aime26-10

Proposed solution

I need to find the maximum possible value of the sum of absolute differences between adjacent cells in an 8×8 grid containing numbers 1 through 64.

Let me first understand what we're maximizing:
$$M = \sum_{i=1}^8 \sum_{j=1}^7 (|a_{i,j+1} - a_{i,j}| + |a_{j+1,i} - a_{j,i}|)$$

This sum includes:
- Horizontal adjacent pairs: $|a_{i,j+1} - a_{i,j}|$ for each row $i$ and column $j$
- Vertical adjacent pairs: $|a_{j+1,i} - a_{j,i}|$ for each column $i$ and row $j$

Let me count how many terms there are:
- Horizontal pairs: 8 rows × 7 columns = 56 terms
- Vertical pairs: 7 rows × 8 columns = 56 terms
- Total: 112 terms

To maximize the sum of absolute differences, I want to maximize each individual term $|x - y|$ where $x$ and $y$ are adjacent numbers.

The maximum value of $|x - y|$ for any two distinct numbers from $\{1, 2, \ldots, 64\}$ is $63$ (achieved by $|64 - 1| = 63$).

However, I need to be more careful. Each number can only appear once, and the arrangement must be such that we can achieve maximum differences.

Let me think about this systematically. 

The key insight is that to maximize $M$, I want to arrange numbers so that adjacent cells have numbers that are as far apart as possible in value.

One approach is to consider a "snake" pattern or similar, but let me think more carefully.

Actually, let me reconsider what we're summing:
$$M = \sum_{i=1}^8 \sum_{j=1}^7 |a_{i,j+1} - a_{i,j}| + \sum_{j=1}^7 \sum_{i=1}^8 |a_{j+1,i} - a_{j,i}|$$

This is the sum of absolute differences between all horizontally and vertically adjacent pairs.

To maximize this, I should try to make adjacent numbers as far apart as possible in value.

The maximum possible sum would occur if we could arrange the numbers so that each adjacent pair has difference 63, but that's impossible since we'd need to place 1 and 64 next to each other repeatedly.

Let me think about this more systematically. 

The total number of adjacent pairs is:
- Horizontal adjacent pairs: 8 rows × 7 columns = 56 pairs
- Vertical adjacent pairs: 7 rows × 8 columns = 56 pairs  
- Total: 112 pairs

For each pair, the maximum absolute difference is 63.

But we can't achieve all differences of 63 because that would require placing numbers 1 and 64 next to each other in a way that's impossible given the constraints.

Let me think about this differently. 

The key insight is to consider how many times each number contributes to the sum. Each number appears in at most 4 adjacent pairs (if it's not on an edge).

Actually, let me reconsider what the maximum could be by thinking about the extreme arrangement.

If I arrange numbers so that the largest and smallest numbers are adjacent as much as possible, I can get close to optimal.

But there's a more elegant approach. 

Let's consider that we want to maximize:
$$\sum_{\text{adjacent pairs}} |x - y|$$

The maximum value is achieved by placing numbers in an order that maximizes this sum.

One way to think about it: if I arrange the numbers 1,2,...,64 in a sequence such that adjacent numbers have maximum possible difference, and then arrange them in the grid accordingly.

But there's a simpler approach. 

The maximum value of $M$ is achieved by arranging the numbers so that we maximize the sum of absolute differences between adjacent cells.

Let me think about this as a graph problem. We want to arrange 1,2,...,64 on an 8×8 grid such that the sum of absolute differences along edges is maximized.

The maximum possible value occurs when we can make as many pairs as possible have difference 63.

But let's approach this more carefully by considering what happens if we try to place numbers optimally.

Actually, let me think about it this way:
- We have 112 adjacent pairs
- Each pair contributes at most 63 to the sum
- But we can't achieve 63 for all pairs

The maximum is achieved when we arrange the numbers in a pattern that maximizes the sum.

By careful analysis (which involves arranging numbers optimally), the maximum value of $M$ is:
$$