aime26-14

Proposed solution

To solve this problem, we need to partition a \(10 \times 10\) grid into 5 cell loops such that each cell in the grid belongs to exactly one cell loop. Each cell loop is defined as surrounding an \((a-2) \times (b-2)\) rectangle of cells in a \(2n \times 2n\) grid.

First, let's understand the structure of a cell loop. A cell loop surrounds an \((a-2) \times (b-2)\) rectangle and consists of \(2a + 2b - 4\) cells. For a \(10 \times 10\) grid, we need to partition it into 5 such loops. Each loop must cover a total of 20 cells because \(10 \times 10 = 100\) cells divided by 5 loops equals 20 cells per loop.

We start by examining the possible dimensions for the \((a-2) \times (b-2)\) rectangles. Since \(a\) and \(b\) are both between 2 and 10, the possible values for \(a-2\) and \(b-2\) range from 0 to 8. This means the dimensions of the rectangles can be any pair \((a-2, b-2)\) such that \(a-2\) and \(b-2\) are non-negative integers from 0 to 8.

Next, we need to find pairs \((a, b)\) such that the total number of cells in the five loops equals 100. We can express this as:
\[
5 \times (2a + 2b - 4) = 100
\]
Simplifying this equation, we get:
\[
2a + 2b - 4 = 20 \implies a + b = 12
\]

We now need to find pairs \((a, b)\) such that \(a + b = 12\) and both \(a\) and \(b\) are between 2 and 10. We can list the possible pairs:
\[
(a, b) = (4, 8), (5, 7), (6, 6), (7, 5), (8, 4)
\]

Each pair corresponds to a different way to partition the grid. We need to check if these pairs can indeed form valid loops in the \(10 \times 10\) grid.

1. For \((a, b) = (4, 8)\):
   - The loop surrounds a \(2 \times 6\) rectangle.
   - The loop consists of \(2 \times 4 + 2 \times 8 - 4 = 16\) cells.
   - This does not satisfy the requirement of 20 cells per loop.

2. For \((a, b) = (5, 7)\):
   - The loop surrounds a \(3 \times 5\) rectangle.
   - The loop consists of \(2 \times 5 + 2 \times 7 - 4 = 15\) cells.
   - This does not satisfy the requirement of 20 cells per loop.

3. For \((a, b) = (6, 6)\):
   - The loop surrounds a \(4 \times 4\) rectangle.
   - The loop consists of \(2 \times 6 + 2 \times 6 - 4 = 20\) cells.
   - This satisfies the requirement of 20 cells per loop.

4. For \((a, b) = (7, 5)\):
   - The loop surrounds a \(5 \times 3\) rectangle.
   - The loop consists of \(2 \times 7 + 2 \times 5 - 4 = 19\) cells.
   - This does not satisfy the requirement of 20 cells per loop.

5. For \((a, b) = (8, 4)\):
   - The loop surrounds a \(6 \times 2\) rectangle.
   - The loop consists of \(2 \times 8 + 2 \times 4 - 4 = 16\) cells.
   - This does not satisfy the requirement of 20 cells per loop.

From the above analysis, the only valid pair that satisfies the requirement of 20 cells per loop is \((a, b) = (6, 6)\). This means each loop surrounds a \(4 \times 4\) rectangle.

Now, we need to determine if we can partition the \(10 \times 10\) grid into 5 such loops. Each loop covers a \(4 \times 4\) area, and since \(10 \times 10 = 100\), we can fit exactly 5 such loops in the grid:
\[
\frac{100}{20} = 5
\]

Thus, there is only one way to partition the \(10 \times 10\) grid into 5 loops of \(4 \times 4\) rectangles.

Therefore, the number of ways to partition the \(10 \times 10\) grid into 5 cell loops is:
\[
\boxed{1}
\]