2.
Q1.. each fence/cut adds 1 to the total number of separate areas. from 1 to 64 that's 63 cuts/fences. from one to n^2, n^2 - 1.
Q2. not using a 1x1 area, the next smallest eleven add up to 64 and can easily be placed so they cover a chessboard. A generic solution for an n x n board needs more work.