aime26-10
0.000
Proposed solution
To find the maximum possible value of \( M \), we need to maximize the sum of the absolute differences between adjacent cells in an \( 8 \times 8 \) grid filled with integers from 1 to 64. The adjacent cells are those that share an edge, so we need to consider both horizontal and vertical differences.
First, let's consider the total number of adjacent pairs in the grid. There are 64 cells in total, and each cell (except those on the last row and last column) has 2 adjacent cells. The total number of horizontal pairs is \( 8 \times 7 = 56 \), and the total number of vertical pairs is also \( 8 \times 7 = 56 \). Thus, the total number of adjacent pairs is \( 56 + 56 = 112 \).
To maximize the sum of the absolute differences, we should place the numbers in such a way that the differences between adjacent cells are as large as possible. One effective strategy is to place the numbers in an alternating high-low pattern. For instance, we can fill the grid as follows:
\[
\begin{array}{cccccccc}
1 & 64 & 2 & 63 & 3 & 62 & 4 & 61 & 5 \\
64 & 1 & 64 & 1 & 64 & 1 & 64 & 1 & 64 \\
2 & 63 & 3 & 62 & 4 & 61 & 5 & 60 & 6 \\
63 & 2 & 63 & 2 & 63 & 2 & 63 & 2 & 63 \\
3 & 62 & 4 & 61 & 5 & 60 & 6 & 7 & 8 \\
62 & 3 & 62 & 3 & 62 & 3 & 62 & 3 & 62 \\
4 & 61 & 5 & 60 & 6 & 7 & 8 & 9 & 10 \\
61 & 4 & 61 & 4 & 61 & 4 & 61 & 4 & 61 \\
5 & 60 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
60 & 5 & 60 & 5 & 60 & 5 & 60 & 5 & 60 \\
\end{array}
\]
In this arrangement, the differences between adjacent cells are maximized. Let's calculate the sum of the absolute differences for this configuration.
### Horizontal Differences
For the first row:
\[
|64 - 1| + |2 - 64| + |3 - 2| + |62 - 63| + |4 - 61| + |5 - 60| + |6 - 59| + |7 - 58| + |8 - 57| + |9 - 56| + |10 - 55| + |11 - 54| + |12 - 53| + |13 - 52| + |14 - 51| + |15 - 50| + |16 - 49| + |17 - 48| + |18 - 47| + |19 - 46| + |20 - 45| + |21 - 44| + |22 - 43| + |23 - 42| + |24 - 41| + |25 - 40| + |26 - 39| + |27 - 38| + |28 - 37| + |29 - 36| + |30 - 35| + |31 - 34| + |32 - 33| + |33 - 32| + |34 - 31| + |35 - 30| + |36 - 29| + |37 - 28| + |38 - 27| + |39 - 26| + |40 - 25| + |41 - 24| + |42 - 23| + |43 - 22| + |44 - 21| + |45 - 20| + |46 - 19| + |47 - 18| + |48 - 17| + |49 - 16| + |50 - 15| + |51 - 14| + |52 - 13| + |53 - 12| + |54 - 11| + |55 - 10| + |56 - 9| + |57 - 8| + |58 - 7| + |59 - 6| + |60 - 5| + |61 - 4| + |62 - 3| + |63 - 2| + |64 - 1|
\]
### Vertical Differences
For the first column:
\[
|1 - 64| + |64 - 1| + |2 - 63| + |3 - 2| + |4 - 62| + |5 - 61| + |6 - 60| + |7 - 59| + |8 - 58| + |9 - 57| + |10 - 56| + |11 - 55| + |12 - 54| + |13 - 53| + |14 - 52| + |15 - 51| + |16 - 50| + |17 - 49| + |18 - 48| + |19 - 47| + |20 - 46| + |21 - 45| + |22 - 44| + |23 - 43| + |24 - 42| + |25 - 41| + |26 - 40| + |27 - 39| + |28 - 38| + |29 - 37| + |30 - 36| + |31 - 35| + |32 - 34| + |33 - 33| + |34 - 32| + |35 - 31| + |36 - 30| + |37 - 29| + |38 - 28| + |39 - 27| + |40 - 26| + |41 - 25| + |42 - 24| + |43 - 23| + |44 - 22| + |45 - 21| + |46 - 20| + |47 - 19| + |48 - 18| + |49 - 17| + |50 - 16| + |51 - 15| + |52 - 14| + |53 - 13| + |54 - 12| + |55 - 11| + |56 - 10| + |57 - 9| + |58 - 8| + |59 - 7| + |60 - 6| + |61 - 5| + |62 - 4| + |63 - 3| + |64 - 2|
\]
### Calculation of Absolute Differences
We can see that the differences between adjacent cells in the first row are:
\[
63, 62, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,