1. The problem is to find 49 expressions arranged in a 7x7 grid (cells labeled a to g for rows and columns) such that each expression is a distinct integer and all 49 integers are numerically distinct. 2. To solve this, we need to create expressions for each cell that depend on their row and column indices, say $a_{ij}$ for row $i$ and column $j$, where $i,j=1,2,...,7$. 3. A common approach is to use a formula like $$a_{ij} = 7(i-1) + j$$ which generates integers from 1 to 49 uniquely for each cell. 4. This formula ensures all values are distinct because for different pairs $(i,j)$, the value $7(i-1)+j$ is unique. 5. We can set inequalities to ensure distinctness: for any two cells $(i,j)$ and $(k,l)$, if $(i,j) \neq (k,l)$ then $$7(i-1)+j \neq 7(k-1)+l$$. 6. This satisfies the requirement that all 49 expressions produce numerically distinct integers. 7. Therefore, the expressions for the 49 cells are $$a_{ij} = 7(i-1) + j$$ for $i,j=1,...,7$. Final answer: The expressions $a_{ij} = 7(i-1) + j$ produce 49 distinct integers arranged in a 7x7 grid.