2026 AIME I Problem 11

The integers from 1 to 64 are placed in some order into an 8\times 8 grid of cells with one number in each cell. Let a_{i,j} be the number placed in the cell in row i and column j, and let M be the sum of the absolute differences between adjacent cells. That is, M=\sum^{8}_{i=1}\sum^{7}_{j=1}\left(|a_{i,j+1}-a_{i,j}|+|a_{j+1,i}-a_{j,i}|\right). Find the remainder when the maximum possible value of M is divided by 1000.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.