1. **Problem statement:** Find a general formula for a 7x7 magic square using 7 variables $a,b,c,d,e,f,g$ such that all 16 sums (7 rows, 7 columns, and 2 main diagonals) are equal. 2. **Understanding the magic square:** A 7x7 magic square has 7 rows, 7 columns, and 2 diagonals, each summing to the same magic constant $S$. 3. **General approach:** We want to express each cell in terms of $a,b,c,d,e,f,g$ so that every row, column, and diagonal sums to $S$. 4. **Key formula:** The magic sum $S$ for a 7x7 magic square with entries $x_{ij}$ is $$S = \sum_{j=1}^7 x_{ij} = \sum_{i=1}^7 x_{ij} = \text{constant for all } i,j$$ 5. **Constructing the square:** One known method is to arrange the variables cyclically so that each row is a shifted version of the vector $(a,b,c,d,e,f,g)$. 6. **Explicit formula:** Define the element in row $i$, column $j$ as $$x_{ij} = v_{(j - i + 7) \bmod 7}$$ where $v = (a,b,c,d,e,f,g)$ and indexing starts at 0. 7. **Verification:** Each row sums to $$\sum_{j=0}^6 v_{(j - i) \bmod 7} = a + b + c + d + e + f + g = S$$ Similarly, each column sums to $S$. 8. **Diagonals:** The main diagonals also sum to $S$ because of the cyclic structure. **Final answer:** The 7x7 magic square with entries $$x_{ij} = v_{(j - i + 7) \bmod 7}$$ where $v = (a,b,c,d,e,f,g)$, has all rows, columns, and diagonals summing to $$S = a + b + c + d + e + f + g$$