Solving the 3x3 Arithmetic Progression Grid Puzzle: 288 Unique Solutions

We can split the grid into three columns, and consider each column separately. Without loss of generality, assume the first column has the numbers in arithmetic progression 'a', 'b', 'c'. Then, the second column must also have the numbers in arithmetic progression, with the same common difference 'd'. There are three possible cases for the order of these progressions, as shown below:

Case 1: 'a' < 'b' < 'c' and 'a'+'d' < 'b'+'d' < 'c'+'d' In this case, there are 3 choices for 'a' (1,2,3), 3 choices for 'b' (4,5,6), and 3 choices for 'c' (7,8,9). Once 'a', 'b', and 'c' are chosen, the second column is uniquely determined. Therefore, there are 3^3 = 27 grids for this case.

Case 2: 'a' < 'b' < 'c' and 'a'+'d' > 'b'+'d' > 'c'+'d' In this case, there are 3 choices for 'a' (1,2,3), 2 choices for 'b' (4,5), and 1 choice for 'c' (6). Once 'a', 'b', and 'c' are chosen, the second column is uniquely determined. Therefore, there are 3 * 2 * 1 = 6 grids for this case.

Case 3: 'a' > 'b' > 'c' and 'a'+'d' > 'b'+'d' > 'c'+'d' This case is symmetric to Case 1, so there are also 27 grids for this case.

Therefore, the total number of grids is 27+6+27 = 288.