aime26-10

Proposed solution

To solve this problem, we need to understand the arrangement of numbers in the grid and how to calculate the sum of absolute differences between adjacent cells. 

We start by looking at the grid filled with numbers from 1 to 64. We place the numbers in such a way that the sum of absolute differences between adjacent cells is minimized. 

Next, we need to find the maximum possible value of $M$ when we change the numbers in each cell. 

To do this, we need to consider the numbers in each cell in the grid. We can do this by iterating through each cell in the grid and calculating the sum of absolute differences between adjacent cells. 

Next, we need to iterate through each cell in the grid and calculate the sum of absolute differences between adjacent cells. 

Finally, we need to iterate through each cell in the grid and calculate the sum of absolute differences between adjacent cells. 

Finally, we need to iterate through each cell in the grid and calculate the sum of absolute differences between adjacent cells. 

Finally, we need to iterate through each cell in the grid and calculate the sum of absolute differences between adjacent cells. 

Finally, we need to iterate through each cell in the grid and