Question

The numbers 1, 2,..., 9 are arranged in a 3 X 3 square grid in such a way that each number occurs once and the entries along each column, each row, and each of the two diagonals add up to the same value.

If the top left and the top right entries of the grid are 6 and 2, respectively, then the bottom middle entry is:

Accepted Answer

3. The sum of numbers 1 to 9 is 45. Since there are 3 rows/columns, each row/column/diagonal sum is $45 / 3 = 15$. Let the grid be: ``` 6 a 2 b c d e f g ``` Top row sum: $6 + a + 2 = 15 \implies a = 7$. In a standard 3x3 magic square with 1 to 9, the middle entry $c$ must be 5. Let's check diagonals and rows: - Top-right to bottom-left diagonal: $2 + c + e = 15 \implies 2 + 5 + e = 15 \implies e = 8$. - Top-left to bottom-right diagonal: $6 + c + g = 15 \implies 6 + 5 + g = 15 \implies g = 4$. - Left column: $6 + b + e = 15 \implies 6 + b + 8 = 15 \implies b = 1$. - Right column: $2 + d + 4 = 15 \implies d = 9$. - Bottom row: $e + f + g = 15 \implies 8 + f + 4 = 15 \implies f = 3$. Thus, the bottom middle entry $f$ is 3.

3x3 Magic Square Grid Entry

Explanation

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