aime26-14

Proposed solution

I need to find the number of ways to partition a $10 \times 10$ grid into 5 cell loops, where each loop surrounds a rectangle.

First, let me understand what a cell loop is. An $a \times b$ cell loop surrounds an $(a-2) \times (b-2)$ rectangle and contains $2a + 2b - 4$ cells.

For a $10 \times 10$ grid to be partitioned into 5 loops, I need to determine the possible dimensions of these loops.

Let me think about what sizes of loops can fit in a $10 \times 10$ grid.

If I have an $a \times b$ loop, it surrounds an $(a-2) \times (b-2)$ rectangle. For this to fit within a $10 \times 10$ grid:
- The surrounding rectangle must be at least $0 \times 0$ (empty) and at most $8 \times 8$
- So $a-2 \leq 8$ and $b-2 \leq 8$, which means $a \leq 10$ and $b \leq 10$
- Also, $a \geq 2$ and $b \geq 2$

But I also need to consider that the loops must be arranged so they don't overlap and cover the entire grid.

Let me reconsider the constraint. If I have a loop of size $a \times b$, it surrounds an $(a-2) \times (b-2)$ rectangle, and the total area covered by this loop is:
- The surrounding area: $2a + 2b - 4$ cells
- The inner rectangle: $(a-2)(b-2)$ cells  
- Total: $(a-2)(b-2) + (2a + 2b - 4) = a \cdot b$

So the loop covers exactly $a \cdot b$ cells, which makes sense.

For a $10 \times 10$ grid, I need 5 loops that cover all 100 cells.
So if the loops have dimensions $(a_1, b_1), (a_2, b_2), (a_3, b_3), (a_4, b_4), (a_5, b_5)$, then:
$a_1 b_1 + a_2 b_2 + a_3 b_3 + a_4 b_4 + a_5 b_5 = 100$

Also, each loop must fit within the grid. The maximum dimensions for any loop are $10 \times 10$.

But I need to be more careful about what "fits" means. If I have an $a \times b$ loop in a $10 \times 10$ grid, it surrounds an $(a-2) \times (b-2)$ rectangle. For this to fit, we need:
$(a-2) + 2 = a \leq 10$ and $(b-2) + 2 = b \leq 10$

Actually, let me re-read the problem more carefully.

Looking at the example: a $6 \times 6$ grid partitioned into 4 loops. 

If we have an $a \times b$ loop in a $6 \times 6$ grid:
- It surrounds an $(a-2) \times (b-2)$ rectangle
- The loop itself has area $2a + 2b - 4$

For the example, if we had 4 loops in a $6 \times 6$ grid:
The total area covered by loops should be $36$.

But I think there's an issue with my interpretation. Let me read more carefully.

Actually, looking at the diagram and the definition: "an $a \times b$ cell loop in a $2n \times 2n$ grid of cells to be the $2a + 2b - 4$ cells that surround an $(a - 2) \times (b - 2)$ rectangle of cells."

So the loop itself consists of $2a + 2b - 4$ cells, and it surrounds a rectangle of area $(a-2)(b-2)$.

For a $10 \times 10$ grid partitioned into 5 loops:
The total area covered by loops is $100$.
So I need: $\sum_{i=1}^{5} (2a_i + 2b_i - 4) = 100$
This gives: $\sum_{i=1}^{5} (a_i + b_i - 2) = 50$
Therefore: $\sum_{i=1}^{5} (a_i + b_i) = 60$

Also, each loop must fit in the $10 \times 10$ grid.

For an $a \times b$ loop to fit:
- The surrounding rectangle $(a-2) \times (b-2)$ must fit in a $10 \times 10$ grid
- So $a-2 \leq 10$ and $b-2 \leq 10$, which means $a \leq 12$ and $b \leq 12$
- But also $a \geq 2, b \geq 2$

Wait, this is wrong. The loop itself is not necessarily centered in the grid.

Let me think of it differently. An $a \times b$ cell loop means:
- It's a border around an $(a-2) \times (b-2)$ rectangle
- The total number of cells in the loop is $2a + 2b - 4$
- This loop must fit within the $10 \times 10$ grid

For a loop to fit, we need:
- The loop cannot extend beyond the grid boundaries
- If we place the loop at some position, it needs space for its border

Actually, let me think combinatorially. 

The key insight is that this is asking for partitions of the $10 \times 10$ grid into 5 cell loops.

Looking at the structure more carefully: 
If I have a $10 \times 10$ grid and want to partition it into 5 cell loops, each loop must be a connected region that forms a "frame" around some inner rectangle.

This is actually asking for the number of ways