Problem 1482

1. Assign Values to Unit Squares:
- Write the number $1$ in each unit square tiled by $2 \times 2$ tiles.
- Write the number $0$ in each unit square tiled by $3 \times 1$ tiles (both horizontal and vertical).

2. **Define the Sum $S$:**
- For any possible $1 \times 2$ rectangle (domino), sum up all the numbers on the unit squares covered by that domino.
- Add all these sums together to get the total sum $S$.

3. Double Counting and Modulo Operation:
- Let $N$ be the number of $1$'s on the edges of the grid, excluding the corners.
- Notice that $S \equiv N \pmod{2}$ because each $1 \times 2$ rectangle contributes to $S$ based on the number of $1$'s it covers.

4. **Claim: $N$ is Even:**
- Investigate the cases when $2 \times 2$ tiles are on the edge but do not contain any corners.
- Also, consider the case when a $2 \times 2$ tile contains a corner unit square.
- In each case, each $2 \times 2$ tile adds exactly $2$ $1$'s to $N$, making the total even.

5. Conclusion:
- Since $N$ is even, $S$ must also be even.
- The sum $S$ is equal to the number of neighboring edges between $2 \times 2$ tiles and $3 \times 1$ tiles.
- Therefore, the number of ways to choose a $1 \times 2$ rectangle such that one of its cells is tiled by a $2 \times 2$ mosaic and the other cell is tiled by a $1 \times 3$ mosaic is even.

$\blacksquare$